Differential Evolution with Event-Triggered Impulsive Control
Wei Du, Sunney Yung Sun Leung, Yang Tang, Athanasios V. Vasilakos
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Differential Evolution with Event-TriggeredImpulsive Control
Wei Du, Sunney Yung Sun Leung, Yang Tang, and Athanasios V. Vasilakos
Abstract —Differential evolution (DE) is a simple but powerfulevolutionary algorithm, which has been widely and successfullyused in various areas. In this paper, an event-triggered impulsivecontrol scheme (ETI) is introduced to improve the performanceof DE. Impulsive control, the concept of which derives fromcontrol theory, aims at regulating the states of a network byinstantly adjusting the states of a fraction of nodes at certaininstants, and these instants are determined by event-triggeredmechanism (ETM). By introducing impulsive control and ETMinto DE, we hope to change the search performance of thepopulation in a positive way after revising the positions of someindividuals at certain moments. At the end of each generation,the impulsive control operation is triggered when the updaterate of the population declines or equals to zero. In detail,inspired by the concepts of impulsive control, two types ofimpulses are presented within the framework of DE in thispaper: stabilizing impulses and destabilizing impulses. Stabilizingimpulses help the individuals with lower rankings instantlymove to a desired state determined by the individuals withbetter fitness values. Destabilizing impulses randomly alter thepositions of inferior individuals within the range of the currentpopulation. By means of intelligently modifying the positions ofa part of individuals with these two kinds of impulses, bothexploitation and exploration abilities of the whole populationcan be meliorated. In addition, the proposed ETI is flexibleto be incorporated into several state-of-the-art DE variants.Experimental results over the CEC 2014 benchmark functionsexhibit that the developed scheme is simple yet effective, whichsignificantly improves the performance of the considered DEalgorithms.
Index Terms —Differential evolution, impulsive control, event-triggered mechanism.
I. I
NTRODUCTION
Differential evolution (DE), firstly proposed by Storn andPrice [1, 2], has proven to be a reliable and powerfulpopulation-based evolutionary algorithm for global numericaloptimization. Over the past decade, different variants ofDE have been proposed to handle complicated optimizationproblems in various application fields [3], such as engineering
W. Du is with the Key Laboratory of Advanced Control and Optimizationfor Chemical Processes, Ministry of Education, East China University ofScience and Technology, Shanghai 200237, China and the Institute of Textileand Clothing, The Hong Kong Polytechnic University, Hong Kong, China(e-mail: [email protected]).S. Y. S. Leung is with the Institute of Textile and Clothing,The Hong Kong Polytechnic University, Hong Kong, China (e-mail:[email protected]).Y. Tang is with the Key Laboratory of Advanced Control andOptimization for Chemical Processes, Ministry of Education, East ChinaUniversity of Science and Technology, Shanghai 200237, China (e-mail:[email protected]; [email protected]).Athanasios V. Vasilakos is with the Department of Computer Science,Electrical and Space Engineering, Lulea University of Technology, Lulea97187, Sweden (e-mail: [email protected]). design [4], image processing [5], data mining [6], robot control[7], and so on.Generally, DE employs three main operators: mutation,crossover, and selection at each generation for the popula-tion production [8–10]. The mutation operator provides theindividuals with a sudden change or perturbation, which helpsexplore the search space. In order to increase the diversity ofthe population, the crossover operator is implemented afterthe mutation operation. The selection operator chooses thebetter one between a parent and its offspring, which guaranteesthat the population never deteriorates. In addition to thesethree basic operators, there are three control parameters whichgreatly influence the performance of DE: the mutation scalefactor F , the crossover rate CR , and the population size NP .Most of the current research on DE has focused on four aspectsto enhance the performance of DE: developing novel mutationoperators [11–19], designing new parameter control strategies[11–13, 20–25], improving crossover operator [12, 26–28],and pooling multiple mutation strategies [29–33]. These fourcategories of research on DE are described in detail as follows.1) In recent years, some efficient mutation operators havebeen presented and incorporated into the DE framework. Forinstance, Zhang and Sanderson [13] proposed a new mutationstrategy “DE/current-to- p best” to improve the performanceof the basic DE. Gong and Cai [14] developed a ranking-based mutation operator to assign better individuals to leadthe population. Guo et al. [15] presented a successful-parent-selecting method, which adapts the selection of parents whenstagnation is occurred. 2) Various parameter control schemeshave been introduced to the DE algorithm. In [20] and[13], F and CR can be evolved during the evolution of thepopulation. In [21], an adaptive population tuning schemewas proposed to reassign computing resources in a morereasonable way. 3) Some researchers have made efforts tooptimize the conventional crossover strategy. For example,Islam et al. [12] incorporated a greedy parent selection strategywith the traditional binomial crossover scheme to developa p -best crossover operation. Guo and Yang [28] utilizedeigenvectors of covariance matrix to make the crossoverrotationally invariant, which generates a better search behavior.4) Several DE variants have been put forward, which employsmore than one mutation operator to breed new solutions, suchas EPSDE [29], CoDE [30], SaDE [31], and so on.Despite numerous efforts on improving DE from the abovefour aspects, there are some DE variants which take advantageof ideas from other disciplines. For instance, Rahnamayan et al. [34] presented opposition-based DE (ODE), whichadopts opposition-based learning, a new scheme in machine REPRINT SUBMITTED TO ARXIV 2 intelligence, to speed up the convergence rate of DE. Laeloand Ali [35] made use of the attraction-repulsion concept inelectromagnetism to boost the performance of the original DE.Vasile et al. [36] proposed a novel DE, which is inspiredby discrete dynamical systems. These improvements on DEenlighten us to look through techniques in other areas, whichmight be introduced to the development of DE variants.On another research frontier, as an important component incontrol theory, impulsive control has attracted much attentionin recent years due to its high efficiency. As exemplified in [37,38], impulsive effects can be detected in various dynamicalsystems, like communication networks, electronic systems,biological networks, and so on. Besides, impulsive controlis able to manipulate the states of a network to a desiredvalue by adding impulsive signals to some specific nodesat certain instants. In addition, another effective technique,event-triggered mechanism (ETM), has also been widelyutilized [39–42] in control theory. In ETM, the state of thecontroller is updated when the system’s state exceeds a giventhreshold. By integrating ETM into impulsive control, theoperation of impulsive control can only be activated whensome predefined events are triggered. This way, ETM avoidsthe periodical execution of impulsive control, which efficientlysaves computational resources.Taking a look at how DE works in an optimization problem,the movement of the population in the evolution process can betreated as a complicated multi-agent system in control theory,where individuals in the population can be regarded as nodesin a network. On one hand, in original DE algorithms andsome popular DE variants, it may take a long time for certainindividuals to reach the desired positions. For instance, the“ p best” individual is utilized to guide the search of other in-dividuals in JADE [13]. However, this operation is carried outat each generation and forces the individuals to approach thedesired state slowly, which deteriorates the search performanceof the population in the limited computational resources. Onthe other hand, in many DE variants, like jDE [20], JADE [13],CoDE [30] et al. , the diversity of the population is maintainedonly by mutation and crossover, which are indirect. Inspiredby how impulsive control manipulates a dynamical system, weintroduce the concept of impulsive control into the design ofDE, aiming at increasing the search efficiency and the diversityof the population by instantly letting selected individuals moveclose to the desired positions. Besides, when DE is usedfor an optimization problem, the computational resources areoften limited, measured by the maximum number of functionevaluations (MAX FES). Therefore, it is reasonable to triggerthe instantaneous movement of certain individuals by somepredefined events, which follows the idea of ETM.Motivated by the above discussion, by integrating ETM intoimpulsive control, we introduce an event-triggered impulsivecontrol scheme (ETI) for DE in this paper. Similar to adjustingthe states of some nodes in dynamical systems in controltheory, impulsive control revises the positions of a fractionof population at certain moments, the purpose of which is topositively change the evolution state of the whole population.In detail, two varieties of impulses: stabilizing impulses anddestabilizing impulses are presented to fit into the framework of DE. In addition, based on both the fitness value andthe number of consecutive stagnation generation, a novelmeasure R i is developed to pick the individuals to be injectedwith impulsive controllers. When the update rate ( UR ) ofthe population begins to diminish or reduces to zero, theindividuals with large values of R i will be injected withimpulsive controllers. Stabilizing impulses are adopted to forcea number of individuals with lower rankings in the currentpopulation to get close to the individuals with better fitnessvalues, which increases the exploitation ability of DE. Besides,destabilizing impulses are considered to randomly adjust thepositions of inferior individuals within the range of the currentpopulation, which improves the exploration capability of DE.The major contributions of this paper are mainly threefold:1) an event-triggered impulsive control scheme is introducedinto the DE framework, which aims to improve the searchability of DE; 2) two kinds of impulses, stabilizing anddestabilizing impulses, are developed respectively, to enhancethe exploitation and exploration performance of DE; 3) theproposed scheme is simple but effective, which can improvethe performance of most of the considered representative DEalgorithms in this paper. It is worth pointing out that we havedone some preliminary work in [43], in which an impulsivecontrol framework (IPC) is proposed for DE. IPC differs fromETI in the following three major aspects: 1) ETI includesETM to identify when the individuals should be injected withimpulsive controllers, while no ETM is used in IPC. 2) Twotypes of impulses, stabilizing and destabilizing impulses, areproposed in ETI, while only stabilizing impulses are presentedin IPC. 3) ETI adopts ranking assignment and an adaptivemechanism (described in Section III) to select the individualstaking impulsive control, while IPC just utilizes a non-adaptivepiecewise threshold function for choosing the individuals.This paper is organized as follows. In Section II, the originalDE and the concepts of ETM and impulsive control areintroduced. Our proposed scheme ETI is presented in SectionIII. Experimental results are reported in Section IV. Finally,concluding remarks are made in Section V.II. R ELATED W ORK
In this section, we firstly introduce the original DE. Then theconcepts of event-triggered mechanism and impulsive controlare briefly outlined.A single objective optimization problem can be formulatedas follows (without any loss of generality, in this paper, aminimization problem is considered with a decision space Ω ):minimize f ( x ) , x ∈ Ω , (1)where Ω is a decision space, x = [ x , x , ..., x D ] T is a decisionvector, and D is the dimension size, representing the numberof the decision variables involved in the problem. For eachvariable x j , it should obey a boundary constraint: L j ≤ x j ≤ U j , j = 1 , , ..., D, (2)where L j and U j are the lower and upper bounds for the j thdimension, respectively. REPRINT SUBMITTED TO ARXIV 3
A. DE
DE is a population-based evolutionary algorithm for anumerical optimization problem. It initializes a populationof NP individuals in a D -dimensional search space. Eachindividual represents a potential solution to the optimizationproblem. After initialization, at each generation, three op-erators: mutation, crossover and selection are employed togenerate the offspring for the current population.
1) Mutation:
Mutation is the most consequential operatorin DE. Each vector x i,G in the population at the G th generationis called target vector. A mutant vector called donor vectoris obtained through the differential mutation operation. Forsimplicity, the notation “DE/ a / b ” is used to represent differ-ent mutation operators, where “DE” denotes the differentialevolution, “ a ” stands for the base vector, and “ b ” indicatesthe number of difference vectors utilized. In DE, there are sixmutation operators that are most widely used:i) “DE/rand/1” v i,G = x r , G + F · ( x r , G − x r , G ) , (3)ii) “DE/rand/2” v i,G = x r , G + F · ( x r , G − x r , G )+ F · ( x r , G − x r , G ) , (4)iii) “DE/best/1” v i,G = x best,G + F · ( x r , G − x r , G ) , (5)iv) “DE/best/2” v i,G = x best,G + F · ( x r , G − x r , G )+ F · ( x r , G − x r , G ) , (6)v) “DE/current-to-best/1” v i,G = x i,G + F · ( x best,G − x i,G )+ F · ( x r , G − x r , G ) , (7)vi) “DE/current-to-rand/1” u i,G = x i,G + K · ( x r , G − x i,G )+ ˆ F · ( x r , G − x r , G ) , (8)where x best,G specifies the best individual in the currentpopulation; r , r , r , r and r ∈ { , , ..., NP } , and r = r = r = r = r = i . The parameter F > is called scaling factor , which scales the difference vector. It is worthmentioning that (8) shows the rotation-invariant mutation [44]. K is the combination coefficient, which should be selected witha uniform random distribution from [0, 1] and ˆ F = K · F . Since“DE/current-to-rand/1” contains both mutation and crossover,it is not necessary for the offspring to go through the crossoveroperation.
2) Crossover:
After mutation, a binomial crossover op-eration is implemented to generate the trial vector u i =[ u i , u i , ..., u iD ] T : u ij,G = (cid:26) v ij,G , if rand (0 , ≤ CR or j = j rand ,x ij,G , otherwise , (9)where rand (0 , is a uniform random number in the range [0 , . CR ∈ [0 , is called crossover probability , which determines how much the trial vector is inherited from themutant vector. j rand is an integer randomly selected from to D and newly generated for each i , which ensures at leastone dimension of the trial vector will be different from thecorresponding target vector. If u ij,G is out of the boundary, itwill be reinitialized in the range [ L j , U j ] .
3) Selection:
The selection operator employs a one-to-oneswapping strategy, which picks the better one from each pairof x i,G and u i,G for the next generation: x i,G+1 = (cid:26) u i,G , if f ( u i,G ) ≤ f ( x i,G ) , x i,G , otherwise . (10) B. Event-triggered mechanism (ETM)
Event-triggered mechanism (ETM) is an effective strategyin control theory that determines when the state of a controlleris updated. Typically, a controller’s state is independentof a system’s state except at periodic instants. When thecommunication resource is insufficient, the traditional time-triggered paradigm may not be efficient. While in ETM, thestate of the controller is revised only when a system’s stateexceeds a predefined threshold, or a specified event occurs.This way, ETM is able to reduce the amount of unnecessarycommunications. It is of paramount importance to make use ofETM by devising suitable event-triggering conditions, whichsaves system resources and ensures stable performance at thesame time. One can refer to the references [39–41] and therein.
C. Impulsive control
In various dynamical networks [45], like biological net-works, communication networks, and electronic networks, thestates of networks often undergo abrupt changes at someinstants, which may be due to switching phenomena orcontrol requirements; and these changes can be modelled byimpulsive effects. Usually, impulses can be divided into twocategories: stabilizing and destabilizing impulses [37, 38],which respectively make networks stable and unstable. Fordynamical networks, impulsive control is capable of adjustingthe states of a network by instantaneously regulating the statesof a fraction of nodes at certain instants. Due to the highefficiency of impulsive control, it has attracted increasingattention in recent years. Besides, as shown in [37], if theimpulsive strength of each node is distinct in networks, wecall such kind of impulses as heterogeneous impulses in spacedomain.In order to clearly explain the mechanism of impulsivecontrol, here we consider the following complex nonlineardynamical network model: ˙ x i ( t ) = ˜ f ( x i ( t )) + υ N X j =1 a ij x j ( t ) , (11)where i = 1 , , ..., N, x i ( t ) = [ x i ( t ) , x i ( t ) , ..., x in ( t )] T ∈ R n is the state vector of the i th node at time t ; ˜ f ( x i ( t )) =( ˜ f ( x i ( t )) , ..., ˜ f n ( x in ( t )) T ∈ R n ; υ > denotes thecoupling strength; A = [ a ij ] N × N is the coupling matrix,where a ij is defined as follows: if there is a connection from REPRINT SUBMITTED TO ARXIV 4 node j to node i ( i = j ), then a ij = a ji > ; otherwise a ij = 0 ; for i = j , a ij is defined as follows: a ii = − N X j =1 ,j = i a ij . (12)Assume that the nonlinear dynamical network in (11) canbe forced to the following reference state s ( t ) : ˙ s ( t ) = ˜ f ( s ( t )) . (13)Let e i ( t ) = x i ( t ) − s ( t ) , then we get the error dynamicalsystem: ˙ e i ( t ) = f ( e i ( t )) + υ N X j =1 a ij e j ( t ) , (14)where f ( e i ( t )) = ˜ f ( x i ( t )) − ˜ f ( s ( t )) . Consider heterogeneousimpulsive effects in system (11) or (14), we obtain thefollowing model: (cid:26) ˙ e i ( t ) = f ( e i ( t )) + υ P Nj =1 a ij e j ( t ) , t = t k , k ∈ N + ,e i ( t + k ) = e i ( t − k ) + µ ik e i ( t − k ) , (15)where µ ik denote impulsive strengths; the impulsive instantsequence { t k } ∞ k =1 satisfies < t < t <, ..., < t k <..., lim t →∞ t k = ∞ ; x i ( t − k ) and x i ( t + k ) denote the limit fromthe left and the right at time t k , respectively. Without lossof generality, in this paper, we assume that x i ( t + k ) = x i ( t k ) , i = 1 , , ..., N and t = 0 .III. A N E VENT -T RIGGERED I MPULSIVE C ONTROL S CHEME
In this section, we propose an event-triggered impulsivecontrol scheme (ETI) for DE. In the following, we firstlyintroduce the proposed ETI in detail, which involves fourcomponents, i.e., stabilizing impulses, destabilizing impulses,ranking assignment, and an adaptive mechanism. Afterwards,we combine our approach with DE to develop ETI-DE, thepseudo-code and the computational complexity analysis ofwhich are also given.
