Combining Particle Swarm Optimizer with SQP Local Search for Constrained Optimization Problems
CCombining Particle Swarm Optimizer with SQP LocalSearch for Constrained Optimization Problems
Pelley C., Innocente M., Sienz J. ∗ Keywords: Particle swarm optimization (PSO), Sequential quadratic programming (SQP), Localsearch hybrid, GP-PSO
The combining of a General-Purpose Particle Swarm Optimizer (GP-PSO)with Sequential Quadratic Programming (SQP) algorithm for constrainedoptimization problems has been shown to be highly beneficial to the refine-ment, and in some cases, the success of finding a global optimum solution.It is shown that the likely difference between leading algorithms are in theirlocal search ability. A comparison with other leading optimizers on the testedbenchmark suite, indicate the hybrid GP-PSO with implemented local searchto compete along side other leading PSO algorithms.
Nomenclature
PSO = Particle Swarm OptimizerGP-PSO = General Purpose PSOSQP = Sequential Quadratic ProgrammingGP-PSO-SQP = General-Purpose PSO with SQPGA = Genetic AlgorithmEA = Evolutionary AlgorithmMA = Memetic AlgorithmSA = Simulated AnnealingSI = Swarm IntelligenceDMS-PSO = Dynamic Multi-Swarm PSOPESO+ = Particle Evolutionary Swarm Optimization PlusFE = Function Evaluations ∗ School of Engineering, Swansea University, Singleton Park, SA2 8PP, UK. Pelley C. Email:[email protected]; Innocente M. Email: [email protected]; Sienz J. Email:[email protected] (cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:72)(cid:71)(cid:76)(cid:81)(cid:74)(cid:86)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:27)(cid:87)(cid:75)(cid:3)(cid:36)(cid:54)(cid:48)(cid:50)(cid:3)(cid:56)(cid:46)(cid:3)(cid:38)(cid:82)(cid:81)(cid:73)(cid:72)(cid:85)(cid:72)(cid:81)(cid:70)(cid:72)(cid:3)(cid:82)(cid:81)(cid:3)(cid:40)(cid:81)(cid:74)(cid:76)(cid:81)(cid:72)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:72)(cid:86)(cid:76)(cid:74)(cid:81)(cid:3)(cid:50)(cid:83)(cid:87)(cid:76)(cid:80)(cid:76)(cid:93)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:3)(cid:51)(cid:68)(cid:74)(cid:72)(cid:3)(cid:21)(cid:25)(cid:27) Preprint submitted to the 8th ASMO UK Conference on Engineering Design Optimization
Introduction
The main motivation and inspiration of this investigation is regarding the ‘Law of Sufficiency’ asdescribed by Kennedy et al.[1] (p175)“If a solution is good enough, and it is fast enough, and ischeap enough, then it is sufficient”. Following this line of thought, comparisons between GeneralPurpose PSO and GP-PSO-SQP are made together with two of the leading algorithms.
Research into the field of Swarm Intelligence (SI) in the last two decades has resulted in manyadvantages being exploited for the purpose of optimization. SI fits into the category of mod-ern heuristics, defined here as an algorithm that intends to find a solution to a problem withsuitable computational time without guarantee of optimality. The original formalization of thePSO paradigm and its place amongst other paradigms was first described by Kennedy et al.[2].Kennedy et al. describes PSO as fitting into these categories and to have roots in EvolutionaryAlgorithms (EA) and Genetic Algorithms (GA). Stochastic processes giving it similarity to theformer and an ability to follow a local and neighbourhood best being similar to the crossoveroperator in the later. The advantage of modern heuristics over that of traditional methods isthat they are not problem specific.A general description of the original PSO by Kennedy et al.[2][3] is a randomly initializedswarm within feasible space with randomly initialised velocities. The velocity of each of the n − dimensional particles is accelerated towards its own personal best position and towards thebest of its local neighbourhood with stochastic weighting between the former and the later.Let v ij ( t ) be the velocity at the current time-step and x ( t ) ij be the current solution coordinate,then we have the basic underlying equations driving the GP-PSO: v ij ( t ) = w · v ij ( t − + iw · U (0 , · pbest ( t − ij − x ( t − ij + sw · U (0 , · lbest ( t − ij − x ( t − ij (1) x ( t ) ij = x ( t − ij + v ( t ) ij (2)In Eqn.1, U (0 , is a random number from a uniform random distribution in the interval [0,1],sampled anew for each time it is called; w , iw and sw are the inertia, individuality and socialweights respectively; pbest i and lbest i are the solution coordinate of the i’th particles best everposition and the solution coordinate of the best of the i’th particle’s neighbourhood respectively.A number of the coefficients in PSO, though small in number, are highly problem dependantand as such, the GP-PSO developed by Innocente et al.[4] can be characterised by the featuresdeveloped to overcome this inherent problem and can produce high quality solutions to triggera local search.A brief overview of features of the GP-PSO designed by Innocente et al.