Large Scale Global Optimization Algorithms for IoT Networks: A Comparative Study
Sotirios K. Goudos, Achilles D. Boursianis, Ali Wagdy Mohamed, Shaohua Wan, Panagiotis Sarigiannidis, George K. Karagiannidis, Ponnuthurai N. Suganthan
LLarge Scale Global Optimization Algorithms forIoT Networks: A Comparative Study
Sotirios K. Goudos ∗ , Achilles D. Boursianis ∗ , Ali Wagdy Mohamed † , Shaohua Wan ‡ , Panagiotis Sarigiannidis § ,George K. Karagiannidis ¶ , and Ponnuthurai N. Suganthan (cid:107)∗ ELEDIA@AUTH,School of Physics,Aristotle University of Thessaloniki,Thessaloniki, Greece † Operations Research Department, Faculty of Graduate Studies for Statistical Research,Cairo University, Giza, Egypt,School of Engineering and Applied Sciences, Wireless Intelligent Networks Center (WINC),Nile University, Giza, Egypt ‡ School of Information and Safety Engineering, Zhongnan University of Economics and Law, Wuhan, China § Department of Informatics and Telecommunications Engineering, University of Western Macedonia, Kozani, Greece ¶ School of Electrical and Computer Engineering, Aristotle University of Thessaloniki,Thessaloniki, Greece (cid:107)
School of Electrical Electronic Engineering,Nanyang Technological University, Singapore
Abstract —The advent of Internet of Things (IoT) has bring anew era in communication technology by expanding the currentinter-networking services and enabling the machine-to-machinecommunication. IoT massive deployments will create the problemof optimal power allocation. The objective of the optimizationproblem is to obtain a feasible solution that minimizes the totalpower consumption of the WSN, when the error probabilityat the fusion center meets certain criteria. This work studiesthe optimization of a wireless sensor network (WNS) at higherdimensions by focusing to the power allocation of decentral-ized detection. More specifically, we apply and compare fouralgorithms designed to tackle Large scale global optimization(LGSO) problems. These are the memetic linear populationsize reduction and semi-parameter adaptation (MLSHADE-SPA),the contribution-based cooperative coevolution recursive differ-ential grouping (CBCC-RDG3), the differential grouping withspectral clustering-differential evolution cooperative coevolution(DGSC-DECC), and the enhanced adaptive differential evolution(EADE). To the best of the authors knowledge, this is the firsttime that LGSO algorithms are applied to the optimal powerallocation problem in IoT networks. We evaluate the algorithmsperformance in several different cases by applying them in caseswith 300, 600 and 800 dimensions.
Index Terms —Internet of Things, optimal power allocation,wireless sensor network, large scale global optimization
I. I
NTRODUCTION
Large scale global optimization (LGSO) problems are at-tracting significant attention by the researchers over the lastyears. Several optimization problems in wireless communica-tions can be inherently extended to higher dimensions. TheInternet of Things (IoT) paradigm, as well as the adventof 5G cellular communication systems, will use hundredsof wireless sensor nodes in a limited space that requirean optimal power allocation scheme. These sets of wirelesssensors work together to monitor an area, gather data fromspecific applications, and constitute a wireless sensor network(WSN). There is a huge number of applications that WSNs areutilized. Among the most representative examples are includedthe monitoring of environmental (both indoors/outdoors) pa-rameters, the monitoring of the power grid, the tracking and targeting of military targets, the seismic waves sensing, andthe healthcare and human activity monitoring [1]–[3].In wireless sensor networks, if the detection mechanismis based on a decentralized scheme, each sensor node, aftera local pre-processing of its own observations, transmits asummary of the pre-processed data to the fusion center , i.e.the central node of the wireless sensor architecture. Then, thecentral node applies a final decision based on a set of fewhypotheses. Taking into consideration the given descriptionof the decentralized scheme, the fusion center does not havedirect access to the initial un-processed observation data. Thus,the data rate requirements of the decentralized scheme aresignificantly