A. Our Approach
In our approach, ETM and impulsive control are integratedinto the framework of DE algorithms. ETM identifies whenthe individuals should be injected with impulsive controllers,while impulsive control alters the positions of partial individ-uals when triggering conditions are violated. Specifically, twotypes of impulses, i.e., stabilizing impulses and destabilizingimpulses, are imposed on the selected individuals (sorted byan index according to the fitness value and the number ofconsecutive stagnation generation) when the update rate ( UR )of the population in the current generation decreases or equalsto zero. UR is illustrated by Eq. (16). UR = UPNP , (16)where NP is the population size, and UP is the number ofthe individuals that update in the current generation. On onehand, when UR begins to decrease, stabilizing impulses drive the individuals with lower rankings in the current populationto approach the individuals with better fitness values. Thepurpose of stabilizing impulsive control is to help update someinferior individuals and to enhance the exploitation capabilityof the algorithm. On the other hand, when UR drops to zero orstabilizing impulses fail to take effect, destabilizing impulsesrandomly adjust the positions of the inferior individuals withinthe area of the current population. This operation improves thediversity of the population and hence improves the explorationability of DE. Remark . Given an individual x i , we can also differentiatestabilizing and destabilizing impulses from the perspectiveof the impulsive strength K i = diag { K i1 , K i2 , ..., K iD } , wherediag { K i1 , K i2 , ..., K iD } denotes a diagonal matrix whose diago-nal entries starting in the upper left corner are K i1 , K i2 , ..., K iD :stabilizing impulses ( K ij ∈ [ − , ) and destabilizing impulses( K ij ∈ (0 , ), where D is the dimension and j = 1 , , ..., D .
1) Stabilizing Impulses:
Stabilizing impulses are employedwhen UR begins to decrease. As mentioned before, in controltheory, stabilizing impulses can be employed to regulate thestates of a network to a desired value. Normally, the desiredstate is set as the reference state for the nodes to be injectedwith stabilizing impulsive controllers. In the framework ofDE, stabilizing impulses mainly focus on improving theexploitation ability of DE. In DE algorithms, it is well knownthat good individuals (i.e., with smaller fitness values) usuallycontain useful information, which may be helpful to otherindividuals’ evolution. Hence, these good individuals can beregarded as references. So when stabilizing impulsive controlis triggered during the evolution, we set the individuals withsmaller fitness values in the current generation as the referencestates. The pseudo-code of stabilizing impulsive control isexhibited in Algorithm S.1 of the supplementary file.Assume that x i,G is one of the individuals at the G thgeneration that are chosen to undergo impulsive effects, where x i,G = [ x i , G , x i , G , ..., x iD , G ] T and D is the dimension. We set s i,G as the reference state for x i,G , which is randomly selectedfrom the best individual ( gbest ) or other individuals withsmaller fitness values than x i,G in the current population. Foreach x i,G , a uniform random individual x k,G is firstly chosenfrom the current population. If f ( x i,G ) < f ( x k,G ) , whichmeans the randomly selected individual is worse than x i,G ,then x gbest,G is the reference state for x i,G ; if f ( x i,G ) ≥ f ( x k,G ) ,which means x k,G is better than x i,G in the current population,then x k,G is set as the reference state for x i,G .The error between x i,G and its reference state s i,G at the G thgeneration can be obtained: e i,G = x i,G − s i,G . (17)Then at the end of the G th generation, stabilizing impulsesforce the chosen individuals to approach their reference state.Here we get: x i,G + = x i,G + K i,G · e i,G , (18)where K i,G = diag { K i1,G , K i2,G , ..., K iD,G } is the impulsivestrength for individual x i at the G th generation. K ij,G ∈ ( − , shows that in the j th dimension, x i,G lies on the line between REPRINT SUBMITTED TO ARXIV 5 the reference and the individual itself; K ij,G = 0 meansthat in the j th dimension, x i,G is not injected with impulsivecontrollers; K ij,G = − indicates that in the j th dimension, x i,G reaches the reference state, j = 1 , , ..., D . G + denotes thatstabilizing impulses are imposed on x i,G at the end of the G thgeneration. Every time, DM dimensions of x i,G are selectedin a uniformly random way to be injected with impulsivecontrollers, where DM ∈ { , , ..., D } . When x gbest,G serves asthe reference state, for the selected DM dimensions, the im-pulsive strength ˆ K i,G = diag { K i , G , K i , G , ..., K iDM , G } DM × DM ,and K ij,G is a uniform random number from − to 0, j = 1 , , ..., D ; for the rest D − DM dimensions, ˇ K i,G = diag { , , ..., } ( D − DM ) × ( D − DM ) . When x k,G is as the referencestate, for the selected DM dimensions, the impulsive strength ˆ K i,G = diag {− , − , ..., − } DM × DM ; for the rest D − DM dimensions, ˇ K i,G = diag { , , ..., } ( D − DM ) × ( D − DM ) . K i,G isobtained from combining ˆ K i,G and ˇ K i,G . It is noticed that ζ is a flag to indicate whether stabilizing impulsive control issuccessful to improve the performance: when ζ = 1 , it meansthat the stabilizing impulsive control takes effect, and a newindividual is introduced to the population by replacing an oldone; when ζ = 0 , it shows that the stabilizing impulsivecontrol fails to take effect. Remark . According to [46], if the impulsive strength ofeach node is distinct in networks, such kind of impulses iscalled heterogeneous impulses in space domain. For stabilizingimpulses developed in this paper, the impulsive strengths arenot only heterogeneous in each individual of the populationbut also nonidentical in each dimension of each individual.Hence, it is apparent that our proposed impulses generalizethe heterogeneous impulses in [46]. Apart from enhancingthe performance of DE algorithms, our proposed stabilizingimpulses can also contribute to the design of impulsive controlsystems.
Remark . In [38], if impulses are injected into only afraction of nodes, such kind of impulses is called partial mixedimpulses. In this paper, stabilizing impulses are imposed onnot only a group of individuals in the population, but alsopartial dimensions of each individual. Therefore, our presentedstabilizing impulses can be regarded as a hierarchical partialmixed impulses when compared with the impulses in [38].Besides, the proposed impulses will not only promote thedevelopment of new powerful DEs, but also shed light on thedesign of impulsive control systems.
2) Destabilizing Impulses:
When UR drops to zero ( UR =0 ) or stabilizing impulses fail to take effect ( ζ = 0 ),destabilizing impulses are introduced to provide some ran-domness during the evolution. When destabilizing impulsesare triggered, the selected individuals can be moved to anyposition within the range of the current population. Thepseudo-code of injecting destabilizing impulses is exhibitedin Algorithm S.2 of the supplementary file. Assume that x i,G is one of the individuals at the G th generation thatare chosen to receive destabilizing impulses, where x i,G =[ x i , G , x i , G , ..., x iD , G ] T . min j,G and max j,G are the minimumand maximum values of the j th dimension in the population at the G th generation, j = 1 , , ..., D . The lower and upperbounds of the range of the population at the G th generationare: x L,G = [ min , min , ..., min D,G ] T , (19) x U,G = [ max , max , ..., max D,G ] T . (20)From Eqs. (19)-(20), we can obtain the error between x U,G and x L,G at the G th generation: e i,G = x U,G − x L,G . (21)Then at the end of the G th generation, the positions of thechosen individuals are randomly updated in the specifiedrange. Here we have: x i,G + = x L,G + K i,G · e i,G , (22)where K i,G = diag { K i1,G , K i2,G , ..., K iD,G } is the impulsivestrength for individual x i at the G th generation. Similarly, DM dimensions of x i,G are selected at random to be injectedwith impulses. K ij,G is a uniform random number from 0 to 1, j = 1 , , ..., D . Remark . Similar to our developed stabilizing impulses,destabilizing impulses are imposed on a part of individuals.Thus our proposed destabilizing impulses can be regarded aspartial mixed impulses according to [38]. Besides, randomdimensions of the individual are chosen to be injected withdestabilizing impulses, the impulsive strengths of which rangefrom (0 , . Therefore, destabilizing impulses in this papercan also be understood as generalized heterogeneous impulseswhen compared with the impulses in [46]. Remark . Two kinds of impulses are proposed in theframework of DE. The idea of introducing impulsive controlinto DE comes from the fact that impulsive control takeseffect in dynamical networks. In [37], stabilizing impulses areimposed on partial nodes of a network, and the desired stateis set as the reference state for these nodes. The dynamicalnetwork with stabilizing impulses can be synchronized toa desired state. In DE, stabilizing impulses act as impetus,which forces certain individuals to approach good individuals(references) in the population at certain instants. And thisoperation is expected to facilitate the fast convergence ofthe population. In addition, destabilizing impulses introducedisturbances to a network in multi-agent systems or dynamicalnetworks [38]. Similarly, in DE, a fraction of individuals areinjected with destabilizing impulses at certain moments, whichaims at bring some randomness into the evolution process.In dynamical networks, impulsive control is used to adjustthe states of a network by instantly regulating the states ofa fraction of nodes at certain instants. And in DE, impulsivecontrol is expected to enhance the search performance of thewhole population by instantaneously modifying the positionsof a part of individuals at certain moments.
3) Ranking Assignment:
In the following, we need toconsider which individuals should be injected with impulsivecontrollers. During the evolution process, we consider twomeasures to characterize the status of the individuals. The firstone is the fitness value of each individual, while the second
REPRINT SUBMITTED TO ARXIV 6 one is the number of each individual’s consecutive stagnantgeneration. Fitness value is the most direct index to judgewhether an individual should enter into the next generation ornot. The number of consecutive stagnant generation reflects thedegree of the activity of an individual in the evolution. If anindividual does not evolve for a relatively long time, it mightbe necessary to introduce some additional operations to changeits position. Based on these discussions, in this paper, we rankthe population based on the fitness value and the number ofconsecutive stagnant generation, respectively. ˜ R i is the rankingof x i,G according to the fitness value, and ¯ R i is the ranking of x i,G based on the number of consecutive stagnation generation.These two rankings are both ordered in an ascending way (i.e.,from the best fitness value to the worst and from the smallestnumber of consecutive stagnation generation to the largest).Then we combine ˜ R i and ¯ R i to get R i , which indicates that theindividuals are sorted according to both the fitness value andthe number of consecutive stagnation generation. R i = ˜ R i + ¯ R i . (23) R i not only reflects the fitness value of the individual x i,G butalso delivers the degree of the individual’s activity.When impulsive control is triggered during the evolution,we select the individuals with larger values of R i fromthe population as the candidates to undergo stabilizing ordestabilizing impulses. By specially displacing the individualswith higher rankings (i.e., larger R i ), the evolution status ofthe population can be improved.
4) An adaptive mechanism to determine the number of in-dividuals taking impulsive control:
Finally, in order to furtherimprove the performance of ETI, an adaptive mechanism isproposed to determine the number of the individuals thatshould be injected with impulsive controllers. We firstly dis-cuss the number of individuals ( M ) with stabilizing impulses. LN and UN represent the lower and upper bound of M ,respectively. When stabilizing impulsive control is triggeredfor the first time, M = LN . After x i,G experiences thestabilizing impulse, we get x i,G + . x i,G + can join the currentpopulation instead of x i,G if and only if f ( x i,G + ) < f ( x i,G ) .Every time x i,G is replaced with x i,G + (i.e., ζ = 1 , see step36 in Algorithm S.1 of the supplementary file), M keepsunchanged. If ζ = 0 , M = M + 1 . We aim to increasethe success rate of stabilizing impulsive control by havingmore individuals to be injected with stabilizing impulsivecontrollers. Besides, if a new gbest is generated in thepopulation, M drops to a random integer between [ LN , M ] . Thereason for reducing M to a random integer between [ LN , M ] instead of LN is to increase the times of successful stabilizingimpulsive control, especially in the later stage of the evolution.Next, we explain how to choose the number of individualsthat undergo destabilizing impulses. As introduced above,destabilizing impulses are added in two cases: when UR = 0 or ζ = 0 . Unlike stabilizing impulses, with the purpose ofintroducing some randomness, the selection operation (i.e.,compare the fitness values of x i,G and x i,G + ) will not be usedafter injecting destabilizing impulses, which means that x i,G + replaces x i,G directly. Therefore, in order not to bring toomany individuals with large fitness values into the population, we randomly select the individuals from M candidates to beinjected with destabilizing impulses. The selection process isdescribed in Algorithm S.3 of the supplementary file. Remark . In this paper, stabilizing and destabilizing impulsesare triggered separately based on the status of the individuals.Two measures are used to characterize the status of theindividuals, and one of them is the fitness value of eachindividual. In recent years, fitness control adaptation workseffectively for developing evolutionary algorithms, whichperforms corrections and anti-corrections [25, 47–51]. In theliterature above, the best, worst, and average fitness valuesin the population are utilized to construct some metrics,such as ξ in [47], ψ in [48], χ in [49], and so on. Theseparameters adaptively determine the activation time of eachlocal searcher. Although our method is similar to the fitnesscontrol adaptation, there are some differences between theworks in [25, 47–51] and in our research. Firstly, in ourresearch, we rank the fitness values of the whole population,instead of using some typical values (i.e., the best, worst, andaverage values). Secondly, apart from the ranking of fitnessvalue, we also consider the ranking according to the numberof consecutive stagnation generation of the whole population.These two rankings are combined into one measure R in (22).Thirdly, the metrics ξ , ψ , and χ in [25, 47–51] are used toactivate different local searchers. While in our work, R isthe measure to select individuals as the candidate to undergostabilizing and destabilizing impulses. Therefore, fitness valuesplay different roles in literature [25, 47–51] and in our work.And it can be generally recognized that our work also fits intothe framework of fitness control adaptation. Remark . It is worth mentioning that the essential of ETI isto adjust the search strategies of individuals according to theevolutionary states. In recent studies on DE and particle swarmoptimization (PSO) [22, 52–54], the algorithms proposed alsoselect the search strategies and parameters based on thestates of the individuals. However, our work is quite differentfrom these studies in the following three aspects: 1) Themeasures are different when representing the states of theindividuals. For example, in [22], the measure uses fitness anddistance information; in [52], distance information is utilized;in [53], the measure takes advantage of fitness and positioninformation; in [54], position information is considered; whilewe use fitness and stagnation information in our paper. 2) Thesearch strategies are different in the algorithms developed in[22, 52–54] and our proposed ETI. For instance, [22] and[52] developed parameter adaptation strategies; [53] used amutation operator in PSO; [54] employed a restart strategyafter stagnation; while our research makes certain individualsapproach superior individuals or reinitialized. 3) Our ETI ispresented within a DE framework and cannot be incorporatedinto PSO, which will be explained in Remark 8.
