The Electromagnetism-like (EM) algorithm, developed by Birbil and Fang [31 is a population-based stochastic global optimization algorithm that uses an attraction-repulsion mechanism to move sample points toward optimality. In order to improve the accuracy of the solutions the EM algorithm incorporates a random local search. In this paper we propose: a new local search procedure based on a pattern search method, and a population shrinking strategy to improve efficiency. The proposed method is applied to some test problems and compared with the original EM algorithm.info:eu-repo/semantics/publishedVersio

Fernandes, Edite Manuela da G. P.

Rocha, Ana Maria A. C.

Universidade do Minho: RepositoriUM

A new Electromagnetism-like Algorithm with a PopulationShrinking StrategyANA MARIA A. C. ROCHA and EDITE M.G.P. FERNANDESMinho UniversityDepartment of Production and SystemsCampus de Gualtar, 4710-057 BragaPORTUGAL{arocha;emgpf}@dps.uminho.ptAbstract: The Electromagnetism-like (EM) algorithm, developed by Birbil and Fang [3] is a population-basedstochastic global optimization algorithm that uses an attraction-repulsion mechanism to move sample points towardoptimality. In order to improve the accuracy of the solutions the EM algorithm incorporates a random local search.In this paper we propose a new local search procedure based on a pattern search method, and a population shrinkingstrategy to improve efficiency. The proposed method is applied to some test problems and compared with theoriginal EM algorithm.Key–Words: Global optimization, Electromagnetism-like algorithm, Pattern search method, Population shrinking1 IntroductionWe consider the problem of finding a global solutionof the problem:min f(x)s.t. x ∈ Ω, (1)where f : Rn → R is a nonlinear function andΩ = {x ∈ Rn : −∞ < lk ≤ xk ≤ uk < ∞, k =1, . . . , n} is a bounded feasible region. Further, we as-sume that the derivatives are not available. Recently,Birbil and Fang proposed the electromagnetism-like(EM) algorithm that is a population-based stochas-tic search method for global optimization [3]. Thealgorithm simulates the electromagnetism theory ofphysics by considering each sampled point as anelectrical charge. The method utilizes an attraction-repulsion mechanism to move a population of pointstoward optimality. In order to improve the accuracyof the solutions a typical EM algorithm incorporatesa random local search. Here, we propose a modifi-cation to the EM algorithm using the original patternsearch method of Hooke and Jeeves [5], that is sim-ple to implement and does not require any derivativeinformation. Another modification consists of incor-porating a population shrinking strategy into the EMalgorithm.Thus, in order to solve problem (1), we modifythe EM algorithm proposed in [3] twofold: (i) a pat-tern search method is used to provide at each iterationa local search about the best point of the population;(ii) a population shrinking strategy is implemented toreduce the number of points in the population when-ever the concentration of all points around the bestpoint is considered acceptable.Four sets of experiments are carried out to illus-trate the efficiency of the local pattern search methodand the population shrinking strategy separately.The remainder of the paper is organized as fol-lows. Section 2 briefly describes the original EM al-gorithm and Section 3 introduces the pattern searchmethod to be used as a local search. Section 4 is de-voted to explain the main ideas of the new shrinkingpopulation strategy and Section 5 reports the numeri-cal results. Some conclusions are drawn in Section 6.2 Electromagnetism-like algorithmThe EM algorithm starts with a population of ran-domly generated points from the feasible region. Eachpoint is considered as a charged particle that is re-leased to the space. The charge of each point is re-lated to the objective function value and determinesthe magnitude of attraction or repulsion of the pointover the population. Points with lower objective func-tion values attract others while those with higher func-tion values repel. The charges are used to find a direc-tion for each point to move in subsequent iterations.Throughout the paper, the following notation isadopted: xi ∈ Rn denotes the i th point of a po-pulation; xbest is the point that has the least objectivefunction value; xik ∈ R is the k th (k = 1, . . . , n) co-ordinate of the point xi of the population; m is thenumber of points in the population; MaxIt is themaximum number of EM iterations; LSIt denotes themaximum number of local search iterations; and δ isa local search parameter, δ ∈ [0, 1].The EM mechanism is schematically shown inAlgorithm 1 and relies on four main procedures (Ini-tialize, CalcF, Move and Local).Algorithm 1 (m, MaxIt, LSIt, δ)Initialize()iteration ← 1while termination criteria are not satisfied doF ← CalcF()Move(F)Local(LSIt, δ)iteration ← iteration + 1end whileThe procedure Initialize aims to randomly gene-rate m points from the feasible region. Each coordi-nate of a point (xik) (k = 1, . . . , n) is assumed to beuniformly distributed between the corresponding up-per and lower bounds, i.e., xik = lk+λ(uk−lk) whereλ ∼ U(0, 1). The objective function values are com-puted for all the points in the population, and the bestpoint, xbest, which is the point with the least functionvalue is identified.The CalcF procedure computes the force exertedon a point via other points. First a charge-like value,qi, that determines the power of attraction or repulsionfor the point xi is determined. The charge of the pointis calculated according to the relative efficiency of theobjective function value of the corresponding point inthe population, i.e.,qi = exp(−n f(xi)− f(xbest)∑mk=1(f(xk)− f(xbest))),for i = 1, . . . ,m. The total force vector F i exertedon each point is calculated by adding the individualcomponent forces, F ij , between any pair of points xiand xj ,F i =m∑j 6=iF ij , i = 1, 2, . . . ,mwhereF ij ={(xj − xi) qiqj‖xj−xi‖2 if f(xj) < f(xi)(xi − xj) qiqj‖xj−xi‖2 