[4] is described as follows;The set-up consists of three swarms of different behaviour (different coefficients), complementing2 (cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:72)(cid:71)(cid:76)(cid:81)(cid:74)(cid:86)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:27)(cid:87)(cid:75)(cid:3)(cid:36)(cid:54)(cid:48)(cid:50)(cid:3)(cid:56)(cid:46)(cid:3)(cid:38)(cid:82)(cid:81)(cid:73)(cid:72)(cid:85)(cid:72)(cid:81)(cid:70)(cid:72)(cid:3)(cid:82)(cid:81)(cid:3)(cid:40)(cid:81)(cid:74)(cid:76)(cid:81)(cid:72)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:72)(cid:86)(cid:76)(cid:74)(cid:81)(cid:3)(cid:50)(cid:83)(cid:87)(cid:76)(cid:80)(cid:76)(cid:93)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:3)(cid:51)(cid:68)(cid:74)(cid:72)(cid:3)(cid:21)(cid:25)(cid:28) Preprint submitted to the 8th ASMO UK Conference on Engineering Design Optimization ach other on the others weaknesses whether exploitative or explorative. The neighbourhood isof a ‘forward topology’ as proposed by Innocente et al.[4], similar to that of the ring topologyonly that interconnections are not bidirectional in the former. The intention is to slow clusteringof the swarm and increase exploration of the search space. The size of these neighbourhoods isdynamic in that they change linearly with time so that the search begins highly local (extendingthe search of the space and delaying clustering), then becoming a global search toward the endwith total cooperation between particles.Constraints are handled by the preserving feasibility method with priority rules and pseudo-adaptive relaxation of the tolerances for both equality and inequality constraint violations, asproposed in [4]. Combining the PSO algorithm and SQP gradient by providing the latter with ‘good’ initialsolutions, enables the feature of guaranteed local optima convergence which the PSO alone doesnot have. SQP is known to be one of the most successful methods in non-linear constrainedoptimization, as discussed by a number of authors in the literature, for example, by Victoire etal., who investigated a hybrid PSO-SQP algorithm for an economic dispatch problem [5]. It seemsa logical approach for refined local search hybridization with heuristic algorithms on constrainedoptimization problems. SQP is a quasi-newton method utilising second-order information aboutthe problem to efficiently and accurately converge to a solution. The method considered is anactive-set algorithm for dealing with constrained optimization problems by finding a solution tothe Karush-Kuhn-Tucker (KKT) conditions, which are analogous to finding a point at which thegradient is zero, only considering constraints as well. At each iteration, an approximation of theHessian of the Lagrangian is made using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method.This is then used to formulate a general quadratic problem (QP) whose solution is used todetermine a direction, in which a line search is performed. This is formulized by bound constraintsexpressed as inequality constraints and non-linear constraints linearized. This procedure is thenrepeated until some termination criteria have been met.
Memetic algorithms (MA) were first proposed by Moscato in 1989 (quoted by Petalas et al. in[6]), being inspired by the notion of Memes, proposed by Dawkins in 1976 as a unit of culturalevolution (quoted in [6]). Moscato implemented a hybrid population-based GA with SimulatedAnnealing (SA) as a local search refinement to tackle combinatorial problems including thetravelling salesman problem. The method gained wide acceptance due to its ability to solvedifficult problems. A comparative study was made by Petalas et al.[6] on MAs, concludingthem to be highly superior in effectiveness compared to the stand-alone global algorithm. Ring topology is where each particle shares information with only two other particles in the swarm. NP-hard problems are a particular class of combinatorial problems that no polynomial time algorithm existsand so computation times may tend to exponential computation time in the worst possible cases as described byChristian B [7]. (cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:72)(cid:71)(cid:76)(cid:81)(cid:74)(cid:86)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:27)(cid:87)(cid:75)(cid:3)(cid:36)(cid:54)(cid:48)(cid:50)(cid:3)(cid:56)(cid:46)(cid:3)(cid:38)(cid:82)(cid:81)(cid:73)(cid:72)(cid:85)(cid:72)(cid:81)(cid:70)(cid:72)(cid:3)(cid:82)(cid:81)(cid:3)(cid:40)(cid:81)(cid:74)(cid:76)(cid:81)(cid:72)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:72)(cid:86)(cid:76)(cid:74)(cid:81)(cid:3)(cid:50)(cid:83)(cid:87)(cid:76)(cid:80)(cid:76)(cid:93)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:3)(cid:51)(cid:68)(cid:74)(cid:72)(cid:3)(cid:21)(cid:26)(cid:19) Preprint submitted to the 8th ASMO UK Conference on Engineering Design Optimization mplementations of hybrid based methods are far from uncommon amongst the optimization com-munity. Examples include the training of artificial neural networks for function approximation[8]or the hybrid SQP-PSO designed by Victoire et al.