lower than in a centralized scheme (where all thenodes of the WSN send all the information to the fusion centerdirectly). This is because only a few bits of the observationdata are transmitted [4]. This aforementioned problem, i.e thedistributed detection and fusion under specific constraints inWSNs, has attracted a lot of attention in the literature [5]–[7].In this paper, we assume that the fusion center of a WSNis decentralized and operates on a binary hypothesis testingproblem. Also, the observations of the wireless sensor nodesare correlated. In this context, we can define the goal of theoptimization problem as the minimization of the total power,under the application of specific criteria to the error probabilitydetection of the central node, having the optimal allocation ofthe total power resources.The application of EAs to this problem has already beendiscussed in the literature by utilizing the Particle SwarmOptimization (PSO) [8], the Differential Evolution (DE) [9],a hybrid Biogeography Based Optimization - DifferentialEvolution (BBO-DE) algorithm [10], and a recently introducedhybrid TLBO-Jaya algorithm [11].However, to the best of the authors knowledge, this is thefirst time that LGSO problem-oriented algorithms are appliedto the previously described optimal power allocation problem.The novelty in our work lies also on the fact that we addressfor the first time the optimal power allocation problem in a r X i v : . [ c s . N E ] F e b igher dimensions.In detail, we apply four state-of-art EAs, namely thememetic linear population size reduction and semi-parameteradaptation (MLSHADE-SPA) [12], the contribution-based co-operative coevolution recursive differential grouping (CBCC-RDG3) [13], the differential grouping with spectral clustering-differential evolution cooperative coevolution (DGSC-DECC)[14], and the enhanced adaptive differential evolution (EADE)[15]. We evaluate the above algorithms on the optimal powerallocation problem in IoT networks with different settings andvarious dimensions.II. P ROBLEM D ESCRIPTION
We assume that a WSN, in which a decentralized detectionscheme is utilized, has a fusion center and L sensor nodesthat are spatially distributed. The observed signal vector canbe expressed as x = [ x , x , ..., x L ] T . Also, we assume thatwithin the decentralized scheme two local observation statescan occur, i.e. the absence or the presence of the signal. Thus,each node checks the binary hypothesis problem, by examiningthe hypotheses H (absence of signal) and H (presences ofsignal). The initial probabilities of the hypotheses are definedas P ( H ) = π and P ( H ) = π .If we consider the constant signal detection problem, inwhich an additive Gaussian noise is applied, and having theassumption that the local observation z k is achieved at the k -thnode, the given problem can be formulated as [10], [8] H : z k = v k , k = 1 , , ..., LH : z k = x k + v k , k = 1 , , ..., L (1)Moreover, we assume that the additive local noise, v k , isGaussian distributed noise with mean value equal to zero andvariance expressed as σ v . The signal x k is a known positiveconstant signal, so that x k = m, k = 1 , , .., L , for all sensors.The local observation of signal-to-noise ratio (SNR) is givenby γ = m /σ v . In a vector form, the observation of the SNRis expressed as z = x + v (2)where v = [ v , v , ..., v L ] T is a Gaussian vector of a zeromean value and a covariance matrix Σ v . Each sensor nodeapplies a local decision u k ( z k ) . Additionally, an amplifiedversion of the observation signal is re-transmitted to the fusioncenter by each node of the WSN [10] and is given by u k ( z k ) = g k z k , k = 1 , , .., L (3)where g k denotes the gain at the k -th node that is subjectto attenuation and fading. Therefore, the received signal r k ,which is originated from the k -th node, arrives at the fusioncenter under the two previously mentioned hypotheses [8],[10]. They are expressed as H : r k = n k ; k = 1 , , ..., LH : r k = h k g k x k + n k ; k = 1 , , ..., L (4)where • h k denotes the channel fading coefficient, • g k denotes the gain, • n k = h k g k v k + w k is the effective noise at the fusioncenter having zero mean value, and • w k is the receiver noise that can be analyzed to asequence