B. DE with An Event-Triggered Impulsive Control Scheme
Combining the developed event-triggered impulsive controlscheme (ETI) with DE, the ETI-DE is proposed. The pseudo-code of ETI-DE with “DE/rand/1” mutation operator is given
REPRINT SUBMITTED TO ARXIV 7 in Algorithm 1. From step 7 to step 28, it is the originalDE algorithm with “DE/rand/1” mutation operator. The reststeps in Algorithm 1 illustrate the mechanism of ETI. ETMdetermines the moment to add impulses to the individuals, andimpulsive control modifies the positions of partial individualsat the end of a certain generation. In detail, step 35 to step 42and step 49 to step 56 describe the mechanism of destabilizingimpulses, which are triggered when UR = 0 or ζ = 0 .While step 43 to step 48 shows the details of stabilizingimpulsive control, which is triggered when UR decreases and UR = 0 . These two types of impulses are able to acceleratethe convergence of the population by updating some inferiorindividuals, and improve the diversity of the population byintroducing some randomness to the search. Furthermore, ETIis flexible to be integrated into other advanced DE variants,such as jDE [20], JADE [13], SaDE [31], and so on.In ETI-DEs, stabilizing impulses force the individual x i,G to approach its reference state s i,G , and destabilizing impulsesrandomly adjust the positions of the individuals within the areaof the current population. These two operations are carriedout within the search range of the problem, so they will notgenerate invalid solutions during the evolution process. Inthe following experimental section, when ETI is incorporatedinto other DE variants, we will not change the originalbound constraints handling methods of these DE variants.These methods define that at any stage of the search process,solutions outside the bounds are invalid, just like the situation S1 in [55]. So for the CEC 2014 benchmark functions usedin Section IV, ETI-DEs will not search outside the region [ − , D . Remark . The proposed ETI is presented within a DEframework. Meanwhile, because different evolutionary algo-rithms (EAs) have different structures, ETI cannot be directlyincorporated into other EAs, and some related modificationsare needed on ETI. For example, in genetic algorithm (GA),only a fraction of individuals are selected as parents at eachgeneration, so the number of each individual’s consecutivestagnant generation is meaningless. Therefore, it is necessaryto propose another measure to denote the state of the individu-als. In particle swarm optimization (PSO), each member of theswarm searches the space based on the historical informationof itself ( pbest ) and other members ( gbest ). So ETI may usethe stagnation information of pbest and gbest to be fittedinto PSO. In our future work, we will investigate in detailwhether ETI can work efficiently in other EAs. Accordingto the explanations in [8], in essence, ETI varies the movesand enriches the pool of search moves. In detail, stabilizingimpulses introduce extra moves towards individuals with betterfitness values, the goal of which is to increase the exploitativepressure. Destabilizing impulses bring in more explorativemoves, which helps the population explore the search space.Here, we discuss the complexity of ETI. Generally, theproposed ETI-DE does not significantly increase the overallcomputational complexity of the original DE algorithm. Theadditional complexity of ETI-DE is population sorting whencalculating ˜ R and ¯ R , and implementing impulsive control.The complexity of population sorting is O (2 · NP · log ( NP )) , Algorithm 1
DE with event-triggered impulsive controlscheme Begin /* UR is the update rate of the population in each generation /* UR tp stores the temporary value of UR /* gbest is the best individual of the population in the current generation /* gbest tp stores the temporary value of gbest /* rs records the number of individuals to be injected with destabilizing impulses Set LN = 1 ; UN = NP ; M = LN ; UR = 0 ; F = 0 . ; CR = 0 . Create a random initial population { x i , | i = 1 , , ..., NP } Evaluate the fitness values of the population and record gbest while the maximum evaluation number is not achieved do UR tp = UR ; gbest tp = gbest for i = 1 to NP do Select randomly three individuals r = r = r = i v i,G = x r , G + F · ( x r , G − x r , G ) Check the boundary of v i , G Generate j rand = randi ( D , for j = 1 to D do if j = j rand or rand < CR then u ij,G = v ij,G else u ij,G = x ij,G end if end for Evaluate the fitness value of u i,G if f ( u i,G ) ≤ f ( x i,G ) then x i,G+1 = u i,G end if end for Record the fitness value of the best individual as gbest if gbest < gbest tp then M = randi ([ LN , M ] , end if Calculate ˜ R i and ¯ R i of the population, R i = ˜ R i + ¯ R i Calculate UR of the population if UR = 0 then M = min ( M , UN ) Select M individuals with the largest R i -value as { x i,G | i = 1 , , ..., M } { x i,G | i = 1 , , ..., rs } = Random Selection of Individuals () for i = 1 to rs do x i,G = Injecting Destabilizing Impulsive ()
Evaluate the fitness value of x i,G end for else if UR = 0 and UR < UR tp then M = min ( M , UN ) Select M individuals with largest R i -value as { x i,G | i = 1 , , ..., M } for i = 1 to M do [ x i,G , ζ i,G ] = Stabilizing Impulsive Control () end for if sum ( ζ , ζ , ..., ζ M,G ) = 0 then { x i,G | i = 1 , , ..., rs } = Random Selection of Individuals () for i = 1 to rs do x i,G = Injecting Destabilizing Impulsive ()
Evaluate the fitness value of x i,G M = M + 1 end for end if Record the best individual of current population as gbest tp if gbest tp < gbest then M = randi ([ LN , M ] , end if end if end while End and the maximum complexity of impulsive control is O ( NP · D ) . It is known that the complexity of the original DE is O ( G max · NP · D ) , so the total complexity of ETI-DE is O (2 · G max · NP · ( D + log ( NP )) , which can be regarded as thesame as the original DE. Therefore, our presented scheme doesnot seriously increase the computational cost of the originalDE. IV. E XPERIMENTAL R ESULTS AND A NALYSIS
In this section, we carry out extensive experiments toevaluate the performance of our developed ETI-DE. The
REPRINT SUBMITTED TO ARXIV 8 total 30 benchmark functions presented in the CEC 2014competition on single objective real-parameter numerical op-timization are selected as the test suite [56]. Accordingto their characteristics, the functions can be divided intofour groups: 1) unimodal functions (F01-F03); 2) simplemultimodal functions (F04-F16); 3) hybrid functions (F17-F22); 4) composition functions (F23-F30). More details ofthese functions can be found in [56].
A. Parameter Settings
In the following experiments, we incorporate the proposedevent-triggered impulsive control scheme with two originalDE algorithms and eight state-of-the-art DE variants. Theparameters are set as follows:1) DE/rand/1/bin with F = 0 . , CR = 0 . [16];2) DE/best/1/bin with F = 0 . , CR = 0 . [16];3) jDE with τ = 0 . , τ = 0 . [20];4) JADE with µ F = 0 . , µ CR = 0 . , c = 0 . , p = 0 . [13];5) CoDE with F = [1 . , . , . , CR = [0 . , . , . [30];6) SaDE with LP = 50 [31];7) ODE with F = 0 . , CR = 0 . , J r = 0 . [34];8) EPSDE with F = [0 . , . , . , . , . , . , . , . , . , CR = [0 . , . , . , . , . , . [29];9) SHADE with initial M F = 0 . , M CR = 0 . , H = NP [57];10) OXDE with F = 0 . , CR = 0 . [26].For the incorporated ETI-DE algorithms, the lower andupper bounds of the number of individuals that take impulsivecontrol are set: LN = 1 , UN = NP . The maximum number offunction evaluations (MAX FES) is set to D · . We runeach function optimized by each algorithm 51 times for theexperiments [56]. The simulations are performed on an IntelCore i7 personal computer with 2.10-GHz central processingunit and 8-GB random access memory. Remark . In Algorithm S.1, DM is the number of dimensionsof an individual selected to undergo stabilizing impulsivecontrol. Every time, DM dimensions of an individual arechosen in a uniformly random way to be injected withimpulsive controllers, where DM ∈ { , , ..., D } . In thissection, the sensitivity of DM is studied beforehand bycomparing the performance of ETI-DEs with random DM and with DM = [ D / , D / , D ] . We provide the experimentalresults in Table S.11 of the supplementary file. The resultsshow that in 26 out of 30 cases, ETI-DEs with random DM perform better than those with DM = [ D / , D / , D ] .Therefore, it is reasonable to select random DM dimensionsof an individual to take stabilizing impulsive control, whichintroduces some randomness into the evolution.It is worth mentioning that the above ten algorithms weretested on various benchmark problems, which are differentfrom the CEC 2014 test suite in our paper. To make thecomparisons fair and meaningful, an appropriate tuning ofthe population size must be carried out. Therefore, a setof tests are conducted to select a proper population sizefor each algorithm. In detail, the ten DE algorithms with NP = [30 , , , are applied to optimizing the CEC 2014 test suite 51 times, respectively, and the Holm-Bonferroniprocedure [58] with confidence level 0.05 is used to evaluatethe performance of each algorithm with different NP values.The results are listed in Tables S.1-S.10 of the supplementaryfile. According to the obtained results, for each algorithm,the NP value with the highest rank (highlighted in boldface )is chosen as its population size in the following experi-ments of our research. That is, DE/rand/1/bin: NP = 100 ,DE/best/1/bin: NP = 50 , jDE: NP = 100 , JADE: NP = 100 ,CoDE: NP = 50 , SaDE: NP = 100 , ODE, NP = 100 , EPSDE: NP = 50 , SHADE: NP = 150 , OXDE: NP = 100 .In our experiment studies, three performance evaluationcriteria are used for comparing the performance of eachalgorithm, which are listed below:
1) Error:
The average and standard deviation of the func-tion error value f ( x ) − f ( x ∗ ) are recorded, where x ∗ is theglobal optimum of the test function and x is the best solutionfound by the algorithm in a single run. And error value smallerthan − will be taken as [56].
2) Convergence graphs:
The convergence graphs are plot-ted to illustrate the mean function error values derived fromeach algorithm in the comparison.
3) Wilcoxon rank-sum test:
In order to show the significantdifference between the original DE and its ETI-DE variant, aWilcoxon rank-sum test at 5% significance level is conducted.The cases are marked with “ + / ≈ / − ” when the performance ofthe ETI-DE variant is significantly better than, equal to, andworse than the DE algorithm without the proposed scheme,respectively.
4) Holm-Bonferroni procedure:
In order to complete thestatistical analysis, the Holm-Bonferroni procedure with con-fidence level 0.05 is performed.
B. Comparison with Ten DE Algorithms
In this section, we assess the effectiveness of our developedscheme by comparing ten popular DE algorithms and theircorresponding ETI-based variants. The experimental resultsare provided in Tables S.12-S.14 of the supplementary file.“ + / ≈ / − ” indicates that the performance of DE algorithmswith ETI is significantly better than, equal to, and worsethan those without ETI. The better values compared betweenthe DE variants and their corresponding ETI-based DEs arehighlighted in boldface .From Tables S.12-S.14, we can see that the ten ETI-DEs perform better than their corresponding original DEalgorithms. For example, for all the 30 test functions, ETI-DE/rand/1/bin improves in 16 functions, ties in 9 func-tions, and only loses in 5 functions; when compared withDE/best/1/bin, ETI-DE/best/1/bin wins in 15 cases, ties in 10cases, and merely loses in 5 cases; for jDE, the incorporationof the proposed scheme exhibits superior performance in 13functions, and provides similar performance in 15 functions;for ETI-JADE, it outperforms JADE in 20 out of 30 functions,and ties in 9 functions; for ETI-CoDE, it obtains better resultsin 10 functions, while ties in 17 functions, and just losesin 3 functions; for ETI-SaDE, it wins, ties, and loses in 10,16, and 4 cases, respectively; for ODE, the proposed scheme REPRINT SUBMITTED TO ARXIV 9 improves its performance in 15 functions and only gets worsein 1 function; for ETI-EPSDE, it improves in 21 function,ties in 6 functions, and simply loses in 3 functions; ETI-SHADE wins in 11 cases, ties in 15 cases, and only losesin 4 cases when compared with SHADE; for OXDE, ETIenhances its performance in 17 functions, and merely becomesworse in 3 functions. In general, ETI significantly improvesthe search ability of the ten popular DE variants. Furthermore,in Figs. S.1-S.4 of the supplementary file, we use the boxplot to show the results of JADE, CoDE, SaDE and EPSDEwith and without ETI on CEC 2014 test suite at D = 30 .Combining Tables S.12-S.14 with Figs. S.1-S.4, we can seethe effectiveness of our proposed ETI.The results of the Holm-Bonferroni procedure are givenin Table S.15 of the supplementary file, where we set ETI-SHADE as the reference algorithm. From the rank values inTable S.15, we can find that ETI improves the performance ofall the ten DE variants on the CEC 2014 test suite at D=30.To better illustrate the convergence performance of the tenDE algorithms and their corresponding ETI-DEs, we plot theconvergence curves of these algorithms in Fig. S.5 of thesupplementary file for six selected test functions, which arefrom the four groups of the test suite. From Fig. S.5, wecan observe that our proposed ETI improves the convergenceperformance of the ten original DE algorithms by introducingtwo types of impulses.In summary, the presented ETI is very powerful and theten ETI-DEs possess strong capabilities of rapid convergenceand accurate search for the test functions. The results ofthe Wilcoxon rank-sum test confirm that our scheme is ofparamount importance to improve the performance of theconsidered DE algorithms. C. Effectiveness of Two Types of Impulses
In light of the results shown in Tables S.12-S.14, it canbe seen that the proposed ETI can significantly improve theperformance of the ten DE algorithms. The core of ETIis the use of two types of impulses: stabilizing impulsesand destabilizing impulses. Stabilizing impulses help a partof individuals get close to promising areas, which enhancethe exploitation ability of the algorithm; while destabilizingimpulses increase the diversity of the current population,which improves the exploration capability of the algorithm. Inthis section, we conduct four groups of experiments to examinehow these two kinds of impulses separately take effect forthe DE algorithms. Therefore, we consider the following fourvariants of ETI-DE:1) ETI1-DE: the proposed scheme only with injectingdestabilizing impulses when UR = 0 in each improved DE(steps 43-60 in Algorithm 1 are deleted);2) ETI2-DE: the proposed scheme without injecting desta-bilizing impulses when UR = 0 in each improved DE (steps35-42 in Algorithm 1 are deleted);3) ETI3-DE: the proposed scheme without injecting desta-bilizing impulses when ζ = 0 in each improved DE (steps49-56 in Algorithm 1 are deleted);4) ETI4-DE: the proposed scheme without injecting anydestabilizing impulses both when UR = 0 and ζ = 0 in each improved DE (steps 35-42 and 49-56 in Algorithm 1are deleted).Firstly, ETI1-DEs are compared with the ten ETI-DEsto show the effectiveness of stabilizing impulsive control.Secondly, ETI2-DEs are compared with the ten ETI-DEsto inspect the effectiveness of destabilizing impulses when UR = 0 . Thirdly, ETI3-DEs are compared with the ten ETI-DEs to examine the effectiveness of destabilizing impulseswhen ζ = 0 . Fourthly, ETI4-DEs are compared with the tenETI-DEs to exhibit the effectiveness of destabilizing impulses,which are triggered in two cases: when UR = 0 and when ζ = 0 . For saving space, we only list the win - lose resultsof four types of comparisons (ETI-DEs vs. ETI1-DEs, ETI-DEs vs.
ETI2-DEs, ETI-DEs vs.
ETI3-DEs, ETI-DEs vs.