[5] for the economic dispatch problem. Thelatter being of high interest with its implementation of SQP to the current best solution foundin the swarm ( gbest ), triggered each time the solution has improved. It was argued by Victoireet al.[5] that early on in the PSO search, particles are statistically likely to be of proximity tothe global best but then move away from these areas. For this reason the local search methodwas implemented.One of the most successful algorithms developed for a general purpose optimizer is by Lianget al.[9], that combines and in fact couples the SQP method with their Dynamic Multi-swarmOptimizer (DMS-PSO). Their method describes subpopulations solving their own objectives,being assigned adaptively, and their assignment being periodically changed according to difficulty.For this reason, the number of subpopulations is not necessarily equal to the number of objectivesor constraints. The SQP method is coupled by being called at every ‘ n ’ number of generations,where the positions of the individual best experiences of five randomly chosen particles comprisethe seeds for the local searches. The random choosing of these five particles means that nopreference is made to one over the other, which is consistent with the fact that particles distancesfrom an unknown global optimum give rise to indistinguishability between one ‘good’ particleand another. After a certain percentage of function evaluations are met, every ‘ n ’ generations,a local search is triggered using the gbest solution only (thus ensuring refinement of the finalsolution). A set of problems are used for the purpose of testing the hybrid algorithm’s effectiveness to solvereal-world optimization problems. These test suites include the 13 problems taken from Pulidoet al.[10] (referred hereafter as problems g01-g13) with some added features and performancemeasures in [11] (referred hereafter as problems g14-g24). Measures determining clustering ofthe swarm considered are those formulized by Innocente et al.[12].With respect to tolerance, to remain consistent with problem definitions for CEC06[11], equalityconstraints are relaxed by formulating them as inequalities as shown below: | h j ( x ) | − (cid:23) ≤ (cid:23) = 10 − Inequalities are then defined as; g j ( x ) ≤ ‘good’ particle refers to the quality of a particles solution, considering both its conflict and constraint. (cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:72)(cid:71)(cid:76)(cid:81)(cid:74)(cid:86)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:27)(cid:87)(cid:75)(cid:3)(cid:36)(cid:54)(cid:48)(cid:50)(cid:3)(cid:56)(cid:46)(cid:3)(cid:38)(cid:82)(cid:81)(cid:73)(cid:72)(cid:85)(cid:72)(cid:81)(cid:70)(cid:72)(cid:3)(cid:82)(cid:81)(cid:3)(cid:40)(cid:81)(cid:74)(cid:76)(cid:81)(cid:72)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:72)(cid:86)(cid:76)(cid:74)(cid:81)(cid:3)(cid:50)(cid:83)(cid:87)(cid:76)(cid:80)(cid:76)(cid:93)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:3)(cid:51)(cid:68)(cid:74)(cid:72)(cid:3)(cid:21)(cid:26)(cid:20) Preprint submitted to the 8th ASMO UK Conference on Engineering Design Optimization t should be noted that all constraints once formulated to inequalities have zero tolerance for theGP-PSO, but within the SQP, due to round-off errors, constraints are considered not violated if ≤ − .Tolerances for the SQP are chosen based on a sample of problems of the test suite, where theability of the SQP to converge and terminate at a solution of the desired level of accuracy, resultin the following tolerances set: T olx = 10 − (tolerance on the solution coordinates); T olcon =10 − (tolerance on the constraints); T olf un = 10 − (tolerance on the conflict function).The SQP local search is applied to the final results of the GP-PSO on problems g01-g24 anda determination is made to whether a local search is helpful or not. Secondly, a local searchtriggering is made on each iteration of the GP-PSO on problems g01-g13, to determine thepoints at which the SQP local search becomes successful.With the results obtained from theabove, considerations are made to difficulties met on the 13 problems (where the local searchfails and where might it be sensitive) with particular attnetion made to low dimensional problems.Finally, a comparison between the results obtained for the entire benchmark and the two mostsuccessful algorithms from the CEC06[11] test suite are made to determine the validity of a localsearch hybrid together with the possible computational expense of the algorithms. Comparison between the path of the SQP local search at various points in the solution space ismade with that of the GP-PSO on low dimensional problems including g06, g08 and g11. Allthree functions offer significant insight into the advantages of local search implementation tothe heuristic global optimizing PSO search, together with the drawbacks of traditional methodsalone.In problem g06, the path of the SQP with various initial solutions is shown in Fig.[1,2]. Thechosen starting points are based on varied feasible and infeasible points in the search-space.