of independent and identically distributed com-ponents having a zero mean value and a variance equalto σ w .Therefore, the covariance matrix of the effective noise vector n can be expressed as Σ n = h k g k Σ v g k h k + Σ w (5)where Σ w = σ I is the covariance matrix of the receivernoise and I is the L × L identity matrix. In most of thecases, the observations that are recorded by the sensors arecorrelated. If we assume that between adjacent nodes the spaceis equally distributed in a straight line having a distance of d ,the correlation between noise samples originated by nodes i and j is ρ d | i − j | , with | ρ | ≤ . In 5, Σ v can be expressedusing a symmetric Hermitian Toeplitz matrix [8], [10] in thefollowing form Σ v = σ v ρ d . . . ρ d ( L − ρ d ( L − ρ d . . . ρ d ( L − ρ d ( L − . . . . . . .ρ d ( L − ρ d ( L − . . . ρ d (6)It is written in vector notation as r = Ax + n (7)where • r = [ r , r , ..., r L ] T denotes the received informationvector, • n = [ n , n , ..., n L ] T denotes the noise vector, and • matrix A is given by A = diag ( h g , h g , ..., h L g L ) .The observations r at the fusion center are expressed as [8],[10] H : r ∼ N (0 , Σ n ) H : r ∼ N ( Am , Σ n ) (8)Moreover, the covariance matrix of the noise at the fusioncenter is given by [8], [10] Σ n = AΣ v A + σ I (9)where m is the L -length vector with all components equalto m . Taking into consideration the threshold τ = π /π ,and assuming the minimum probability of Bayesian fusionerror, the optimum Bayesian decision rule is mathematicallyformulated as [8] δ ( r ) = (cid:26) if T ( r ) ≥ ln τ if T ( r ) < ln τ (10)The optimal procedure to apply the decision between thetwo hypotheses denotes the threshold rule on a log-likelihoodratio (LLR) of the observation vector. Thus, the LLR for thedetection problem can be written as [10] T ( r ) = m e T f A Σ − r − m e T A Σ − Ae (11)or the given detection problem, the LLR distribution underthe two hypotheses is given by [8], [10] H : T ( r ) ∼ N (cid:18) − m e T A Σ − Ae , m e T A Σ − Ae (cid:19) (12) H : T ( r ) ∼ N (cid:18) m e T A Σ − Ae , m e T A Σ − Ae (cid:19) (13)Based on [8], [10], [11], if we further assume that the twohypotheses are equally probable, then the threshold is τ = 1 .The fusion error probability, i.e. the fusion center selects H when H is true, or H when H is true, is expressed as P e = Q (cid:18) (cid:113) m e T A Σ − Ae (cid:19) (14)where Q ( . ) denotes the Gaussian Q-function. Therefore, theoptimal power allocation problem is to obtain a set of L optimal sensor gain values g = ( g , g , ..., g L ) , which min-imize the total power, by keeping the fusion error probabilitysmaller than a specified criterion ε , [10]. The mathematicalformulation of the optimization problem can be expressed as min f ( g ) = (cid:80) L(cid:96) =1 g (cid:96) ζ ( g ) = Q (cid:18) (cid:113) m e T A Σ − Ae (cid:19) − ε ≤ ψ l ( g ) = − g (cid:96) ≤ , l = 1 , , ..., L (15)If we apply a penalty function to the above optimizationproblem, we can combine the objective and constraint func-tions to a single objective function. As in [11], we utilize adynamically modified penalty function, because of its effec-tiveness against a static penalty function approach [16]. As aresult, the objective function is formulated as F ( g ) = f ( g ) + It (cid:20) L (cid:80) i =0 θ ( q i ( g )) q i ( g ) λ ( q i ( g )) (cid:21) q i ( g ) = (cid:26) max { , ζ ( g ) } , if i = 0max { , ψ i ( g ) } , otherwise λ ( x ) = (cid:26) if x < otherwiseθ ( x ) = if x ≤ . if . < x ≤ otherwise (16)where • It denotes the current iteration number of the algorithm, • θ ( q i ( g )) is a multi-stage assignment function, and • λ ( q i ( g )) represents the power of the penalty function.III. L ARGE S CALE G LOBAL O PTIMIZATION ALGORITHMS
The algorithms that we will apply for the optimal powerallocation problem can be classified into two categories; thosethat solve the problem directly, either by using several algo-rithms and their hybridization, and those that use a decompo-sition strategy, as well as a variable grouping strategy. The firstcategory includes the MLSHADE-SPA [12] and EADE [15]algorithms, whereas CBCC-RDG3 [13] and DGSC-DECC[14] algorithms fall into the second category. (a)(b)(c)(d)Fig. 1. Convergence rate graph for L = 300 , γ = 10 dB , ρ = 0 . , a ) ε =0 . , b ) ε = 0 . , c ) ε = 0 . , d ) ε = 0 . .ABLE IA LGORITHMS AVERAGE RESULTS FOR L = 300 SENSORS , γ = 10 dB . T HE SMALLER VALUES ARE IN BOLD FONT . D=300 ρ ε