ETI4-DEs) according to the Wilcoxon rank-sum test in Table S.16of the supplementary file.Based on the win - lose results in Table S.16, the followingconclusions can be drawn:1) The first two rows indicate the positive contribution ofstabilizing impulses. Stabilizing impulses rearrange the loca-tion distribution of the population by making the individualswith higher rankings (i.e., larger R ) reach the areas close tothe individuals with better fitness values, which increases thesearch efficiency of the ten DEs.2) The rest six rows confirm the effectiveness of destabiliz-ing impulses. In detail, the third and fourth rows demonstratethat destabilizing impulses triggered by the condition UR = 0 is of great importance to almost all the ten algorithms. UR = 0 means that the whole population stops updatingat the current generation, which is quite unfavorable for theevolution. Destabilizing impulses force the inferior individualsto leave their previous positions, in order to pull the wholepopulation out of the impasse. The fifth and sixth rowsdisplay the effect of injecting destabilizing impulses whenthe stabilizing impulses fail to take effect. Specifically, ETI-DEs wins in 4 cases, ties in 3 cases, and loses in 3 cases.The last two rows show that without destabilizing impulses,the performance of the ten algorithms deteriorates. In controltheory, destabilizing impulses introduce disturbances to adynamical system. Similarly, in DE, destabilizing impulsesbring some randomness into the evolution process when thepopulation reaches an impasse. D. Effectiveness of Random Selection of the Reference Statein Stabilizing Impulses
In ETI, when x i,G is chosen to undergo stabilizing impulses,the reference state s i,G is randomly selected from the currentbest individual ( gbest ) or other individuals with better fitnessvalues than x i,G in the current population (see Section III.A).This operation not only introduces the information of elitist,but also avoids the premature convergence for the populationduring the evolution. In order to show the effectiveness of thisoperation, in this section, we compare the performance of ETI-DE and an ETI-DE modified in the following way (referredas ETIgb-DE): the reference state s i,G is only selected from gbest in stabilizing impulses. The detailed experimental dataare provided in Tables S.17-S.19 of the supplementary file. REPRINT SUBMITTED TO ARXIV 10
The results show that ETI-DEs are significantly better thanETIgb-DEs on most test functions. In ETI-DEs, the referencestate of stabilizing impulses is set as gbest or other individualswith better fitness values than x i,G , instead of merely gbest .The setting optimizes the state of the whole population andincreases the diversity of stabilizing impulses at the same time,which avoids the premature convergence of the population.Therefore, we can conclude that it does make the search moreeffective when s i,G is randomly selected from gbest or otherindividuals with better fitness values. E. Comparison with Other Restart Strategies
To the best of our knowledge, restart strategies directlyreplace the selected individuals with other individuals, withoutcomparing the fitness values of them. Till now, different restartstrategies have been proposed in DE [36, 54, 59–62]. It isnoticed that in our approach, destabilizing impulses serveas a restart strategy, which randomly adjust the positionsof the inferior individuals within the area of the currentpopulation. Therefore, our presented ETI can also be viewedas a restart-based strategy. In this section, firstly, we illustratethe differences between some popular restart strategies in DEand ours; secondly, we compare our ETI with a latest restartstrategy published in 2015.To better illustrate the difference between other restartstrategies and destabilizing impulses presented in this paper,we list the details of each strategy in Table S.20 of thesupplementary file. In [36], if no improvement is observed for n samples sample points, a restart mechanism might be activated.Accordingly, a bubble is defined around the best individualwithin the cluster x best ; then local and global restart strategiesare performed inside and outside the bubble, respectively.However, the size of the bubble ∆ is quite critical to theperformance of the restart strategies. In [59], when the currentpopulation converges at a local optimum, a restart is activated.In detail, the newly generated individuals are forced awayfrom the δ hypersphere neighborhood areas of previous localoptima. Similarly, the neighborhood size δ is also important.In [60], the restart strategy takes effect when the predefinedclusters are “dead”. The strategy consists of two operations:restart by DE/rand/2 and restart by reinitializing certainindividuals within the search range. Specified probabilityvalues are assigned to each operation, and it is necessary todetermine these values beforehand. In [61], when stagnation isdiagnosed, the algorithm performs a restart by increasing thepopulation size by a predefined multiplier k and starting anindependent search. In [62], when the population diversity ispoor or the population stagnates by measuring the Euclideandistances between individuals of a population, the individualsare restarted within the initial search space, which is sampledby a random number randN j,G with normal distribution.In [54], a new diversity enhance mechanism named auto-enhanced population diversity (AEPD) is proposed, which isan improved version of [62]. When population convergenceor stagnation is identified by AEPD, some individuals arereinitialized. Our ETI is inspired by the idea of event-triggeredmechanism (ETM) and impulsive control in control theory, and two kinds of impulses are developed to enhance the exploita-tion and exploration performance of DE. Selected inferiorindividuals are restarted by being injected with destabilizingimpulses when UR drops to zero or stabilizing impulses failto take effect (see Section III). Furthermore, the proposedETI sheds light on the understandings of ETM and impulsivecontrol in evolutionary computation, which broadens theapplications of ETM and impulsive control in wider areas.Compared with the other methods in Table S.20, AEPD in [54]and our ETI are easy to implement: 1) they do not introduceany calculation of distances of individuals; 2) they do notuse the neighborhood, which avoids determining the valueof neighborhood size. The computational complexity of theoriginal DE is O ( G max · NP · D ) . While for AEPD and ETI, itis O ( G max · ((3 · NP +2) · D )) and O (2 · G max · NP · ( D + log ( NP )) ,respectively, both of which do not seriously increase thecomputational cost of the original DE.In the following, we compare the performance of AEPDpublished in [54] in 2015 and ETI by applying them to jDEand JADE, and check their performance on the CEC 2014benchmark functions. The parameters of jDE and JADE aregiven in Section IV.A. It is worth noting that in AEPD [54],the population size NP is set to 20. And in our ETI-jDE andETI-JADE, NP is set to 100. Therefore, the experiments aredivided into three groups: 1) to compare the performance ofAEPD-DEs and ETI-DEs with NP = 20 ; 2) to compare theperformance of AEPD-DEs and ETI-DEs with NP = 100 ; 3)to compare the performance of AEPD-DEs with NP = 20 and ETI-DEs with NP = 100 . The detailed experimentaldata are provided in Tables S.21-S.22 of the supplementaryfile. The upper half of Table S.21 demonstrates the superiorperformance of AEPD-jDE and AEPD-JADE with NP =20 ; the lower half of Table S.21 confirms the outstandingperformance of ETI-jDE and ETI-JADE with NP = 100 .For Table S.22, AEPD-DEs and ETI-DEs use the NP valuesrecommended in [54] ( NP = 20 ) and in our paper ( NP = 100 ),respectively. The results show that ETI-jDE and ETI-JADEwith NP = 100 perform better than AEPD-jDE and AEPD-JADE with NP = 20 , which identifies the effectiveness of ourETI. Compared with AEPD, our proposed ETI takes advantageof the concepts of impulsive control and ETM in controltheory, which optimizes the state of the whole populationby instantly altering the positions of partial individuals. Thestabilizing and destabilizing impulses are triggered at certainmoments, which enhances the exploitation and explorationabilities respectively, and saves the computational resources. F. Effectiveness of Ranking Assignment
As introduced before, we select the individuals to undergoimpulsive control by ranking the population based on twoindices: fitness value and the number of consecutive stagnantgeneration. These two indices identify the state of eachindividual during the evolution process. Hence, ˜ R i and ¯ R i are acquired, which are integrated into R i . In this section,we carry out three classes of experiments to demonstrate theeffectiveness of the ranking assignment. The first experimentcompares the performance of ranking the population according REPRINT SUBMITTED TO ARXIV 11 to R i and only according to ˜ R i ; the second experiment displaysthe difference between utilizing R i and merely utilizing ¯ R i ; thelast experiment picks random individuals to be injected withimpulses instead of using R i . The corresponding three variantsof ETI-DEs are as follows: R1-ETI-DE, R2-ETI-DE, and NoR-ETI-DE. These three variants are compared with ETI-DEs,the results of which ( win - lose results of ETI-DEs vs. R1-ETI-DEs, ETI-DEs vs.
R2-ETI-DEs, ETI-DEs vs.
NoR-ETI-DEsaccording to the Wilcoxon rank-sum test) are exhibited inTable S.23 of the supplementary file. Similar to the analysisof the last experiment, we also obtain three conclusions fromTable S.23:1) From the first two rows, it is observed that the individualsinjected with impulsive controllers cannot be merely selectedaccording to fitness values. The number of consecutive stag-nant generation is also needed to be taken into consideration.For an individual, which has been updated in recent gener-ations, although its fitness value is inferior, it may be neara promising area. Therefore, it is still necessary to keep thisindividual in the population.2) From the third and fourth rows, it is noted that theindividuals with impulsive control cannot be chosen by onlyconsidering the number of consecutive stagnant generationeither. For an individual with superior fitness value, whichstagnates in recent generations, it may still be helpful for theupdate of other individuals.3) The last two rows further explain the necessity of usingboth fitness values and the number of consecutive stagnantgeneration to rank the individuals, which will be added im-pulses later. These two measures reflect the two most importantcharacteristics of an individual in the evolution process. Byadding impulses to the inferior individuals graded by thesetwo measures, the search performance of the population canbe enhanced.
Remark . R i is the sum of ˜ R i and ¯ R i , where ˜ R i and ¯ R i indicate the rankings of the individual according to the fitnessvalue and the number of consecutive stagnation generation,respectively. In the following, we use jDE and ETI-jDE onF02 (unimodal) and F13 (multimodal) to show the impact ofETI on the population during the evolution process. In Fig.S.6 of the supplementary file, we plot the evolution of tenrandom individuals’ R i by jDE and ETI-jDE on F02 and F13,respectively. From the figure, it can be observed that R i byETI-jDE changes more frequently than that by jDE, whichverifies that the introduction of ETI enhances the movementof the population. G. Parameter Sensitivity Study
In the proposed ETI, there are three parameters: pr , LN ,and UN . pr represents the probability of selecting individualsto be injected with destabilizing impulses. LN and UN arethe lower and upper bounds of the number of individuals withstabilizing impulses. In ETI, pr is set as . , LN is set as , and UN is set as the same as the population size NP . Here, weset pr = [0 . , . , . , LN = [1 , . NP , . NP , . NP ] , and UN = [0 . NP , . NP , . NP , NP ] . And we compare the tenDE variants and their corresponding ETI-DEs with different parameter values, in order to investigate the effects of theparameters on the performance of ETI-DEs.In Figs. S.7-S.9 of the supplementary file, we use bar graphto show the number of functions that ETI-DEs with differentparameter values are significantly better than, equal to, andworse than the original DE algorithms, respectively.For pr , most ETI-DEs (e.g., ETI-DE/best/1/bin, ETI-jDE,ETI-JADE, ETI-SaDE, ETI-EPSDE, and ETI-OXDE) with pr = 0 . have more winning functions than that with pr = [0 . , . . Besides, most ETI-DEs (e.g., ETI-jDE, ETI-JADE, ETI-CoDE, ETI-ODE, ETI-SHADE, and ETI-OXDE)with pr = 0 . have fewer losing functions than that with pr = [0 . , . . Then we add the win/tie/lose numbers forall the algorithms when using the same value of pr . And wefind that pr = 0 . is a little bit better than pr = [0 . , . .A larger pr denotes that more individuals will be injectedwith destabilizing impulses. And the introduction of morerandomness may be harmful to the current research. Therefore,we set pr = 0 . in the proposed ETI.For LN , along with the growth of LN , we can find a trendfor nine of the total ten ETI-DEs except ETI-DE/best/1/bin:the number of winning functions decreases and the numberof losing function increases. And for ETI-DE/best/1/bin, itsperformance is not sensitive to the change of LN . Then we addthe win/tie/lose numbers for all the algorithms when using thesame value of LN . And we find that LN = 1 is much betterthan LN = [0 . NP , . NP , . NP ] . A larger LN indicates thatmore individuals are injected with stabilizing impulses, whichwill disrupt the ongoing search. Therefore, we choose LN = 1 for our proposed ETI.For UN , we find that most ETI-DEs (e.g., ETI-DE/best/1/bin, ETI-JADE, ETI-SaDE, ETI-ODE, ETI-EPSDE,ETI-SHADE, and ETI-OXDE) with UN = NP show betterperformance than that with UN = 0 . NP . In general, theresults indicate that too small values of UN (e.g. . NP ) areworse than large values (e.g. . NP , . NP and NP ). Andalong with the increase of UN , the performance of ETI-DEswith different UN values becomes close. UN is the upperbound of the number of individuals with stabilizing impulses.In the experiments, we find that when ETI is adopted, it ismore difficult for the number of individuals injected withstabilizing impulsive controllers to reach UN as UN becomeslarger. And this explains why the performance of ETI-DEsbecomes similar along with the increase of UN . We add the win/tie/lose numbers for all the algorithms when using thesame value of UN . And we find that it is a good choice toselect UN = NP in our paper. H. Scalability Study