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Figure 1: g06 GP-PSO path (full search-space)
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Figure 2: g06 GP-PSO path (region of interest)5 (cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:72)(cid:71)(cid:76)(cid:81)(cid:74)(cid:86)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:27)(cid:87)(cid:75)(cid:3)(cid:36)(cid:54)(cid:48)(cid:50)(cid:3)(cid:56)(cid:46)(cid:3)(cid:38)(cid:82)(cid:81)(cid:73)(cid:72)(cid:85)(cid:72)(cid:81)(cid:70)(cid:72)(cid:3)(cid:82)(cid:81)(cid:3)(cid:40)(cid:81)(cid:74)(cid:76)(cid:81)(cid:72)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:72)(cid:86)(cid:76)(cid:74)(cid:81)(cid:3)(cid:50)(cid:83)(cid:87)(cid:76)(cid:80)(cid:76)(cid:93)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:3)(cid:51)(cid:68)(cid:74)(cid:72)(cid:3)(cid:21)(cid:26)(cid:21)
Preprint submitted to the 8th ASMO UK Conference on Engineering Design Optimization he results indicate that the SQP tends to directions unexpected depending on the problemfeatures. In this case, the edge of the search-space is seen as an attractive area since it offersan improved conflict and a lower maximum constraint violation. It takes a number of iterationsbefore this local optimal region is escaped, where the search for a feasible region of the search-space and a degradation of the conflict is observed to reach this feasible region. Since thisapproach takes the SQP path through a local optimal region of the search-space, it is apparentthat the initial starting solution to the local search will determine the final converged solution inmore complicated problems (suggesting that an erratic success/failure may occur in some cases).
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Figure 3: g06 GP-PSO path (region of interest),Red curve showing the global best history andBlue curve the centre of gravity history.The path of the GP-PSO as shown in Fig.3indicates a similarity with the SQP path,with the best solution history of the swarmfollowing the edge of the search-space for atime before convergence to the global opti-mum occurs. This similarity is however ex-pected, since the PSO is statistically likelyto follow the path of the gradient to op-timality. It is noticed that since the PSOrelies on its diversity and statistical likeli-hood in finding a better solution other thanthe boundary edge, that the SQP offers tobe a more efficient method in dealing withthis function. The local search method isclearly a more efficient method in dealingwith this function.Problem g08 offers to be a highly inter-esting function, in that it is clearly multi-modal, and as such a local search becomeseasily trapped. This is clearly shown in Fig.4. There are a number of suboptimal regions withinthe feasible search-space and the conflict is rather flat (difference between the extremities withinthe region is small). The SQP can hop over the global best in the pursuit of satisfying theconstraints. It is also apparent that the SQP may be led to a solution that may cause an errorwith a divide by zero even though the formulization of this problem allows solution coordinatesof (0,0). Initial starting points near the global optimum still do not guarantee convergence on theglobal optimum since following the curvature of the constraint function leads it away, indicatinga lack of knowledge of the function due to a highly localised approximation of the conflict andconstraint function in the lagrangian. There are at least 6 suboptimal regions within the feasiblesearch-space that the SQP may become trapped. The SQP also often finds itself stuck in thesuboptimal regions at the left boundary of the search-space.The path of the GP-PSO as shown in Fig.5, demonstrates its similarity to the SQP path in thatit is pulled toward the edge of the search-space. Since diversity is kept, the best solution foundby the swarm is quickly shifted to a region of feasibility by other members of the swarm searching6 (cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:72)(cid:71)(cid:76)(cid:81)(cid:74)(cid:86)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:27)(cid:87)(cid:75)(cid:3)(cid:36)(cid:54)(cid:48)(cid:50)(cid:3)(cid:56)(cid:46)(cid:3)(cid:38)(cid:82)(cid:81)(cid:73)(cid:72)(cid:85)(cid:72)(cid:81)(cid:70)(cid:72)(cid:3)(cid:82)(cid:81)(cid:3)(cid:40)(cid:81)(cid:74)(cid:76)(cid:81)(cid:72)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:72)(cid:86)(cid:76)(cid:74)(cid:81)(cid:3)(cid:50)(cid:83)(cid:87)(cid:76)(cid:80)(cid:76)(cid:93)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:3)(cid:51)(cid:68)(cid:74)(cid:72)(cid:3)(cid:21)(cid:26)(cid:22)
Preprint submitted to the 8th ASMO UK Conference on Engineering Design Optimization he solution space. The path of the PSO demonstrated its success in finding a global optimumin a search-space where the local SQP optimizer is prone to fail.Figure 4: g08 SQP path (full search-space), witha 3D plot of small region of the search-space tobetter understand the problem features.