MLSHADE-SPA DGSC-DECC CBCC-RDG3 EADE
LGORITHMS AVERAGE RESULTS FOR L = 600 , SENSORS , ρ = 0 , γ = 10 dB . T HE SMALLER VALUES ARE IN BOLD FONT . D ε
MLSHADE-SPA DGSC-DECC CBCC-RDG3 EADE
600 0.1
VERAGE R ANKINGS ACHIEVED BY F RIEDMAN TEST . Algorithm Average Rank Normalized values Rank
MLSHADE-SPA 1.75 1.31 2DGSC-DECC 2.96 2.22 3CBCC-RDG3 1.33 1.00 1EADE 3.96 2.97 4TABLE IVW
ILCOXON SIGNED - RANK TEST BETWEEN
CBCC-RDG3
AND THE OTHERALGORITHMS .T HE BOLD FONT INDICATES VALUES BELOW SIGNIFICANCELEVEL . CBCC-RDG3 vs p-value
MLSHADE-SPA
DGSC-DECC
EADE
A. Enhanced adaptive differential evolution
Mohamed [15] proposed an enhanced adaptive differentialevolution (EADE) algorithm, which is a non-Cooperative Co-evolution method, to provide feasible solutions in large scaleoptimization problems. EADE includes two novel contribu-tions. Firstly, in order to benefit from the available informa-tion of the whole population, a novel mutation scheme hasbeen introduced. The proposed scheme utilizes two vectors,which are randomly selected, of the top and bottom 100p%individuals in the current population of size
N P . The third vector is randomly selected from the middle [NP-2(100p%)]individuals. A combination between the mutation rule andthe basic mutation strategy
DE/rand/1/bin is applied. It isworth noting that the probability between the mutation rules isequal ( . ). Secondly, a novel self-adaptive scheme has beenintroduced that utilizes the gradual change of the crossover ratevalues. This scheme, during the evolution process, can benefitfrom the previous experience of the individuals in the searchspace. This feature, on its turn, can balance effectively thetrade-off between the convergence speed and the populationdiversity. B. MLSHADE-SPA
Hadi et al. [12] proposed an LSHADE-SPA memetic frame-work, which was the runner up in CEC2018 Large Scale globaloptimization competition. MLSHADE-SPA exhibits hybridcharacteristics by combining population-based algorithms andlocal search. LSHADE-SPA [17], EADE [15], and ANDE [18]are utilized as population-based algorithms for global explo-ration, whereas a modified version of MTS (MMTS) is utilizedas a local search algorithm for local exploitation. In order tofurther enhance the framework’s performance, the ’divide andconquer’ concept is applied. This concept is processed withoutany prior assumptions of the optimized problems’ structure;the dimensions of the problems are divided into groups in arandom way and each of groups is solved independently. a)(b)(c)(d)Fig. 2. Convergence rate graph for L = 300 , γ = 10 dB , ρ = 0 . , a ) ε =0 . , b ) ε = 0 . , c ) ε = 0 . , d ) ε = 0 . . (a)(b)(c)(d)Fig. 3. Convergence rate graph for L = 600 , γ = 10 dB , ρ = 0 , a ) ε =0 . , b ) ε = 0 . , c ) ε = 0 . , d ) ε = 0 . .a)(b)(c)(d)Fig. 4. Convergence rate graph for L = 800 , γ = 10 dB , ρ = 0 , a ) ε =0 . , b ) ε = 0 . , c ) ε = 0 . , d ) ε = 0 . . C. CBCC-RDG3
A ’divide-and-conquer’ approach is proposed in [13] toaddress large-scale optimization problems having componentswith overlapping behavior. The main idea behind the ’divide-and-conquer’ approach is to decompose overlapping problemsby modifying the Recursive Differential Grouping (RDG)method. In order to do so, the linkage to shared by multiplecomponents variables must be broken. The previously men-tioned procedure of decomposition by utilizing RDG methodcan be described using the following three steps. The firststep is to distinguish the interaction between the decisionvariables and the selected variable z i , in order to classifythem into a subset Z . The second step distinguishes andgroups the interaction between the decision variables and anyother variable in Z recursively, in order Z to becomingindependent from the remaining variables. The third steprepeats the two previous steps until all variables have beengrouped. As a result, in the case of an overlapping problem,an assignment into a single group for all the decision variableswill occur, since all the variables will be linked.RDG3, which is designed for overlapping problems, modi-fies the