In the aforementioned experiments, we evaluate the perfor-mance of the algorithms by running them on 30 test functionswith D = 30 from CEC 2014 test suite. The results showthat our proposed ETI can improve the performance of theoriginal DE variants. In this section, we perform the scalabilitystudy of the ETI to further examine its effectiveness. Thedimension of the test functions is set as D = 50 and D = 100 .The detailed experimental data and the results of the Holm-Bonferroni procedure are provided in Tables S.24-S.31 of the REPRINT SUBMITTED TO ARXIV 12 supplementary file. Furthermore, in Figs. S.10-S.13 of thesupplementary file, we use the box plot to show the resultsof JADE, CoDE, SaDE and EPSDE with and without ETI onCEC 2014 test suite at D = 100 . From the results, we canfind out that our developed scheme still works effectively forproblems with large dimension ( D = 50 and D = 100 ). I. Working Mechanism of ETI
The above experimental results reveal the effectiveness ofour proposed ETI. In this subsection, we will investigate theworking mechanism of ETI based on some experiments.In DE, the update rate ( UR ) of the population reflectsthe degree of the activity of the population in the evolutionprocess. The decrease of UR indicates that the populationgradually encounters stagnation. The worst situation is that UR drops to zero before the global best solution is found.As introduced in Section III, two types of impulses, i.e.,stabilizing and destabilizing impulses, are imposed on theselected individuals when UR of the population in the currentgeneration decreases or equals to zero. When UR declines,certain individuals will instantly approach the reference statesby means of stabilizing impulses. And we hope new superiorindividuals can be found, and help other individuals update.When UR reduces to zero, selected individuals will beinstantaneously displaced to other areas with the help ofdestabilizing impulses. These individuals in the new positionsare expected to influence the stagnant population in a positiveway.In the following, we use jDE and ETI-jDE on the Shiftedand Rotated Katsuura Function (i.e., F12 of the CEC 2014test suite) to explain how ETI works. The results are givenin Fig. S.14. Figs. S.14(a)-S.14(b) show the change of thefitness value of gbest when F12 is optimized by jDE and ETI-jDE, respectively. Figs. S.14(c)-S.14(d) display the change ofthe value of UR when F12 is optimized by jDE and ETI-jDE, respectively. Take a look at the status of the populationfrom the 1700th generation to the 1750th generation. In Fig.S.14(c), the generations when UR = 0 are marked by reddots; and we find that UR drops to zero very frequently, whichmeans the whole population lacks the impetus to evolve. InFig. S.14(d), the generations when UR = 0 are also markedby red dots, and red circles are used to mark the momentswhen UR reduces. By observing the magnified figure in Fig.S.14(d), UR does not drop to zero frequently, and we attributeit to the destabilizing impulses in ETI. When UR decreasesin the current generation, it seldom reduces again in the nextgeneration due to the stabilizing impulses in ETI. Stabilizingand destabilizing impulses increase the degree of activity ofthe population during the evolution process, and hence helpthe update of gbest (see Fig. S.14(b)).The advantages of ETI can be summarized as follows:1) Stabilizing impulses in ETI force certain individualsapproach the reference states, which enhances the exploitationability of the algorithm.2) Destabilizing impulses in ETI bring in more explorativemoves, which increases the exploration ability of the algo-rithm. 3) Event-triggered mechanism (ETM) determines the mo-ment of adding two types of impulses, which avoids theperiodical execution of impulsive control and saves the com-putational resources.4) ETI is easy to implement and flexible to be incorporatedinto other state-of-the-art DE variants.V. C ONCLUSION
In this paper, an event-triggered impulsive control scheme(ETI) has been proposed to improve the search performance ofDE algorithms. There are four components in ETI: stabilizingimpulses, destabilizing impulses, ranking assignment, and anadaptive mechanism. Firstly, stabilizing impulses aims to forcethe selected individuals to approach some promising areasinstantly in the search domain, which facilitates the con-vergence of the population. Secondly, destabilizing impulsesinstantaneously alter the positions of inferior individuals in acertain area, which maintains the diversity of the population.Thirdly, ranking assignment is used to select the individualsto be injected with impulsive controllers, which is based onthe fitness value and the number of consecutive stagnantgeneration of each individual. Fourthly, an adaptive mech-anism is presented to determine the number of individualstaking impulsive control. Besides, ETM identifies the momentof imposing these two kinds of impulses. Meanwhile, theproposed ETI does not significantly increase the computationalcomplexity of DE algorithms.Extensive experiments have been carried out based on theCEC 2014 test suite. Firstly, ETI has been incorporated intotwo original DE algorithms and eight state-of-the-art DEvariants. A series of results demonstrate that ETI can greatlyimprove the performance of these DE algorithms. Then severalcomparative experiments have been conducted to show theeffectiveness of two types of impulses, random selection ofthe reference state in stabilizing impulses, and ranking assign-ment. Besides, we compare ETI with other restart strategies,and investigate the influence of three parameters. Then thepresented ETI-DEs have also shown their superiority in high-dimensional problems. Finally, the working mechanism of ETIhas been investigated based on some experiments. It is worthmentioning that the experimental results shed light on theunderstandings of ETM and impulsive control in evolutionarycomputation, which broadens the applications of ETM andimpulsive control in wider areas.A
CKNOWLEDGMENT
The authors would like to thank Dr. Y. Wang, Dr. M. Yang,and Dr. J. Wang for providing the source code of CoDE,AEPD, and OXDE, respectively.R
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REPRINT SUBMITTED TO ARXIV 15
Supplementary file of ETI-DE A LGORITHM C APTIONS • Algorithm S.1
Stabilizing Impulsive Control (). • Algorithm S.2
Injecting Destabilizing Impulses (). • Algorithm S.3
Random Selection of Individuals ().T
ABLE C APTIONS • Table S.1
Holm test on the fitness, reference algorithm= DE/rand/1/bin ( NP = 100 , rank= ) for CEC 2014test suite at D = 30 . • Table S.2
Holm test on the fitness, reference algorithm =DE/best/1/bin ( NP = 100 , rank=2.47) for CEC 2014 testsuite at D = 30 . • Table S.3
Holm test on the fitness, reference algorithm= jDE ( NP = 100 , rank= ) for CEC 2014 test suite at D = 30 . • Table S.4
Holm test on the fitness, reference algorithm= JADE ( NP = 100 , rank= ) for CEC 2014 test suiteat D = 30 . • Table S.5
Holm test on the fitness, reference algorithm= CoDE ( NP = 100 , rank=2.23) for CEC 2014 test suiteat D = 30 . • Table S.6
Holm test on the fitness, reference algorithm= SaDE ( NP = 100 , rank= ) for CEC 2014 test suiteat D = 30 . • Table S.7
Holm test on the fitness, reference algorithm= ODE ( NP = 100 , rank= ) for CEC 2014 test suiteat D = 30 . • Table S.8
Holm test on the fitness, reference algorithm =EPSDE ( NP = 100 , rank=2.70) for CEC 2014 test suiteat D = 30 . • Table S.9
Holm test on the fitness, reference algorithm =SHADE ( NP = 100 , rank=3.13) for CEC 2014 test suiteat D = 30 . • Table S.10
Holm test on the fitness, reference algorithm= OXDE ( NP = 100 , rank= ) for CEC 2014 test suiteat D = 30 . • Table S.11
WIN - LOSE results of ETI-DEs with random DM and with DM = [ D / , D / , D ] (in Section IV.A)for functions F01-F30 at D = 30 . • Table S.12
Experimental results of DE/rand/1/bin,DE/best/1/bin, jDE, JADE and the related ETI-basedvariants for functions F01-F30 at D = 30 . • Table S.13
Experimental results of CoDE, SaDE, ODE,EPSDE and the related ETI-based variants for functionsF01-F30 at D = 30 . • Table S.14
Experimental results of SHADE, OXDE andthe related ETI-based variants for functions F01-F30 at D = 30 . • Table S.15
Holm test on the fitness, reference algorithm =ETI-SHADE (rank=14.47) for functions F01-F30 at D =30 . • Table S.16
WIN - LOSE results of ETI1-DEs, ETI2-DEs, ETI3-DEs, ETI4-DEs (in Section IV.C) and theircounterparts for functions F01-F30 at D = 30 . • Table S.17
Experimental results of ETI-DE/rand/1/bin,ETI-DE/best/1/bin, ETI-jDE, ETI-JADE and the relatedETIgb-DEs for functions F01-F30 at D = 30 . • Table S.18
Experimental results of ETI-CoDE, ETI-SaDE, ETI-ODE, ETI-EPSDE and the related ETIgb-DEsfor functions F01-F30 at D = 30 . • Table S.19
Experimental results of ETI-SHADE, ETI-OXDE and the related ETIgb-DEs for functions F01-F30at D = 30 . • Table S.20
Different restart strategies in DE. • Table S.21
Experimental results of ETI-DEs and AEPD-DEs for functions F01-F30 at D = 30 . • Table S.22
Experimental results of ETI-DEs and AEPD-DEs for functions F01-F30 at D = 30 . • Table S.23
WIN - LOSE results of R1-ETI-DEs, R2-ETI-DEs, NoR-ETI-DEs (in Section IV.F) and ETI-DEs forfunctions F01-F30 at D = 30 . • Table S.24
Experimental results of DE/rand/1/bin,DE/best/1/bin, jDE, JADE and the related ETI-basedvariants for functions F01-F30 at D = 50 . • Table S.25
Experimental results of CoDE, SaDE, ODE,EPSDE and the related ETI-based variants for functionsF01-F30 at D = 50 . • Table S.26
Experimental results of SHADE, OXDE andthe related ETI-based variants for functions F01-F30 at D = 50 . • Table S.27
Experimental results of DE/rand/1/bin,DE/best/1/bin, jDE, JADE and the related ETI-basedvariants for functions F01-F30 at D = 100 . • Table S.28
Experimental results of CoDE, SaDE, ODE,EPSDE and the related ETI-based variants for functionsF01-F30 at D = 100 . • Table S.29
Experimental results of SHADE, OXDE andthe related ETI-based variants for functions F01-F30 at D = 100 . • Table S.30
Holm test on the fitness, reference algorithm =ETI-SHADE (rank=15.70) for functions F01-F30 at D =50 . • Table S.31
Holm test on the fitness, reference algorithm =ETI-SHADE (rank=15.50) for functions F01-F30 at D =100 . F IGURE C APTIONS • Fig. S.1
Box plots for the results of JADE with/withoutETI on CEC 2014 test suite at D = 30 : 1–JADE; 2–ETI-JADE. • Fig. S.2
Box plots for the results of CoDE with/withoutETI on CEC 2014 test suite at D = 30 : 1–CoDE; 2–ETI-CoDE. • Fig. S.3
Box plots for the results of SaDE with/withoutETI on CEC 2014 test suite at D = 30 : 1–SaDE; 2–ETI-SaDE. • Fig. S.4
Box plots for the results of EPSDE with/withoutETI on CEC 2014 test suite at D = 30 : 1–EPSDE; 2–ETI-EPSDE. REPRINT SUBMITTED TO ARXIV 16 • Fig. S.5
Evolution of the mean function error valuesobtained from the algorithms versus the number of FESon six 30-dimensional test functions. (a) F02; (b) F11;(c) F15; (d) F19; (e) F20; (f) F26. • Fig. S.6
Evolution of R i by jDE and ETI-jDE on F02and F13 at D = 30 . (a) Evolution of ten randomindividuals’ R i by jDE on F02. (b) Evolution of tenrandom individuals’ R i by ETI-jDE on F02. (c) Evolutionof ten random individuals’ R i by jDE on F13. (d)Evolution of ten random individuals’ R i by ETI-jDE onF13. • Fig. S.7
The number of functions that ETI-DEs withdifferent pr values ( LN = 1 , UN = 100 ) are significantlybetter than, equal to and worse than the original DEs onCEC 2014 test suite at D = 30 . (The results of addingthe win/tie/lose numbers for all the algorithms when usingthe same value of pr : pr = 0 . / / pr = 0 . / / pr = 1 . / / .) • Fig. S.8