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Figure 5: g08 GP-PSO path (region of interest),Red curve showing the global best history andBlue curve the centre of gravity history.In problem g11, it is clearly shown in Fig.[6,7] that a local search on this simple function does notguarantee global optima convergence. Depending on the initial starting solution, it is possiblefor the SQP to become trapped at the local optimum at the centre of the search-space. It ishowever unlikely for this to happen unless the starting solution is nearly on top of it. The wayin which the SQP escapes this region if near it, is to improve its conflict at the cost of increasingits constraint violation and then to do the vice versa, and so, zigzag its way to optimality. Dueto the central region of sub-optimality it takes a number of iterations to escape this region withrespect to other areas of the search-space. The formulation of the equalities and inequalities isbeneficial in the case of this function, since it allows the solution to deteriorate a little, howeverequality relaxation may be harmful as well as beneficial. It may open a suboptimal region forwhich the SQP may become trapped or on the other hand, open regions normally separated bya worsening solution thus making them accessible.It is also useful to understand how the SQP might deal with multimodal problems (where in thiscase, this vertically symmetric function offers two equally ‘good’ global optima solutions). It isnoticed that depending on the initial solution provided to the SQP, the chosen global optimumis determined (since the SQP method is a deterministic algorithm). For initial solutions at theleft or right border edge of the solution space it is noticed that the initial path of the SQP take ittoward the corners where the constraints are satisfied. This convex function is highly appropriatein demonstrating the success of the local search SQP to find the global optimum.The path of the GP-PSO as shown in Fig.8 demonstrates the effect of two equally good attractorson the swarm behaviour. Unlike the path of the SQP, the swarm’s best solution fluctuatesbetween the two global optima since the diversity of the swarm causes better solutions to be7 (cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:72)(cid:71)(cid:76)(cid:81)(cid:74)(cid:86)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:27)(cid:87)(cid:75)(cid:3)(cid:36)(cid:54)(cid:48)(cid:50)(cid:3)(cid:56)(cid:46)(cid:3)(cid:38)(cid:82)(cid:81)(cid:73)(cid:72)(cid:85)(cid:72)(cid:81)(cid:70)(cid:72)(cid:3)(cid:82)(cid:81)(cid:3)(cid:40)(cid:81)(cid:74)(cid:76)(cid:81)(cid:72)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:72)(cid:86)(cid:76)(cid:74)(cid:81)(cid:3)(cid:50)(cid:83)(cid:87)(cid:76)(cid:80)(cid:76)(cid:93)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:3)(cid:51)(cid:68)(cid:74)(cid:72)(cid:3)(cid:21)(cid:26)(cid:23)
Preprint submitted to the 8th ASMO UK Conference on Engineering Design Optimization −1 −0.5 0 −0.6−0.4−0.200.2
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Figure 6: g11 problem for SQP applied at vari-ous starting positions (full search-space)
Figure 7: g11 problem for SQP applied at vari-ous starting positions (region of interest)found on either side of this vertically symmetric function. The centre of gravity solution of theswarm begins at the centre with the swarm having been randomly initialized, and then it tendsto a point which is vertically cantered and horizontally in line with the two solutions. This isdue to the two attractors being equally ‘good’. After a time, a more refined solution on eitherside will become less and less frequent, at which point the stochastic choice of the global optimawill cause the centre of gravity to drift toward it. It is again apparent that the SQP deals withthis function much more efficiently to the GP-PSO. −1 −0.5 0 −0.4−0.200.2
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Figure 8: g11 GP-PSO path (region of interest),Red curve showing the global best history andBlue curve the centre of gravity history.As shown by the low dimensional probleminvestigation, features of problems suggestthat a MA to be highly beneficial, howeverthat both complement each other on eachothers weaknesses. The PSO relies on itsstatistical likelihood to find a ‘good’ solu-tion while the local search is shown to bereliant on the initial solution it is providedwith.Combining SQP with the GP-PSO, indi-cates much improvement of its solutionsover the standalone GP-PSO algorithm.GP-PSO is found to fail on problems g05,g07, g09, g10. The success of the SQP onall of these problems when applied to theGP-PSO final iteration is then a significantresult. An investigation into the point atwhich the SQP becomes successful is made 8 (cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:72)(cid:71)(cid:76)(cid:81)(cid:74)(cid:86)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:27)(cid:87)(cid:75)(cid:3)(cid:36)(cid:54)(cid:48)(cid:50)(cid:3)(cid:56)(cid:46)(cid:3)(cid:38)(cid:82)(cid:81)(cid:73)(cid:72)(cid:85)(cid:72)(cid:81)(cid:70)(cid:72)(cid:3)(cid:82)(cid:81)(cid:3)(cid:40)(cid:81)(cid:74)(cid:76)(cid:81)(cid:72)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:72)(cid:86)(cid:76)(cid:74)(cid:81)(cid:3)(cid:50)(cid:83)(cid:87)(cid:76)(cid:80)(cid:76)(cid:93)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:3)(cid:51)(cid:68)(cid:74)(cid:72)(cid:3)(cid:21)(cid:26)(cid:24)
Preprint submitted to the 8th ASMO UK Conference on Engineering Design Optimization y applying the local search at each itera-tion.Problems g05, g10 and g13 have erratic success rates when a local search is applied due to theirsensitivity to scaling. Problems g03, g06, g09 and g11 however are successful from the very firstiteration.The results of SQP implementation to the final solution of the GP-PSO is shown in Table.3 andthe success and feasibility rate shown in comparison to that of the stand-alone GP-PSO togetherwith two of the leading optimizers shown in Table.1. The results indicate that the SQP givesnot only an improved refined solution but also in some problems leads to a solution where noneare found by the stand-alone GP-PSO.