above procedure by further examining the interactionbetween the set Z and the rest of the available variables(excluding Z ) to identify those that interact with z i indirectly.If the algorithm finds any interaction, the interacting decisionvariables are moved to Z . RDG3 repeats these steps untilthere is no detection of interaction between Z and the restof the available variables. The decision variables in Z areconsidered to be a consolidated group.The RDG3 method moves on to the next decision variable z i that has not been distinguished and grouped. The aboveprocess is repeated until all the decision variables have beenclassified in groups. RDG3 continues to further divide theseparable variables into small groups with an interval e s . Uponcompletion of RDG3 method, the identified separable and con-solidated variable groups are returned. The contribution-basedCC (CBCC) basic idea is to allocate available computationalresources to components, based on how they contribute to theoverall fitness improvement. This is grouped with RDG3 toform the CBCC-RDG3 algorithm [13]. The selected solveris the covariance matrix adaptation- evolutionary strategy(CMA-ES) [19], which utilizes the round-robin scheme tooptimize sub-problems. The computational resources to eachsun-problem, by utilizing the round-robin scheme, are equallydistributed. CBCC-RDG3 was the winner in IEEE CEC 2019LSGO competition. D. DGSC-DECC algorithm
The authors in [14] present a differential grouping withspectral clustering (DGSC) algorithm. The basic concept ofDifferential Grouping (DG) is to find the interaction rela-tionship between the available variables according to theirdifferential values. On the other hand, Spectral Clustering (SC)is graph partition problem, which is based on spectral graphtheory [20], to provide optimal solutions. Spectral clusteringforms an undirected weighted graph, by considering all theata as edge-connected points in space. The weight of theedge originates from the similarity value between the twopoints. Based on these concepts, the authors in [14] assumethat the decision variables are points in space and considerthe interaction relationship between these variables as theweight of the edge. DGSC combines differential groupingwith spectral clustering, where the similarity matrix of spectralclustering comes from the differential values of grouping.The final result is the grouping of the problem’s decisionvariables. The sub-component optimizer that is utilized, isdescribed in [21]. This resulted algorithm is called DGSC-DECC. DGSC-DECC was the runner-up in IEEE CEC 2019LSGO competition.IV. N
UMERICAL R ESULTS
In this section, the numerical results of the algorithms arepresented. We apply four LSGO algorithms to different WSNconfigurations, the MLSHADE-SPA [12], the EADE [15], theCBCC-RDG3 [13], and the DGSC-DECC [14]. For all of theselected algorithms, we set the population size equal to and the maximum number of objective function evaluation to , . All the selected algorithms are executed for inde-pendent trials. During the initialization of each algorithm, thepopulation is randomly selected, based on the lower and upperboundaries. The number of objective-function evaluations isapplied as the stopping criterion to the optimization problem.The initial position (solution in the optimization process) ofeach member of the population in each dimension is selectedwithin the range [0, 15].We apply the objective function (cost function) defined in(16). Additionally, and without losing the generality of theproblem, we introduce the assumption that the coefficients h i ,which describe the channel fading, have a Rayleigh distribu-tion with a unit mean value. Moreover, we also assume thatthey are ranked in a descending order h ≥ h ≥ ... ≥ h L .The corresponding simulations of the first case are per-formed by applying the following parameters • Number of sensor nodes L = { } , • Different fusion error threshold values ε = { . , . , . , . } , and • Correlation factor values ρ = { , . , . , . } .Both values of different fusion error threshold and correlationfactor are applied for each of the sensor nodes. The SN R isset to a fixed value of dB. Considering all the above, weobtain different optimization cases for each ρ value. As aresult, the