The number of functions that ETI-DEs with dif-ferent LN values ( pr = 0 . , UN = 100 ) are significantlybetter than, equal to and worse than the original DEs onCEC 2014 test suite at D = 30 . (The results of adding the win/tie/lose numbers for all the algorithms when usingthe same value of LN : LN = 1 : 181 / / LN =20 : 170 / / LN = 50 : 149 / / LN = 80 :133 / / .) • Fig. S.9
The number of functions that ETI-DEs withdifferent UN values ( pr = 0 . , LN = 1 ) are significantlybetter than, equal to and worse than the original DEs onCEC 2014 test suite at D = 30 . (The results of adding the win/tie/lose numbers for all the algorithms when usingthe same value of UN : UN = 5 : 152 / / UN =20 : 172 / / UN = 50 : 175 / / UN = 100 :181 / / .) • Fig. S.10
Box plots for the results of JADE with/withoutETI on CEC 2014 test suite at D = 100 : 1–JADE; 2–ETI-JADE. • Fig. S.11
Box plots for the results of CoDE with/withoutETI on CEC 2014 test suite at D = 100 : 1–CoDE; 2–ETI-CoDE. • Fig. S.12
Box plots for the results of SaDE with/withoutETI on CEC 2014 test suite at D = 100 : 1–SaDE; 2–ETI-SaDE. • Fig. S.13
Box plots for the results of EPSDE with/withoutETI on CEC 2014 test suite at D = 100 : 1–EPSDE; 2–ETI-EPSDE. • Fig. S.14
Working mechanism of ETI by jDE and ETI-jDE on F12 at D = 30 . (a) Change of the fitness value of gbest of F12 optimized by jDE. (b) Change of the fitnessvalue of gbest of F12 optimized by ETI-jDE. (c) Changeof the value of UR of F12 optimized by jDE. (d) Changeof the value of UR of F12 optimized by ETI-jDE. REPRINT SUBMITTED TO ARXIV 17
Algorithm S.1
Stabilizing Impulsive Control () Begin /* x i,G is the individual that undergoes stabilizing impulsive control /* ζ i,G is a flag to indicate whether stabilizing impulsive control is to improvethe fitness value /* rand(a,b) uniformly generate a random number belonging to the interval (a,b) /* DM is the number of dimensions selected to undergo stabilizing impulsivecontrol in x i,G A = { , , ..., D } ; B = ∅ ; C = ∅ Randomly select an individual x k,G from the current population if f ( x i,G ) < f ( x k,G ) then Set x gbest,G as the reference state s i,G Generate B by randomly selecting DM elements from A for j = 1 to D do if j ∈ B then K ij,G = rand ( − , else K ij,G = 0 end if end for K i,G = diag { K i1,G , K i2,G , ..., K iD,G } D × D , i = 1 , , ..., NP e i,G = x i,G − s i,G x i,G + = x i,G + K i,G · e i,G else Set x k,G as the reference state s i,G Generate C by randomly selecting DM elements from A for j = 1 to D do if j ∈ C then K ij,G = − else K ij,G = 0 end if end for K i,G = diag { K i1,G , K i2,G , ..., K iD,G } D × D , i = 1 , , ..., NP e i,G = x i,G − s i,G x i,G + = x i,G + K i,G · e i,G end if if f ( x i,G + ) ≤ f ( x i,G ) then x i,G = x i,G + ζ i,G = 1 else ζ i,G = 0 end if End
Algorithm S.2
Injecting Destabilizing Impulses () Begin /* x i,G is the individual that undergoes destabilizing impulses /* min j,G and max j,G are the minimum and maximum values of the j th dimensionin the population at the G th generation /* rand(a,b) uniformly generate a random number belonging to the interval (a,b) x L,G = [ min , min , ..., min D,G ] T x U,G = [ max , max , ..., max D,G ] T for j = 1 to D do K ij,G = rand (0 , end for K i,G = diag { K i1,G , K i2,G , ..., K iD,G } D × D , i = 1 , , ..., NP e i,G = x U,G − x L,G x i,G + = x L,G + K i,G · e i,G End
Algorithm S.3
Random Selection of Individuals () Begin /* { x i,G | i = 1 , , ..., M } are the candidates that undergo destabilizing impulses /* pr is the probability for selecting individuals to be injected with destabilizingimpulses /* ǫ is a flag for judging whether it is necessary to increase pr for i = 1 to M do r i = rand end for pr = 0 . , ǫ = 0 while ǫ = 0 do for i = 1 to M do if r i < pr then x i,G will undergo destabilizing impulsive control later ǫ = 1 end if end for pr = pr + 0 . pr = min ( pr , . end while End
REPRINT SUBMITTED TO ARXIV 18
TABLE S.1H
OLM TEST ON THE FITNESS , REFERENCE ALGORITHM = DE/
RAND /1/
BIN ( NP = 100 , RANK = ) FOR
CEC 2014
TEST SUITE AT D = 30 . j Optimizer Rank z j p j δ / j Hypothesis NP = 30 ) 1.87 -3.30E+00 4.83E-04 5.00E-02 Rejected2 DE/rand/1/bin ( NP = 50 ) 2.77 -6.00E-01 2.74E-01 2.50E-02 Accepted3 DE/rand/1/bin ( NP = 150 ) 2.40 -1.70E+00 4.46E-02 1.67E-02 AcceptedTABLE S.2H OLM TEST ON THE FITNESS , REFERENCE ALGORITHM = DE/
BEST /1/
BIN ( NP = 100 , RANK =2.47)
FOR
CEC 2014
TEST SUITE AT D = 30 . j Optimizer Rank z j p j δ / j Hypothesis NP = 30 ) 2.77 9.00E-01 8.16E-01 5.00E-02 Accepted2 DE/best/1/bin ( NP = 50 ) NP = 150 ) 1.73 -2.20E+00 1.39E-02 1.67E-02 RejectedTABLE S.3H OLM TEST ON THE FITNESS , REFERENCE ALGORITHM = J DE ( NP = 100 , RANK = ) FOR
CEC 2014
TEST SUITE AT D = 30 . j Optimizer Rank z j p j δ / j Hypothesis NP = 30 ) 2.20 -1.60E+00 5.48E-02 5.00E-02 Accepted2 jDE ( NP = 50 ) NP = 150 ) 2.33 -1.20E+00 1.15E-01 1.67E-02 AcceptedTABLE S.4H OLM TEST ON THE FITNESS , REFERENCE ALGORITHM = JADE ( NP = 100 , RANK = ) FOR
CEC 2014
TEST SUITE AT D = 30 . j Optimizer Rank z j p j δ / j Hypothesis NP = 30 ) 1.67 -4.00E+00 3.17E-05 5.00E-02 Rejected2 JADE ( NP = 50 ) 2.53 -1.40E+00 8.08E-02 2.50E-02 Accepted3 JADE ( NP = 150 ) 2.80 -6.00E-01 2.74E-01 1.67E-02 AcceptedTABLE S.5H OLM TEST ON THE FITNESS , REFERENCE ALGORITHM = C O DE ( NP = 100 , RANK =2.23)
FOR
CEC 2014
TEST SUITE AT D = 30 . j Optimizer Rank z j p j δ / j Hypothesis NP = 30 ) 3.27 3.10E+00 9.99E-01 5.00E-02 Accepted2 CoDE ( NP = 50 ) NP = 150 ) 1.00 -3.70E+00 1.08E-04 1.67E-02 Rejected REPRINT SUBMITTED TO ARXIV 19
TABLE S.6H
OLM TEST ON THE FITNESS , REFERENCE ALGORITHM = S A DE ( NP = 100 , RANK = ) FOR
CEC 2014
TEST SUITE AT D = 30 . j Optimizer Rank z j p j δ / j Hypothesis NP = 30 ) 1.73 -3.70E+00 1.08E-04 5.00E-02 Rejected2 SaDE ( NP = 50 ) 2.53 -1.30E+00 9.68E-02 2.50E-02 Accepted3 SaDE ( NP = 150 ) 2.77 -6.00E-01 2.74E-01 1.67E-02 AcceptedTABLE S.7H OLM TEST ON THE FITNESS , REFERENCE ALGORITHM = ODE ( NP = 100 , RANK = ) FOR
CEC 2014
TEST SUITE AT D = 30 . j Optimizer Rank z j p j δ / j Hypothesis NP = 30 ) 1.57 -4.80E+00 7.93E-07 5.00E-02 Rejected2 ODE ( NP = 50 ) 2.87 -9.00E-01 1.84E-01 2.50E-02 Accepted3 ODE ( NP = 150 ) 2.40 -2.30E+00 1.07E-02 1.67E-02 RejectedTABLE S.8H OLM TEST ON THE FITNESS , REFERENCE ALGORITHM = EPSDE ( NP = 100 , RANK =2.70)
FOR
CEC 2014
TEST SUITE AT D = 30 . j Optimizer Rank z j p j δ / j Hypothesis NP = 30 ) 2.30 -1.20E+00 1.15E-01 5.00E-02 Accepted2 EPSDE ( NP = 50 ) NP = 150 ) 2.20 -1.50E+00 6.68E-02 1.67E-02 AcceptedTABLE S.9H OLM TEST ON THE FITNESS , REFERENCE ALGORITHM = SHADE ( NP = 100 , RANK =3.13)
FOR
CEC 2014
TEST SUITE AT D = 30 . j Optimizer Rank z j p j δ / j Hypothesis NP = 30 ) 1.37 -5.30E+00 5.79E-08 5.00E-02 Rejected2 SHADE ( NP = 50 ) 2.17 -2.90E+00 1.90E-03 2.50E-02 Rejected3 SHADE ( NP = 150 ) OLM TEST ON THE FITNESS , REFERENCE ALGORITHM = OXDE ( NP = 100 , RANK = ) FOR
CEC 2014
TEST SUITE AT D = 30 . j Optimizer Rank z j p j δ / j Hypothesis NP = 30 ) 1.90 -3.50E+00 2.33E-04 5.00E-02 Rejected2 OXDE ( NP = 50 ) 2.70 -1.10E+00 1.36E-01 2.50E-02 Accepted3 OXDE ( NP = 150 ) 2.33 -2.20E+00 1.39E-02 1.67E-02 RejectedTABLE S.11 WIN - LOSE
RESULTS OF
ETI-DE
S WITH RANDOM DM AND WITH DM = [ D / , D / , D ] ( IN S ECTION
IV.A)
FOR FUNCTIONS
F01-F30 AT D = 30 . ETI-DE/rand/1/bin ETI-DE/best/1/bin random
DMvs. DM = D / random DMvs. DM = 2 D / random DMvs. DM = D random DMvs. DM = D / random DMvs. DM = 2 D / random DMvs. DM = D ETI-jDE ETI-JADE random
DMvs. DM = D / random DMvs. DM = 2 D / random DMvs. DM = D random DMvs. DM = D / random DMvs. DM = 2 D / random DMvs. DM = D ETI-CoDE ETI-SaDE random
DMvs. DM = D / random DMvs. DM = 2 D / random DMvs. DM = D random DMvs. DM = D / random DMvs. DM = 2 D / random DMvs. DM = D ETI-ODE ETI-EPSDE random
DMvs. DM = D / random DMvs. DM = 2 D / random DMvs. DM = D random DMvs. DM = D / random DMvs. DM = 2 D / random DMvs. DM = D ETI-SHADE ETI-OXDE random
DMvs. DM = D / random DMvs. DM = 2 D / random DMvs. DM = D random DMvs. DM = D / random DMvs. DM = 2 D / random DMvs. DM = D REPRINT SUBMITTED TO ARXIV 20
TABLE S.12E
XPERIMENTAL RESULTS OF
DE/
RAND /1/
BIN , DE/
BEST /1/
BIN , J DE, JADE
AND THE RELATED
ETI-
BASED VARIANTS FOR FUNCTIONS
F01-F30 AT D = 30 . Function DE/rand/1/bin ETI-DE/rand/1/bin DE/best/1/bin ETI-DE/best/1/bin
F01 ± ± − ± ± + F02 0.000E+00 ± ± ≈ ± ± ≈ F03 0.000E+00 ± ± ≈ ± ± ≈ F04 ± ± − ± ± + F05 2.094E+01 ± ± + ± ± + F06 ± ± ≈ ± ± ≈ F07 3.383E-04 ± ± ≈ ± ± ≈ F08 1.256E+02 ± ± + ± ± − F09 1.780E+02 ± ± + ± ± + F10 5.153E+03 ± ± + ± ± − F11 6.817E+03 ± ± + ± ± + F12 2.426E+00 ± ± + ± ± + F13 3.525E-01 ± ± + ± ± + F14 2.834E-01 ± ± + ± ± ≈ F15 1.548E+01 ± ± + ± ± + F16 1.242E+01 ± ± + ± ± + F17 1.331E+03 ± ± + ± ± + F18 5.407E+01 ± ± + ± ± − F19 4.529E+00 ± ± + ± ± + F20 3.354E+01 ± ± + ± ± + F21 6.480E+02 ± ± + ± ± + F22 ± ± − ± ± − F23 3.152E+02 ± ± ≈ ± ± ≈ F24 2.185E+02 ± ± ≈ ± ± ≈ F25 2.027E+02 ± ± ≈ ± ± + F26 1.003E+02 ± ± + ± ± + F27 3.592E+02 ± ± ≈ ± ± ≈ F28 ± ± ≈ ± ± ≈ F29 ± ± − ± ± − F30 ± ± − ± ± ≈ + / ≈ / − - 16/9/5 - 15/10/5 Function jDE ETI-jDE JADE ETI-JADE
F01 7.607E+04 ± ± ≈ ± ± ≈ F02 0.000E+00 ± ± ≈ ± ± ≈ F03 0.000E+00 ± ± ≈ ± ± + F04 4.419E+00 ± ± ≈ ± ± ≈ F05 2.031E+01 ± ± + ± ± + F06 9.958E+00 ± ± + ± ± + F07 0.000E+00 ± ± ≈ ± ± + F08 0.000E+00 ± ± ≈ ± ± ≈ F09 4.833E+01 ± ± + ± ± + F10 ± ± − ± ± − F11 2.467E+03 ± ± + ± ± + F12 4.379E-01 ± ± + ± ± + F13 2.983E-01 ± ± + ± ± + F14 2.849E-01 ± ± + ± ± + F15 5.760E+00 ± ± + ± ± + F16 9.988E+00 ± ± + ± ± + F17 2.421E+03 ± ± ≈ ± ± + F18 ± ± ≈ ± ± + F19 4.667E+00 ± ± + ± ± + F20 1.183E+01 ± ± + ± ± + F21 2.704E+02 ± ± ≈ ± ± + F22 1.385E+02 ± ± + ± ± + F23 3.152E+02 ± ± ≈ ± ± ≈ F24 ± ± ≈ ± ± ≈ F25 2.034E+02 ± ± ≈ ± ± ≈ F26 1.003E+02 ± ± + ± ± + F27 3.719E+02 ± ± ≈ ± ± + F28 ± ± − ± ± + F29 ± ± ≈ ± ± ≈ F30 1.594E+03 ± ± ≈ ± ± ≈ + / ≈ / − - 13/15/2 - 20/9/1REPRINT SUBMITTED TO ARXIV 21 TABLE S.13E
XPERIMENTAL RESULTS OF C O DE, S A DE, ODE, EPSDE
AND THE RELATED
ETI-
BASED VARIANTS FOR FUNCTIONS
F01-F30 AT D = 30 . Function CoDE ETI-CoDE SaDE ETI-SaDE
F01 ± ± − ± ± − F02 0.000E+00 ± ± ≈ ± ± ≈ F03 0.000E+00 ± ± ≈ ± ± ≈ F04 1.246E+00 ± ± ≈ ± ± ≈ F05 2.055E+01 ± ± + ± ± + F06 ± ± − ± ± ≈ F07 0.000E+00 ± ± ≈ ± ± ≈ F08 0.000E+00 ± ± ≈ ± ± ≈ F09 3.488E+01 ± ± ≈ ± ± + F10 1.082E+02 ± ± + ± ± ≈ F11 3.138E+03 ± ± + ± ± + F12 8.274E-01 ± ± + ± ± + F13 3.849E-01 ± ± + ± ± + F14 2.660E-01 ± ± + ± ± + F15 6.638E+00 ± ± + ± ± + F16 1.119E+01 ± ± + ± ± + F17 ± ± ≈ ± ± ≈ F18 ± ± ≈ ± ± − F19 4.330E+00 ± ± + ± ± + F20 9.196E+00 ± ± ≈ ± ± ≈ F21 ± ± ≈ ± ± ≈ F22 ± ± ≈ ± ± ≈ F23 3.152E+02 ± ± ≈ ± ± − F24 ± ± ≈ ± ± ≈ F25 2.030E+02 ± ± ≈ ± ± ≈ F26 1.004E+02 ± ± + ± ± + F27 3.967E+02 ± ± ≈ ± ± ≈ F28 8.066E+02 ± ± ≈ ± ± ≈ F29 ± ± ≈ ± ± − F30 ± ± − ± ± ≈ + / ≈ / − - 10/17/3 - 10/16/4 Function ODE ETI-ODE EPSDE ETI-EPSDE
F01 ± ± ≈ ± ± + F02 1.112E+03 ± ± + ± ± ≈ F03 0.000E+00 ± ± ≈ ± ± ≈ F04 6.494E+00 ± ± ≈ ± ± − F05 2.079E+01 ± ± + ± ± + F06 7.431E-01 ± ± ≈ ± ± + F07 ± ± − ± ± + F08 4.823E+01 ± ± + ± ± ≈ F09 3.137E+01 ± ± ≈ ± ± + F10 3.028E+03 ± ± + ± ± − F11 3.015E+03 ± ± + ± ± + F12 9.704E-01 ± ± + ± ± + F13 3.238E-01 ± ± + ± ± + F14 2.564E-01 ± ± + ± ± + F15 6.645E+00 ± ± + ± ± + F16 1.190E+01 ± ± + ± ± + F17 1.499E+03 ± ± + ± ± + F18 1.117E+01 ± ± ≈ ± ± + F19 3.214E+00 ± ± + ± ± + F20 3.754E+01 ± ± + ± ± + F21 7.235E+02 ± ± + ± ± + F22 ± ± ≈ ± ± − F23 3.152E+02 ± ± ≈ ± ± + F24 ± ± ≈ ± ± ≈ F25 ± ± ≈ ± ± ≈ F26 1.003E+02 ± ± + ± ± + F27 ± ± ≈ ± ± + F28 ± ± ≈ ± ± ≈ F29 6.989E+02 ± ± ≈ ± ± + F30 ± ± ≈ ± ± ++ / ≈ / − - 15/14/1 - 21/6/3REPRINT SUBMITTED TO ARXIV 22 TABLE S.14E
XPERIMENTAL RESULTS OF
SHADE, OXDE
AND THE RELATED
ETI-
BASED VARIANTS FOR FUNCTIONS
F01-F30 AT D = 30 . Function SHADE ETI-SHADE OXDE ETI-OXDE
F01 7.131E+00 ± ± + ± ± − F02 0.000E+00 ± ± ≈ ± ± ≈ F03 0.000E+00 ± ± ≈ ± ± ≈ F04 0.000E+00 ± ± ≈ ± ± − F05 ± ± − ± ± + F06 8.833E-02 ± ± ≈ ± ± ≈ F07 0.000E+00 ± ± ≈ ± ± + F08 0.000E+00 ± ± ≈ ± ± + F09 2.482E+01 ± ± + ± ± + F10 ± ± ≈ ± ± + F11 ± ± ≈ ± ± + F12 ± ± − ± ± + F13 1.723E-01 ± ± + ± ± + F14 ± ± − ± ± + F15 3.819E+00 ± ± + ± ± + F16 ± ± ≈ ± ± + F17 9.119E+02 ± ± + ± ± + F18 1.737E+01 ± ± + ± ± + F19 4.331E+00 ± ± + ± ± + F20 ± ± − ± ± + F21 1.762E+02 ± ± ≈ ± ± + F22 1.211E+02 ± ± + ± ± ≈ F23 3.152E+02 ± ± ≈ ± ± ≈ F24 ± ± ≈ ± ± ≈ F25 2.034E+02 ± ± + ± ± ≈ F26 1.002E+02 ± ± + ± ± + F27 3.088E+02 ± ± ≈ ± ± ≈ F28 8.170E+02 ± ± + ± ± ≈ F29 7.190E+02 ± ± ≈ ± ± ≈ F30 ± ± ≈ ± ± − + / ≈ / − - 11/15/4 - 17/10/3 TABLE S.15H
OLM TEST ON THE FITNESS , REFERENCE ALGORITHM = ETI-SHADE (
RANK =14.47)
FOR FUNCTIONS
F01-F30 AT D = 30 . j Optimizer Rank z j p j δ / j Hypothesis
TABLE S.16
WIN - LOSE
RESULTS OF
ETI1-DE S , ETI2-DE S , ETI3-DE S , ETI4-DE S ( IN S ECTION
IV.C)
AND THEIR COUNTERPARTS FOR FUNCTIONS
F01-F30 AT D = 30 . ETI-DEs vs.
ETI1-DE/rand/1/bin vs.
ETI1-DE/best/1/bin vs.
ETI1-jDE vs.
ETI1-JADE vs.
ETI1-CoDE vs.
ETI1-SaDE vs.
ETI1-ODE vs.
ETI1-EPSDE vs.
ETI1-SHADE vs.
ETI1-OXDE
ETI-DEs vs.
ETI2-DE/rand/1/bin vs.
ETI2-DE/best/1/bin vs.
ETI2-jDE vs.
ETI2-JADE vs.
ETI2-CoDE vs.
ETI2-SaDE vs.
ETI2-ODE vs.
ETI2-EPSDE vs.
ETI2-SHADE vs.
ETI2-OXDE
ETI-DEs vs.
ETI3-DE/rand/1/bin vs.
ETI3-DE/best/1/bin vs.
ETI3-jDE vs.
ETI3-JADE vs.
ETI3-CoDE vs.
ETI3-SaDE vs.
ETI3-ODE vs.
ETI3-EPSDE vs.
ETI3-SHADE vs.
ETI3-OXDE
ETI-DEs vs.
ETI4-DE/rand/1/bin vs.
ETI4-DE/best/1/bin vs.
ETI4-jDE vs.
ETI4-JADE vs.
ETI4-CoDE vs.
ETI4-SaDE vs.
ETI4-ODE vs.
ETI4-EPSDE vs.
ETI4-SHADE vs.