PROBLEM
GP-PSO GP-PSO-SQP PESO+ DMS-PSO
SUCCESS FEASIBLE SUCCESS FEASIBLE SUCCESS FEASIBLE SUCCESS FEASIBLE g01 100% 100% 100% 100% 100% 100% 100% 100% g02 70% 100% 70% 100% 56% 100%
84% 100%g03 70% 100% 100% 100% g07 0% 100% 100% 100% 96% 100% 100% 100% g08 100% 100% 100% 100% 100% 100%
70% 70% 16% 100% 100% 100%g11 100% 100% g13 90% 100% 100% 100% 100% 100% 100% 100% g14 0% 100% 100% 100% 0% 100%
90% 90% 0% 100% 0% 100%g18
20% 100% 100% 100% 92% 100% 100% 100% g19 0% 100% 100% 100% 0% 100% 100% 100% g20 NA 0% NA 0% NA 0%
NA 0%g21 0% 0% 0% 0%
0% 100% 100% 100%g22 0% 0%
0% 0% 0% 0% 0% 0g23 0% 0%
Table 1:
Final success rates and feasibility rates using the end values (corresponding to 10,000 iterationsin the case of the GP-PSO and GP-PSO-SQP.
Problems g05, g07, g09, g10, g14, g19 and g23 are shown to improve considerably in their solutionquality when the local search is applied. It is likely that the guaranteed local optima convergence,overcomes the inherent limitation of the stochastic swarm, where it is shown to refine or evenfind the solution (to within accuracy of results defined by CEC06). Since the swarm relies onits diversity, when it is lost (depending on the features of the problem), the swarms centre ofgravity solution and the best solution may not ever converge to the global optimum as the swarmbecomes as one particle and loses momentum. The local search implementation then allows thelocal convergegence within machine precision thus overcoming this limitation. The most notableproblems however in its implementation are in problems g05, g07, g10 and g17, where scalingand overshooting (termination criteria) are found to be a major issue in the outcome of theconverged solution. It is also noted that GP-PSO struggles considerably in obtaining feasiblesolutions to problems g21-g23, however the SQP method results in a 100% on problem g23. Thelack a feasible solution in problems g21-22 results in total failure of the SQP to find a feasiblesolution either. PESO+ also fails on these two problems but DMS-PSO is successful on problem9 (cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:72)(cid:71)(cid:76)(cid:81)(cid:74)(cid:86)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:27)(cid:87)(cid:75)(cid:3)(cid:36)(cid:54)(cid:48)(cid:50)(cid:3)(cid:56)(cid:46)(cid:3)(cid:38)(cid:82)(cid:81)(cid:73)(cid:72)(cid:85)(cid:72)(cid:81)(cid:70)(cid:72)(cid:3)(cid:82)(cid:81)(cid:3)(cid:40)(cid:81)(cid:74)(cid:76)(cid:81)(cid:72)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:72)(cid:86)(cid:76)(cid:74)(cid:81)(cid:3)(cid:50)(cid:83)(cid:87)(cid:76)(cid:80)(cid:76)(cid:93)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:3)(cid:51)(cid:68)(cid:74)(cid:72)(cid:3)(cid:21)(cid:26)(cid:25)
Preprint submitted to the 8th ASMO UK Conference on Engineering Design Optimization
GP-PSO GP-PSO loc SQP PESO+
DMS-PSOg01 5.5E+04 3.1E+04 9.2E+01 g05 – 0.0E+00 – 4.5E+05 2.9E+04 g06 4.1E+04 0.0E+00 4.0E+01 5.7E+04 – 7.2E+04 7.2E+02 4.5E+05 2.6E+04 g11 1.0E+04 0.0E+00 4.0E+01 4.5E+05 1.5E+04 g12 9.1E+03 6.1E+03 4.1E+01 8.1E+03 – 2.5E+04g15 3.8E+04 g17 7.4E+04 8.2E+04 1.5E+03 – – g18 4.5E+04 2.0E+04 2.0E+02 2.1E+05 – 2.2E+04g20 NA NA NA
NA NAg21 – – – – 1.4E+05g22 – – – – – g23 – 3.7E+04 2.6E+02 – 2.1E+05 g24 1.2E+04 7.4E+00 2.9E+01 2.0E+04
Table 2:
Mean number of FES to achieve the fixed accuracy level (( f (¯ x ) − f (¯ x ∗ ) ≤ e − loc represents the mean FE at which the local search becomes successful. SQP is the mean number ofFEs required at the point at which it becomes successful. With respect to computational expense, Table.2 indicates the mean function evalutations (FE)on the bench mark problems, between the three algorithms (GP-PSO, DMS-PSO and PESO+)together with the FEs required from the point at which a local search becomes successful. Whencompared on problems successful between all three algorithms (GP-PSO with local search im-plementation included for completeness), the mean FEs put the DM-PSO in the lead with theGP-PSO closely behind (with the GP-PSO’s success likely being due its not having to calculatethe objective function outside the feasible space). PESO+ has indicated that it requires a greatdeal more FEs on average with respect to the other optimizers, however it is a considerably morerobust optimizer than the standard GP-PSO with its success on a greater number of problemsof the test suite. From the application of the SQP, it is clear that on average, the number of FEsrequired before error is attained is halfed with respect to that required for the basic GP-PSO, andthat the number of FEs by the local search itself is on average only 2% the total computational10 (cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:72)(cid:71)(cid:76)(cid:81)(cid:74)(cid:86)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:27)(cid:87)(cid:75)(cid:3)(cid:36)(cid:54)(cid:48)(cid:50)(cid:3)(cid:56)(cid:46)(cid:3)(cid:38)(cid:82)(cid:81)(cid:73)(cid:72)(cid:85)(cid:72)(cid:81)(cid:70)(cid:72)(cid:3)(cid:82)(cid:81)(cid:3)(cid:40)(cid:81)(cid:74)(cid:76)(cid:81)(cid:72)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:72)(cid:86)(cid:76)(cid:74)(cid:81)(cid:3)(cid:50)(cid:83)(cid:87)(cid:76)(cid:80)(cid:76)(cid:93)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:3)(cid:51)(cid:68)(cid:74)(cid:72)(cid:3)(cid:21)(cid:26)(cid:26)
Preprint submitted to the 8th ASMO UK Conference on Engineering Design Optimization ost. A trigger is however required in order to switch between the global and local search.
GP-PSO SQPPROBLEM OPTIMUM Conflict Max Constraint
SQPconf SQPconsBEST -15.000000 0.0E+00 -15.000000 0.0E+00g01 -15.000000 AVERAGE -15.000000 0.0E+00 -15.000000 0.0E+00STDEV -0.803619 3.1E-15g02 -0.803619 AVERAGE -0.800309 0.0E+00 -0.800316 4.3E-15STDEV 5.3E-03 0.0E+00 5.3E-03 2.2E-15
BEST -1.000495 -5.2E-06 -1.000500 7.0E-16g03 -1.000500 AVERAGE -1.000102 -5.4E-07 -1.000500 5.2E-16STDEV 9.9E-04 1.6E-06 -30665.538672 AVERAGE -30665.538672 0.0E+00 -30665.538672 0.0E+00STDEV
BEST -6961.813876 0.0E+00 -6961.813876 0.0E+00g06 -6961.813876 AVERAGE -6961.813876 0.0E+00 -6961.813876 0.0E+00STDEV 1.9E-12 0.0E+00 -0.095825 -1.7E-01g08 -0.095825 AVERAGE -0.095825 0.0E+00 -0.095825 -1.7E-01STDEV 1.4E-17 0.0E+00 1.4E-17 6.3E-10
BEST 680.630911 0.0E+00 680.630057 0.0E+00g09 680.630057 AVERAGE
BEST -1.000000 0.0E+00 -1.000000 -6.2E-02g12 -1.000000 AVERAGE -1.000000 0.0E+00 -1.000000 -6.2E-02STDEV 0.0E+00 0.0E+00
STDEV 3.8E-05 2.0E-10 5.0E-17 2.2E-15BEST -47.723001 -9.4E-06 -47.764888 2.1E-16g14 -47.764888 AVERAGE -47.606122 -9.9E-07 -47.764888 1.9E-16STDEV 1.5E-01 3.0E-06 1.1E-14 7.0E-17
BEST 961.715023 -3.9E-14 961.715022 3.3E-15g15 961.715022 AVERAGE -1.905155 AVERAGE -1.905155 0.0E+00 -1.905155 0.0E+00
STDEV 4.7E-16 0.0E+00 4.7E-16 0.0E+00BEST 8853.539675 -1.1E-09
BEST -0.866014 0.0E+00 -0.866025 6.8E-15g18 -0.866025
AVERAGE -0.858576 0.0E+00 -0.866025 1.4E-15STDEV 1.2E-02 0.0E+00
STDEV 1.4E+00 0.0E+00 4.1E-15 2.9E-15BEST
BEST 1113.283037 – NaN NaNg21 193.724510
AVERAGE 1372.222391 – NaN NaNSTDEV
STDEV 9.4E+03 – NaN NaNBEST -2016.651110 1.7E+00 -400.055100 3.3E-15g23 -400.055100 AVERAGE -966.309692 2.3E+00 -400.055100 8.4E-15STDEV 8.6E+02 2.6E-01 2.2E-13 1.2E-14
BEST -5.508013 0.0E+00 -5.508013 0.0E+00g24 -5.508013
AVERAGE -5.508013 0.0E+00 -5.508013 0.0E+00STDEV
Table 3:
Results of the SQP supplied with the solutions of the GP-PSO. ‘NaN’ in problems g21 andg22 signify the lack of a suitable initial solution provided to the local search. (cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:72)(cid:71)(cid:76)(cid:81)(cid:74)(cid:86)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:27)(cid:87)(cid:75)(cid:3)(cid:36)(cid:54)(cid:48)(cid:50)(cid:3)(cid:56)(cid:46)(cid:3)(cid:38)(cid:82)(cid:81)(cid:73)(cid:72)(cid:85)(cid:72)(cid:81)(cid:70)(cid:72)(cid:3)(cid:82)(cid:81)(cid:3)(cid:40)(cid:81)(cid:74)(cid:76)(cid:81)(cid:72)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:72)(cid:86)(cid:76)(cid:74)(cid:81)(cid:3)(cid:50)(cid:83)(cid:87)(cid:76)(cid:80)(cid:76)(cid:93)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:3)(cid:51)(cid:68)(cid:74)(cid:72)(cid:3)(cid:21)(cid:26)(cid:27) Preprint submitted to the 8th ASMO UK Conference on Engineering Design Optimization
Conclusions
Combining an SQP local search to the GP-PSO algorithm as designed by Innocente et al.[4] hasshown that it overcomes the inherent limitations in swarm dynamics of a stochastic swarm. Itis also made apparent by the author of this document that an algorithm based solely on thefeatures of a global optimizer is unlikely to achieve the best possible results across all problems.For this reason, a hybrid (Memetic algorithm) which implements individual learning, indicatesconsiderable improvement of the solution quality, as well as the finding of the global solution insome cases, where swarm dynamics limit the statistical likelihood of it being found.The GP-PSO-SQP has shown that it can compete with the leading optimizers, with its successrates across the problems of the test-suite. However, difficulties have been identified in itsimplementation, with its sensitivity to scaling and termination criteria. The computation expenseof the algorithms by comparing problems successful on all three of the algorithms, indicates thatthe GP-PSO and the DMS-PSO to be at a similar level, and though PESO+ indicates a muchhigher FE count, its higher success on the range of problems tested overcomes any computationaladvantage that the GP-PSO may offer. The local search implementation however indicates thata great computational saving may be possible with suitable switching criteria and only a 2%computational expense on average of the total FEs by its implementation.
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Proceedings of the6th ASMO UK Conference on Engineering Design Optimization , page 203, Oxford, 2006. (cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:72)(cid:71)(cid:76)(cid:81)(cid:74)(cid:86)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:27)(cid:87)(cid:75)(cid:3)(cid:36)(cid:54)(cid:48)(cid:50)(cid:3)(cid:56)(cid:46)(cid:3)(cid:38)(cid:82)(cid:81)(cid:73)(cid:72)(cid:85)(cid:72)(cid:81)(cid:70)(cid:72)(cid:3)(cid:82)(cid:81)(cid:3)(cid:40)(cid:81)(cid:74)(cid:76)(cid:81)(cid:72)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:72)(cid:86)(cid:76)(cid:74)(cid:81)(cid:3)(cid:50)(cid:83)(cid:87)(cid:76)(cid:80)(cid:76)(cid:93)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:3)(cid:51)(cid:68)(cid:74)(cid:72)(cid:3)(cid:21)(cid:26)(cid:28)(cid:51)(cid:85)(cid:82)(cid:70)(cid:72)(cid:72)(cid:71)(cid:76)(cid:81)(cid:74)(cid:86)(cid:3)(cid:82)(cid:73)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)(cid:27)(cid:87)(cid:75)(cid:3)(cid:36)(cid:54)(cid:48)(cid:50)(cid:3)(cid:56)(cid:46)(cid:3)(cid:38)(cid:82)(cid:81)(cid:73)(cid:72)(cid:85)(cid:72)(cid:81)(cid:70)(cid:72)(cid:3)(cid:82)(cid:81)(cid:3)(cid:40)(cid:81)(cid:74)(cid:76)(cid:81)(cid:72)(cid:72)(cid:85)(cid:76)(cid:81)(cid:74)(cid:3)(cid:39)(cid:72)(cid:86)(cid:76)(cid:74)(cid:81)(cid:3)(cid:50)(cid:83)(cid:87)(cid:76)(cid:80)(cid:76)(cid:93)(cid:68)(cid:87)(cid:76)(cid:82)(cid:81)(cid:15)(cid:3)(cid:3)(cid:51)(cid:68)(cid:74)(cid:72)(cid:3)(cid:21)(cid:26)(cid:28)