number of dimensions of the total problem resultsin 300.Table I lists the average results of all executions of thealgorithms. We notice that CBCC-RDG3 outperforms theother algorithms in 12 out of the 16 cases. MLSHADE-SPA performs better in four cases having ε = 0 . . However,MLSHADE-SPA obtains a solution in all cases. The other al-gorithms fail to produce acceptable solutions and the objectivefunction values are significantly high.In the second case, the following parameters are modified • Number of sensor nodes L = { , } , • Different fusion error threshold values ε = { . , . , . , . } , and • Correlation factor ρ = 0 (Uncorrelated case).In this case, we set the population size to , whereas themaximum number of objective function evaluations remains , . Taking into consideration the given parameters, theoptimization problem becomes more difficult for the selectedalgorithms to solve. The comparative results of the algorithmsare listed in Table II. Once again, we notice that CBCC-RDG3performs better in six out of the eight cases, however it fails toobtain an acceptable solution in two cases. MLSHADE-SPAmanages to obtain a feasible solution in all cases. EADE andDGSC-DECC fail to obtain acceptable solutions and achievehigher objective function values. In this case, the algorithms’results seem to be worse than the previous one.A comparison between the algorithms in terms of two non-parametric statistical tests is also performed. The rankingsaccording to Friedman test are reported in Table III. It isnoteworthy that the CBCC-RDG3 obtained the best rank of . , having the MLSHADE-SPA the second best rank of . .The Wilcoxon signed-rank test between the CBCC-RDG3 andthe other selected algorithms is listed in Table IV. The p -values below 0.05, which denoted the level of significance, areindicated in boldface. Again, we can observe that the CBCC-RDG3 for the optimal power allocation optimization problemis performed significantly better than all the other algorithms.The convergence rate plots for L = { } and ρ = { . , . } are depicted in Figs. 1 and 2, respectively. We noticethat the CBCC-RDG3 converges faster than the other algo-rithms in all of the given cases except two. MLSHADE-SPAconverges with a similar speed in all cases and achieves a smallvalue of the objective function. Both algorithms converge atsimilar speed at higher objective function values. Moreover,Figs. 3 and 4 portray the corresponding convergence rategraphs for L = { , } and ρ = 0 . We can observe that theplots are similar with the previous case. The only differenceis that the CBCC-RDG3 fails to obtain an acceptable solutionin two cases, which is evident by Figs. 3d and 4d.Concluding our results, it can be clearly seen thatMLSHADE-SPA is more stable than CBCC-RDG3 al-gorithm with the different fusion error threshold ε = { . , . , . , . } , as it reaches very promising solutionsin all cases. Besides, having a fusion error threshold ε = { . } in all dimensions, MLSHADE-SPA outperforms CBCC-RDG3algorithm, which proves that MLSHADE-SPA can reach theglobal solution in stochastic environment, i.e. it is able to solvevery complicated stochastic real-world problems.V. C ONCLUSION
In this paper, we have applied LSGO algorithms to theoptimal power allocation problem in IoT networks. We havecompared results with a small number of objective functionevaluations and different numbers of sensors. Overall, CBCC-RDG3 is a powerful optimizer that obtains good results in mostof the cases. However, MLSHADE-SPA obtains a solution invery case, which could be very useful in a real-case of wire-less communications, where decisions in real-time are oftendesirable. EADE and DGSC-DECC failed to obtain acceptablesolutions, which indicates that more objective function evalua-tions are probably required. DGSC-DECC performance can beexplained since this algorithm is designed to perform well withoverlapping variables, which is not the case here. In our futurework, we will apply LSGO algorithms to more demandingoptimization problems in wireless communications.A
CKNOWLEDGMENT
This project has received funding from the EuropeanUnion’s Horizon 2020 research and innovation programmeunder grant agreement No. 957406 (TERMINET).R
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