ETI4-OXDE
TABLE S.17E
XPERIMENTAL RESULTS OF
ETI-DE/
RAND /1/
BIN , ETI-DE/
BEST /1/
BIN , ETI- J DE, ETI-JADE
AND THE RELATED
ETI GB -DE S FOR FUNCTIONS
F01-F30 AT D = 30 . Function ETI-DE/rand/1/bin ETIgb-DE/rand/1/bin ETI-DE/best/1/bin ETIgb-DE/best/1/bin
F01 ± ≈ ± ± ≈ ± F02 0.000E+00 ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± ± + ± ± − ± F06 6.459E-01 ± ≈ ± ± + ± ± ≈ ± ± ≈ ± ± − ± ± + ± ± − ± ± ≈ ± ± + ± ± ≈ ± ± + ± ± − ± F12 ± + ± ± − ± F13 ± + ± ± + ± ± ≈ ± ± ≈ ± ± + ± ± ≈ ± F16 ± + ± ± ≈ ± F17 ± + ± ± ≈ ± ± + ± ± ≈ ± F19 ± + ± ± ≈ ± ± + ± ± + ± ± + ± ± ≈ ± ± − ± ± − ± F23 3.152E+02 ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± F26 ± + ± ± ≈ ± F27 3.542E+02 ± ≈ ± ± ≈ ± F28 8.111E+02 ± ≈ ± ± ≈ ± ± ≈ ± ± − ± F30 6.597E+02 ± ≈ ± ± ≈ ± + / ≈ / − Function ETI-jDE ETIgb-jDE ETI-JADE ETIgb-JADE
F01 ± ≈ ± ± ≈ ± F02 0.000E+00 ± ≈ ± ± ≈ ± ± ≈ ± ± + ± ± ≈ ± ± ≈ ± ± + ± ± + ± ± ≈ ± ± − ± F07 0.000E+00 ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± ± + ± ± ≈ ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± − ± ± − ± F15 ± + ± ± + ± ± + ± ± + ± ± ≈ ± ± ≈ ± F18 4.974E+01 ± ≈ ± ± ≈ ± ± + ± ± + ± ± ≈ ± ± + ± ± − ± ± ≈ ± F22 8.190E+01 ± − ± ± ≈ ± F23 3.152E+02 ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± F25 ± + ± ± ≈ ± ± + ± ± + ± ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± F29 8.497E+02 ± ≈ ± ± ≈ ± F30 1.586E+03 ± ≈ ± ± ≈ ± + / ≈ / − TABLE S.18E
XPERIMENTAL RESULTS OF
ETI-C O DE, ETI-S A DE, ETI-ODE, ETI-EPSDE
AND THE RELATED
ETI GB -DE S FOR FUNCTIONS
F01-F30 AT D = 30 . Function ETI-CoDE ETIgb-CoDE ETI-SaDE ETIgb-SaDE
F01 7.441E+04 ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± F05 ± + ± ± + ± ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± F08 ± + ± ± ≈ ± ± ≈ ± ± ≈ ± F10 ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± ≈ ± ± + ± ± + ± ± + ± ± ≈ ± ± ≈ ± ± ≈ ± F19 ± + ± ± + ± ± ≈ ± ± ≈ ± F21 1.424E+02 ± ≈ ± ± ≈ ± F22 1.467E+02 ± − ± ± ≈ ± F23 3.152E+02 ± ≈ ± ± ≈ ± ± ≈ ± ± − ± F25 2.030E+02 ± ≈ ± ± ≈ ± F26 ± + ± ± + ± ± ≈ ± ± ≈ ± F28 8.046E+02 ± ≈ ± ± ≈ ± F29 6.322E+02 ± ≈ ± ± ≈ ± F30 6.285E+02 ± ≈ ± ± ≈ ± + / ≈ / − Function ETI-ODE ETIgb-ODE ETI-EPSDE ETIgb-EPSDE
F01 ± ≈ ± ± ≈ ± F02 ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± F05 ± + ± ± + ± ± ≈ ± ± + ± ± ≈ ± ± ≈ ± F08 3.366E+01 ± − ± ± ≈ ± F09 2.273E+01 ± ≈ ± ± − ± F10 1.250E+03 ± − ± ± + ± ± ≈ ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± − ± F15 ± ≈ ± ± ≈ ± F16 ± + ± ± + ± ± + ± ± + ± ± ≈ ± ± + ± ± + ± ± + ± ± + ± ± ≈ ± ± + ± ± ≈ ± ± ≈ ± ± − ± F23 3.152E+02 ± ≈ ± ± ≈ ± ± + ± ± ≈ ± ± − ± ± ≈ ± ± + ± ± + ± ± ≈ ± ± + ± ± ≈ ± ± ≈ ± ± ≈ ± ± + ± ± ≈ ± ± − ± + / ≈ / − TABLE S.19E
XPERIMENTAL RESULTS OF
ETI-SHADE, ETI-OXDE
AND THE RELATED
ETI GB -DE S FOR FUNCTIONS
F01-F30 AT D = 30 . Function ETI-SHADE ETIgb-SHADE ETI-OXDE ETIgb-OXDE
F01 4.551E+00 ± ≈ ± ± ≈ ± F02 0.000E+00 ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± F05 ± + ± ± + ± ± ≈ ± ± ≈ ± F07 0.000E+00 ± ≈ ± ± ≈ ± ± + ± ± − ± F09 ± + ± ± − ± F10 ± + ± ± ≈ ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± ≈ ± ± ≈ ± ± + ± ± + ± ± + ± ± + ± ± ≈ ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± ≈ ± ± ≈ ± F23 3.152E+02 ± ≈ ± ± ≈ ± ± ≈ ± ± ≈ ± F25 2.030E+02 ± ≈ ± ± ≈ ± ± + ± ± + ± ± + ± ± ≈ ± ± ≈ ± ± ≈ ± F29 7.181E+02 ± ≈ ± ± ≈ ± F30 1.156E+03 ± ≈ ± ± ≈ ± + / ≈ / − TABLE S.20D
IFFERENT RESTART STRATEGIES IN
DE.
Reference Restart Strategy Parameter [35] local restart in a bubble surrounding the best individual within the cluster x best , and global restartsampling outside the bubble n samples , ∆ [52] restart by reinitializing certain individuals away from previous local optima δ, δ vib , δ fit , λ, interval restart [53] restart by DE/rand/2 and by reinitializing certain individuals within the search range cc , M1 , M2 , S , K [54] restart by increasing the population size k , c [55] restart according to Eqs. (4)-(9) in [55] UN [56] restart according to Eqs. (15)-(20) in [56] T , a our method restart according to Eqs. (18)-(21) in this paper pr , LN , UN REPRINT SUBMITTED TO ARXIV 26
TABLE S.21E
XPERIMENTAL RESULTS OF
ETI-DE
S AND
AEPD-DE
S FOR FUNCTIONS
F01-F30 AT D = 30 . Function ETI-jDE ( NP = 20 ) AEPD-jDE ( NP = 20 ) ETI-JADE ( NP = 20 ) AEPD-JADE ( NP = 20 ) F01 4.733E+05 ± − ± ± ≈ ± F02 2.537E-08 ± − ± ± ≈ ± ± ≈ ± ± ≈ ± F04 ± + ± ± − ± F05 ± + ± ± + ± ± − ± ± − ± F07 5.476E-02 ± − ± ± ≈ ± F08 9.053E+00 ± − ± ± − ± F09 4.457E+01 ± − ± ± − ± F10 6.549E+01 ± − ± ± − ± F11 2.497E+03 ± − ± ± − ± F12 ± + ± ± + ± ± ≈ ± ± − ± F14 2.713E-01 ± − ± ± − ± F15 4.949E+00 ± ≈ ± ± ≈ ± F16 1.007E+01 ± − ± ± − ± F17 2.791E+04 ± − ± ± ≈ ± F18 2.700E+03 ± − ± ± − ± F19 6.429E+00 ± ≈ ± ± ≈ ± ± ≈ ± ± − ± F21 1.588E+04 ± − ± ± ≈ ± ± − ± ± − ± F23 3.152E+02 ± − ± ± + ± ± − ± ± − ± F25 2.079E+02 ± − ± ± − ± F26 1.100E+02 ± ≈ ± ± + ± ± − ± ± − ± F28 9.248E+02 ± − ± ± − ± F29 3.408E+05 ± − ± ± − ± F30 3.068E+03 ± − ± ± ≈ ± + / ≈ / − Function ETI-jDE ( NP = 100 ) AEPD-jDE ( NP = 100 ) ETI-JADE ( NP = 100 ) AEPD-JADE ( NP = 100 ) F01 ± + ± ± + ± ± + ± ± ≈ ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± ≈ ± ± ≈ ± F08 0.000E+00 ± ≈ ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± ≈ ± F18 4.974E+01 ± − ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± ≈ ± ± + ± ± + ± ± + ± ± ≈ ± ± ≈ ± ± ≈ ± F25 ± ≈ ± ± − ± F26 ± + ± ± + ± ± + ± ± − ± F28 ± ≈ ± ± + ± ± + ± ± ≈ ± F30 1.586E+03 ± ≈ ± ± ≈ ± + / ≈ / − TABLE S.22E
XPERIMENTAL RESULTS OF
ETI-DE
S AND
AEPD-DE
S FOR FUNCTIONS
F01-F30 AT D = 30 . Function ETI-jDE ( NP = 100 ) AEPD-jDE ( NP = 20 ) ETI-JADE ( NP = 100 ) AEPD-JADE ( NP = 20 ) F01 ± + ± ± + ± ± ≈ ± ± ≈ ± ± ≈ ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± ≈ ± ± − ± ± ≈ ± ± + ± ± + ± ± − ± ± + ± ± + ± ± + ± ± + ± ± + ± ± ≈ ± ± − ± F15 ± + ± ± + ± ± − ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± ≈ ± ± ≈ ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± + ± ± ≈ ± ± + ± ± + ± ± + ± + / ≈ / − TABLE S.23
WIN - LOSE
RESULTS OF
R1-ETI-DE S , R2-ETI-DE S , N O R-ETI-DE S ( IN S ECTION
IV.F)
AND
ETI-DE
S FOR FUNCTIONS
F01-F30 AT D = 30 . ETI-DEs vs.
R1-ETI-DE/rand/1/bin vs.
R1-ETI-DE/best/1/bin vs.
R1-ETI-jDE vs.
R1-ETI-JADE vs.
R1-ETI-CoDE vs.
R1-ETI-SaDE vs.
R1-ETI-ODE vs.
R1-ETI-EPSDE vs.
R1-ETI-SHADE vs.
R1-ETI-OXDE
ETI-DEs vs.
R2-ETI-DE/rand/1/bin vs.
R2-ETI-DE/best/1/bin vs.
R2-ETI-jDE vs.
R2-ETI-JADE vs.
R2-ETI-CoDE vs.
R2-ETI-SaDE vs.
R2-ETI-ODE vs.
R2-ETI-EPSDE vs.
R2-ETI-SHADE vs.
R2-ETI-OXDE
ETI-DEs vs.
NoR-ETI-DE/rand/1/bin vs.
NoR-ETI-DE/best/1/bin vs.
NoR-ETI-jDE vs.
NoR-ETI-JADE vs.
NoR-ETI-CoDE vs.
NoR-ETI-SaDE vs.
NoR-ETI-ODE vs.
NoR-ETI-EPSDE vs.
NoR-ETI-SHADE vs.
NoR-ETI-OXDE
TABLE S.24E
XPERIMENTAL RESULTS OF
DE/
RAND /1/
BIN , DE/
BEST /1/
BIN , J DE, JADE
AND THE RELATED
ETI-
BASED VARIANTS FOR FUNCTIONS
F01-F30 AT D = 50 . Function DE/rand/1/bin ETI-DE/rand/1/bin DE/best/1/bin ETI-DE/best/1/bin
F01 1.399E+06 ± ± + ± ± + F02 ± ± − ± ± − F03 ± ± ≈ ± ± + F04 ± ± ≈ ± ± + F05 2.113E+01 ± ± + ± ± + F06 ± ± ≈ ± ± − F07 1.450E-04 ± ± + ± ± ≈ F08 1.942E+02 ± ± + ± ± + F09 3.517E+02 ± ± + ± ± + F10 9.467E+03 ± ± + ± ± + F11 1.299E+04 ± ± + ± ± + F12 3.243E+00 ± ± + ± ± + F13 4.575E-01 ± ± + ± ± + F14 3.369E-01 ± ± + ± ± ≈ F15 3.150E+01 ± ± + ± ± + F16 2.211E+01 ± ± + ± ± + F17 ± ± ≈ ± ± + F18 1.342E+02 ± ± + ± ± ≈ F19 1.191E+01 ± ± + ± ± + F20 9.887E+01 ± ± + ± ± + F21 2.630E+03 ± ± + ± ± + F22 ± ± ≈ ± ± + F23 3.440E+02 ± ± + ± ± ≈ F24 2.704E+02 ± ± ≈ ± ± − F25 ± ± ≈ ± ± + F26 1.005E+02 ± ± + ± ± − F27 ± ± ≈ ± ± − F28 ± ± − ± ± ≈ F29 ± ± ≈ ± ± + F30 ± ± ≈ ± ± ≈ + / ≈ / − - 18/10/2 - 19/6/5 Function jDE ETI-jDE JADE ETI-JADE
F01 ± ± ≈ ± ± ≈ F02 ± ± − ± ± ≈ F03 ± ± − ± ± + F04 ± ± − ± ± ≈ F05 2.043E+01 ± ± + ± ± + F06 1.879E+01 ± ± + ± ± + F07 0.000E+00 ± ± ≈ ± ± + F08 0.000E+00 ± ± ≈ ± ± ≈ F09 9.724E+01 ± ± + ± ± + F10 ± ± − ± ± − F11 5.118E+03 ± ± + ± ± + F12 4.531E-01 ± ± + ± ± + F13 3.836E-01 ± ± + ± ± + F14 3.334E-01 ± ± + ± ± + F15 1.206E+01 ± ± + ± ± + F16 1.842E+01 ± ± + ± ± + F17 2.468E+04 ± ± + ± ± ≈ F18 4.649E+02 ± ± + ± ± ≈ F19 1.365E+01 ± ± + ± ± + F20 5.216E+01 ± ± + ± ± + F21 1.094E+04 ± ± ≈ ± ± ≈ F22 5.653E+02 ± ± + ± ± ≈ F23 3.440E+02 ± ± + ± ± + F24 2.686E+02 ± ± ≈ ± ± ≈ F25 ± ± ≈ ± ± ≈ F26 ± ± − ± ± − F27 4.371E+02 ± ± ≈ ± ± ≈ F28 ± ± ≈ ± ± + F29 ± ± − ± ± ≈ F30 ± ± − ± ± ++ / ≈ / − - 15/8/7 - 16/12/2REPRINT SUBMITTED TO ARXIV 29 TABLE S.25E
XPERIMENTAL RESULTS OF C O DE, S A DE, ODE, EPSDE
AND THE RELATED
ETI-
BASED VARIANTS FOR FUNCTIONS
F01-F30 AT D = 50 . Function CoDE ETI-CoDE SaDE ETI-SaDE
F01 4.059E+05 ± ± ≈ ± ± ≈ F02 ± ± − ± ± ≈ F03 ± ± − ± ± − F04 3.357E+01 ± ± ≈ ± ± − F05 2.079E+01 ± ± + ± ± + F06 4.088E+00 ± ± ≈ ± ± ≈ F07 ± ± − ± ± ≈ F08 4.359E+00 ± ± + ± ± ≈ F09 7.236E+01 ± ± ≈ ± ± + F10 6.008E+02 ± ± + ± ± + F11 4.433E+03 ± ± + ± ± + F12 1.231E+00 ± ± + ± ± + F13 4.437E-01 ± ± + ± ± + F14 2.944E-01 ± ± + ± ± + F15 6.494E+00 ± ± + ± ± + F16 2.053E+01 ± ± ≈ ± ± + F17 1.616E+04 ± ± ≈ ± ± ≈ F18 8.667E+01 ± ± ≈ ± ± ≈ F19 1.054E+01 ± ± + ± ± + F20 ± ± ≈ ± ± ≈ F21 ± ± ≈ ± ± ≈ F22 ± ± ≈ ± ± ≈ F23 3.440E+02 ± ± + ± ± ≈ F24 ± ± − ± ± ≈ F25 ± ± ≈ ± ± ≈ F26 1.024E+02 ± ± + ± ± ≈ F27 ± ± ≈ ± ± ≈ F28 ± ± ≈ ± ± ≈ F29 9.610E+02 ± ± ≈ ± ± ≈ F30 ± ± ≈ ± ± ≈ + / ≈ / − - 11/15/4 - 10/18/2 Function ODE ETI-ODE EPSDE ETI-EPSDE
F01 2.160E+06 ± ± + ± ± ≈ F02 5.682E+03 ± ± ≈ ± ± − F03 ± ± ≈ ± ± − F04 9.261E+01 ± ± ≈ ± ± − F05 2.104E+01 ± ± + ± ± + F06 2.120E+00 ± ± ≈ ± ± + F07 4.724E-03 ± ± + ± ± − F08 8.073E+01 ± ± ≈ ± ± − F09 5.367E+01 ± ± ≈ ± ± + F10 6.508E+03 ± ± + ± ± + F11 7.749E+03 ± ± + ± ± + F12 1.791E+00 ± ± + ± ± + F13 4.409E-01 ± ± + ± ± + F14 3.229E-01 ± ± + ± ± + F15 2.003E+01 ± ± + ± ± + F16 2.164E+01 ± ± + ± ± + F17 ± ± ≈ ± ± + F18 8.829E+01 ± ± ≈ ± ± ≈ F19 9.496E+00 ± ± + ± ± + F20 1.006E+02 ± ± + ± ± + F21 2.500E+03 ± ± + ± ± + F22 ± ± ≈ ± ± ≈ F23 3.440E+02 ± ± + ± ± + F24 2.706E+02 ± ± ≈ ± ± ≈ F25 ± ± ≈ ± ± ≈ F26 ± ± − ± ± + F27 3.914E+02 ± ± ≈ ± ± + F28 1.085E+03 ± ± ≈ ± ± − F29 ± ± − ± ± + F30 8.421E+03 ± ± ≈ ± ± ++ / ≈ / − - 14/14/2 - 19/5/6REPRINT SUBMITTED TO ARXIV 30 TABLE S.26E
XPERIMENTAL RESULTS OF
SHADE, OXDE
AND THE RELATED
ETI-
BASED VARIANTS FOR FUNCTIONS
F01-F30 AT D = 50 . Function SHADE ETI-SHADE OXDE ETI-OXDE
F01 ± ± − ± ± + F02 0.000E+00 ± ± ≈ ± ± − F03 0.000E+00 ± ± ≈ ± ± ≈ F04 ± ± ≈ ± ± ≈ F05 ± ± − ± ± + F06 1.029E+00 ± ± ≈ ± ± ≈ F07 1.450E-04 ± ± + ± ± + F08 0.000E+00 ± ± ≈ ± ± + F09 3.834E+01 ± ± ≈ ± ± + F10 ± ± − ± ± + F11 ± ± ≈ ± ± + F12 ± ± ≈ ± ± + F13 2.702E-01 ± ± + ± ± + F14 ± ± − ± ± + F15 7.457E+00 ± ± + ± ± + F16 1.743E+01 ± ± + ± ± + F17 2.272E+03 ± ± + ± ± ≈ F18 1.349E+02 ± ± + ± ± + F19 1.510E+01 ± ± + ± ± + F20 1.142E+02 ± ± + ± ± + F21 1.108E+03 ± ± + ± ± + F22 3.661E+02 ± ± + ± ± − F23 3.440E+02 ± ± + ± ± + F24 ± ± ≈ ± ± ≈ F25 2.122E+02 ± ± + ± ± ≈ F26 1.062E+02 ± ± + ± ± + F27 ± ± ≈ ± ± ≈ F28 1.112E+03 ± ± ≈ ± ± ≈ F29 8.461E+02 ± ± ≈ ± ± − F30 8.994E+03 ± ± ≈ ± ± ≈ + / ≈ / − - 13/13/4 - 18/9/3REPRINT SUBMITTED TO ARXIV 31 TABLE S.27E
XPERIMENTAL RESULTS OF
DE/
RAND /1/
BIN , DE/
BEST /1/
BIN , J DE, JADE
AND THE RELATED
ETI-
BASED VARIANTS FOR FUNCTIONS
F01-F30 AT D = 100 . Function DE/rand/1/bin ETI-DE/rand/1/bin DE/best/1/bin ETI-DE/best/1/bin
F01 4.492E+06 ± ± + ± ± + F02 2.156E+04 ± ± ≈ ± ± − F03 ± ± ≈ ± ± + F04 ± ± ≈ ± ± + F05 2.131E+01 ± ± + ± ± + F06 ± ± − ± ± − F07 4.833E-04 ± ± + ± ± + F08 1.126E+02 ± ± ≈ ± ± + F09 8.142E+02 ± ± + ± ± + F10 1.589E+04 ± ± + ± ± + F11 2.989E+04 ± ± + ± ± + F12 3.952E+00 ± ± + ± ± + F13 5.664E-01 ± ± + ± ± + F14 3.467E-01 ± ± + ± ± ≈ F15 7.304E+01 ± ± + ± ± + F16 4.645E+01 ± ± + ± ± + F17 4.617E+05 ± ± + ± ± + F18 ± ± ≈ ± ± ≈ F19 9.509E+01 ± ± + ± ± + F20 ± ± ≈ ± ± + F21 1.204E+05 ± ± + ± ± + F22 3.755E+03 ± ± + ± ± + F23 3.482E+02 ± ± + ± ± − F24 ± ± ≈ ± ± − F25 ± ± − ± ± + F26 ± ± ≈ ± ± + F27 ± ± − ± ± − F28 ± ± ≈ ± ± ≈ F29 1.776E+03 ± ± ≈ ± ± + F30 ± ± ≈ ± ± ≈ + / ≈ / − - 16/11/3 - 21/4/5 Function jDE ETI-jDE JADE ETI-JADE
F01 ± ± ≈ ± ± − F02 ± ± − ± ± ≈ F03 3.568E-05 ± ± ≈ ± ± + F04 ± ± ≈ ± ± ≈ F05 2.066E+01 ± ± + ± ± + F06 6.800E+01 ± ± + ± ± + F07 0.000E+00 ± ± ≈ ± ± − F08 ± ± ≈ ± ± ≈ F09 2.463E+02 ± ± + ± ± ≈ F10 ± ± − ± ± − F11 1.345E+04 ± ± + ± ± + F12 6.235E-01 ± ± + ± ± + F13 4.881E-01 ± ± + ± ± + F14 3.491E-01 ± ± + ± ± + F15 3.077E+01 ± ± + ± ± + F16 4.062E+01 ± ± + ± ± + F17 1.568E+05 ± ± + ± ± ≈ F18 ± ± ≈ ± ± ≈ F19 9.250E+01 ± ± ≈ ± ± + F20 3.267E+02 ± ± + ± ± ≈ F21 6.641E+04 ± ± + ± ± ≈ F22 1.799E+03 ± ± + ± ± ≈ F23 3.482E+02 ± ± + ± ± + F24 3.741E+02 ± ± ≈ ± ± ≈ F25 ± ± ≈ ± ± ≈ F26 1.963E+02 ± ± ≈ ± ± ≈ F27 9.959E+02 ± ± + ± ± ≈ F28 ± ± ≈ ± ± ≈ F29 1.630E+03 ± ± + ± ± ≈ F30 ± ± − ± ± − + / ≈ / − - 16/11/3 - 11/15/4REPRINT SUBMITTED TO ARXIV 32 TABLE S.28E
XPERIMENTAL RESULTS OF C O DE, S A DE, ODE, EPSDE
AND THE RELATED
ETI-
BASED VARIANTS FOR FUNCTIONS
F01-F30 AT D = 100 . Function CoDE ETI-CoDE SaDE ETI-SaDE
F01 ± ± ≈ ± ± − F02 ± ± − ± ± ≈ F03 ± ± ≈ ± ± − F04 ± ± ≈ ± ± ≈ F05 2.066E+01 ± ± + ± ± + F06 2.909E+01 ± ± ≈ ± ± ≈ F07 1.450E-04 ± ± + ± ± + F08 6.277E+01 ± ± + ± ± − F09 1.734E+02 ± ± + ± ± + F10 1.984E+03 ± ± + ± ± + F11 1.146E+04 ± ± ≈ ± ± + F12 8.670E-01 ± ± + ± ± + F13 ± ± ≈ ± ± + F14 2.911E-01 ± ± + ± ± + F15 1.570E+01 ± ± + ± ± + F16 4.218E+01 ± ± + ± ± + F17 ± ± ≈ ± ± ≈ F18 7.257E+02 ± ± ≈ ± ± ≈ F19 9.047E+01 ± ± ≈ ± ± + F20 ± ± ≈ ± ± ≈ F21 8.179E+04 ± ± ≈ ± ± ≈ F22 1.700E+03 ± ± ≈ ± ± − F23 3.482E+02 ± ± + ± ± − F24 3.764E+02 ± ± ≈ ± ± ≈ F25 ± ± ≈ ± ± ≈ F26 ± ± − ± ± ≈ F27 ± ± ≈ ± ± ≈ F28 2.184E+03 ± ± + ± ± ≈ F29 ± ± ≈ ± ± − F30 ± ± − ± ± ≈ + / ≈ / − - 11/16/3 - 11/13/6 Function ODE ETI-ODE EPSDE ETI-EPSDE
F01 3.109E+06 ± ± ≈ ± ± − F02 ± ± ≈ ± ± − F03 1.644E+03 ± ± ≈ ± ± + F04 ± ± ≈ ± ± ≈ F05 2.130E+01 ± ± + ± ± + F06 2.449E+01 ± ± ≈ ± ± + F07 1.389E-02 ± ± ≈ ± ± − F08 ± ± ≈ ± ± + F09 2.004E+02 ± ± ≈ ± ± + F10 1.244E+04 ± ± + ± ± + F11 1.526E+04 ± ± + ± ± + F12 2.882E+00 ± ± + ± ± + F13 5.335E-01 ± ± + ± ± + F14 3.420E-01 ± ± + ± ± + F15 5.464E+01 ± ± + ± ± + F16 4.593E+01 ± ± + ± ± + F17 7.537E+05 ± ± + ± ± + F18 ± ± ≈ ± ± ≈ F19 9.844E+01 ± ± + ± ± + F20 1.110E+03 ± ± ≈ ± ± + F21 1.522E+05 ± ± + ± ± + F22 2.015E+03 ± ± ≈ ± ± ≈ F23 3.482E+02 ± ± ≈ ± ± + F24 ± ± ≈ ± ± ≈ F25 ± ± ≈ ± ± ≈ F26 2.001E+02 ± ± + ± ± ≈ F27 ± ± ≈ ± ± + F28 ± ± ≈ ± ± ≈ F29 ± ± ≈ ± ± + F30 ± ± − ± ± ++ / ≈ / − - 12/17/1 - 20/7/3REPRINT SUBMITTED TO ARXIV 33 TABLE S.29E
XPERIMENTAL RESULTS OF
SHADE, OXDE
AND THE RELATED
ETI-
BASED VARIANTS FOR FUNCTIONS
F01-F30 AT D = 100 . Function SHADE ETI-SHADE OXDE ETI-OXDE
F01 ± ± − ± ± − F02 0.000E+00 ± ± ≈ ± ± ≈ F03 5.657E-04 ± ± + ± ± ≈ F04 ± ± ≈ ± ± ≈ F05 2.035E+01 ± ± + ± ± + F06 2.354E+01 ± ± + ± ± ≈ F07 ± ± − ± ± + F08 0.000E+00 ± ± ≈ ± ± ≈ F09 ± ± − ± ± + F10 ± ± − ± ± + F11 1.014E+04 ± ± + ± ± + F12 2.872E-01 ± ± + ± ± + F13 3.702E-01 ± ± + ± ± + F14 ± ± − ± ± + F15 2.127E+01 ± ± + ± ± + F16 ± ± − ± ± + F17 ± ± − ± ± + F18 8.545E+02 ± ± + ± ± ≈ F19 9.300E+01 ± ± + ± ± + F20 5.636E+02 ± ± + ± ± − F21 ± ± ≈ ± ± ≈ F22 ± ± − ± ± + F23 3.482E+02 ± ± + ± ± + F24 3.894E+02 ± ± ≈ ± ± ≈ F25 ± ± ≈ ± ± ≈ F26 2.001E+02 ± ± ≈ ± ± − F27 ± ± ≈ ± ± ≈ F28 2.136E+03 ± ± ≈ ± ± ≈ F29 1.170E+03 ± ± ≈ ± ± − F30 ± ± ≈ ± ± ≈ + / ≈ / − - 11/11/8 - 14/12/4 TABLE S.30H
OLM TEST ON THE FITNESS , REFERENCE ALGORITHM = ETI-SHADE (
RANK =15.70)
FOR FUNCTIONS
F01-F30 AT D = 50 . j Optimizer Rank z j p j δ / j Hypothesis
TABLE S.31H
OLM TEST ON THE FITNESS , REFERENCE ALGORITHM = ETI-SHADE (
RANK =15.50)
FOR FUNCTIONS
F01-F30 AT D = 100 . j Optimizer Rank z j p j δ / j Hypothesis −14 −13 −3 Fig. S.1. Box plots for the results of JADE with/without ETI on CEC 2014 test suite at D = 30 : 1–JADE; 2–ETI-JADE. −14 −14 −14 −11 Fig. S.2. Box plots for the results of CoDE with/without ETI on CEC 2014 test suite at D = 30 : 1–CoDE; 2–ETI-CoDE. −14 −12 Fig. S.3. Box plots for the results of SaDE with/without ETI on CEC 2014 test suite at D = 30 : 1–SaDE; 2–ETI-SaDE. REPRINT SUBMITTED TO ARXIV 36 −13 −10 Fig. S.4. Box plots for the results of EPSDE with/without ETI on CEC 2014 test suite at D = 30 : 1–EPSDE; 2–ETI-EPSDE. −20 −15 −10 −5 FES M ea n f it n e ss : f( x ) − f( x* ) DE/rand/1/binETI−DE/rand/1/binDE/best/1/binETI−DE/best/1/binjDEETI−jDEJADEETI−JADECoDEETI−CoDESaDEETI−SaDEODEETI−ODEEPSDEETI−EPSDESHADEETI−SHADEOXDEETI−OXDE (a) FES M ea n f it n e ss : f( x ) − f( x* ) DE/rand/1/binETI−DE/rand/1/binDE/best/1/binETI−DE/best/1/binjDEETI−jDEJADEETI−JADECoDEETI−CoDESaDEETI−SaDEODEETI−ODEEPSDEETI−EPSDESHADEETI−SHADEOXDEETI−OXDE (b) FES M ea n f it n e ss : f( x ) − f( x* ) DE/rand/1/binETI−DE/rand/1/binDE/best/1/binETI−DE/best/1/binjDEETI−jDEJADEETI−JADECoDEETI−CoDESaDEETI−SaDEODEETI−ODEEPSDEETI−EPSDESHADEETI−SHADEOXDEETI−OXDE (c) FES M ea n f it n e ss : f( x ) − f( x* ) DE/rand/1/binETI−DE/rand/1/binDE/best/1/binETI−DE/best/1/binjDEETI−jDEJADEETI−JADECoDEETI−CoDESaDEETI−SaDEODEETI−ODEEPSDEETI−EPSDESHADEETI−SHADEOXDEETI−OXDE (d) FES M ea n f it n e ss : f( x ) − f( x* ) DE/rand/1/binETI−DE/rand/1/binDE/best/1/binETI−DE/best/1/binjDEETI−jDEJADEETI−JADECoDEETI−CoDESaDEETI−SaDEODEETI−ODEEPSDEETI−EPSDESHADEETI−SHADEOXDEETI−OXDE (e) FES M ea n f it n e ss : f( x ) − f( x* ) DE/rand/1/binETI−DE/rand/1/binDE/best/1/binETI−DE/best/1/binjDEETI−jDEJADEETI−JADECoDEETI−CoDESaDEETI−SaDEODEETI−ODEEPSDEETI−EPSDESHADEETI−SHADEOXDEETI−OXDE (f)Fig. S.5. Evolution of the mean function error values obtained from the algorithms versus the number of FES on six 30-dimensional test functions. (a) F02;(b) F11; (c) F15; (d) F19; (e) F20; (f) F26.
REPRINT SUBMITTED TO ARXIV 37 R i (a) R i (b) R i (c) R i (d)Fig. S.6. Evolution of R i by jDE and ETI-jDE on F02 and F13 at D = 30 . (a) Evolution of ten random individuals’ R i by jDE on F02. (b) Evolution often random individuals’ R i by ETI-jDE on F02. (c) Evolution of ten random individuals’ R i by jDE on F13. (d) Evolution of ten random individuals’ R i byETI-jDE on F13.Fig. S.7. The number of functions that ETI-DEs with different pr values ( LN = 1 , UN = NP ) are significantly better than, equal to and worse than theoriginal DEs on CEC 2014 test suite at D = 30 . (The results of adding the win/tie/lose numbers for all the algorithms when using the same value of pr : pr = 0 . / / pr = 0 . / / pr = 1 . / / .) REPRINT SUBMITTED TO ARXIV 38
Fig. S.8. The number of functions that ETI-DEs with different LN values ( pr = 0 . , UN = NP ) are significantly better than, equal to and worse than theoriginal DEs on CEC 2014 test suite at D = 30 . (The results of adding the win/tie/lose numbers for all the algorithms when using the same value of LN : LN = 1 : 148 / / LN = 0 . NP : 139 / / LN = 0 . NP : 125 / / LN = 0 . NP : 106 / / .)Fig. S.9. The number of functions that ETI-DEs with different UN values ( pr = 0 . , LN = 1 ) are significantly better than, equal to and worse than theoriginal DEs on CEC 2014 test suite at D = 30 . (The results of adding the win/tie/lose numbers for all the algorithms when using the same value of UN : UN = 0 . NP : 138 / / UN = 0 . NP : 145 / / UN = 0 . NP : 146 / / UN = NP : 148 / / .) REPRINT SUBMITTED TO ARXIV 39
123 x 10 −9 −14 Fig. S.10. Box plots for the results of JADE with/without ETI on CEC 2014 test suite at D = 100 : 1–JADE; 2–ETI-JADE. −3 Fig. S.11. Box plots for the results of CoDE with/without ETI on CEC 2014 test suite at D = 100 : 1–CoDE; 2–ETI-CoDE.