if f(xi) ≤ f(xj).The Move procedure uses the normalized totalforce vector exerted on the point xi, so that feasibilitycan be maintained, to move it in the direction of theforce by a random step length λ, i.e.,xi = xi + λF i‖F i‖(RNG),for i = 1, . . . ,m and i 6= best, where RNG is avector with components that define the allowed rangeof movement toward the lower bound lk, or the up-per bound uk, for each coordinate k. The randomparameter λ is assumed to be uniformly distributedbetween 0 and 1. Note that the best point, xbest, is notmoved and is carried to the subsequent iterations.Finally, the Local procedure presented in [3] isa random line search algorithm and is applied coordi-nate by coordinate to the best point only to explore theneighborhood of that point in the population. First,based on the parameter δ, the procedure computesthe maximum feasible step length, δ(maxk(uk− lk)).This quantity is used to guarantee that the local searchgenerates only feasible points. Second, for each co-ordinate k, the best point is assigned to a temporarypoint y to store the initial information. Next, a ran-dom number is selected as a step length and the pointy is moved along that direction. If an improvementis observed, within LSIt iterations, the best point isreplaced by y and the search along that coordinate kends. The reader is referred to [2, 3, 4] for details.3 Pattern search Local procedureBirbil and Fang [3] show that the Local procedure iscrucial in improving the accuracy of the average func-tion values although at the cost of the number of func-tion evaluations required. Another local search that issimple and does not use derivative information is thepattern search (PS) method, see for example [1, 7, 10].The research about pattern search methods is flou-rishing. This method has been first applied to uncons-trained optimization and then successfully extendedto bound constrained [8], as well as to equality cons-trained problems [9]. Here, in the EM context, theoriginal Hook and Jeeves pattern search algorithm isapplied, at each iteration, to the current best point inthe population [5, 10]. This algorithm is based on twomoves: the exploratory move and the pattern move.The exploratory move carries out a coordinatesearch about the best point, with a step length δ. If atthe new point, y, f(y) < f(xbest), the iteration is suc-cessful. Otherwise, the iteration is unsuccessful and δshould be reduced. If the previous iteration was suc-cessful, the vector y−xbest defines a promising direc-tion and a pattern move is then implemented, meaningthat the exploratory move is carried out about the pointy + (y − xbest) instead of y. Then, if the coordinatesearch is successful, the returned point is accepted asthe new point; otherwise, the pattern move is rejectedand the coordinate search is carried out about y. Here,a factor of 0.1 is used to reduce δ when the iteration isunsuccessful and the minimum step length allowed is1E-08.Our implementation of the pattern search methoduses an exact penalty-like technique to maintain fea-sibility, i.e., problem (1) is replaced by the followingminF (x) ≡{f(x) if x ∈ Ω,∞ otherwise,meaning that any generated point that is infeasible isrejected, since the corresponding function value is∞.4 A population shrinking strategyThe population shrinking strategy is the main contri-bution of this paper. The purpose of implementinga devise that is able to strategically reduce the num-ber of points in the population, here denoted by shrin-king the population, is to reduce the overall numberof objective function evaluations without affecting theaccuracy of the results. So, a strategy is designedto shrink the population as the iterative process pro-gresses. The crucial point here is to decide when toshrink the population. When the concentration of allpoints in the population around the best point is con-sidered acceptable, it seems that some points could bediscarded without affecting the convergence rate to thesolution. The points that will remain in the populationshould be the best ones.One way to measure the concentration of thepoints in the population around the best point is tocompute the standard deviation (SD) of the functionvalues with respect to the best value,SD =√∑mi=1(f(xi)− f(xbest))2m.This quantity is used to decide when to shrink thepopulation. Although one may think that the SD de-creases monotonically as the iterative process con-verges to the solution, this is not always true. Theidea is to shrink the population at a particular iterationl, when the SD(l) is below 10% of the SD of a refe-rence iteration (SDref ). Here, a reference iteration isthe first iteration that was carried out with the samepopulation of the iteration l. Thus, the population isshrunk ifdeviation ratio =SD(l)SDref< 0.1 .Table 1: Numerical resultsTf code mf fbest ItEM feGP orig - 3.000004 8 256orig-shri 5 3.000073 11 179ps - 3.000004 5 306ps-shri 5 3.000013 6 350H orig - 0.000038 8 265orig-shri 10 0.000042 9 231ps - 0.000068 3 209ps-shri 10 0.000068 3 209R orig - 0.008941 500 10533orig-shri 5 0.000053 380 2690ps - 0.000685 500 31520ps-shri 10 0.000079 72 3678M orig-shri 20 -1.913215 9 284ps-shri 20 -1.913220 5 317S orig-shri 20 0.000002 12 308ps-shri 20 0.000001 4 269Whenever the population is shrunk, the referenceiteration changes. Although different shrinking fac-tors have been tested, the constant 0.5 proved to beefficient. The shrinking process as described above isnot activated if the number of points in the popula-tion is less or equal to 2n. In the Algorithm 1 context,this shrinking procedure appears after the Local pro-cedure.5 Numerical resultsComputational tests were performed on a PC with a3GHz Pentium IV microprocessor and 1Gb of memo-ry. We compare the performance of the two versionsof the EM algorithm, as described in Sections 2 and3, with the corresponding versions with the proposedshrinking strategy, using 5 well-known test functionsselected from the literature. The algorithms terminatewhen the number of EM iterations exceeds MaxIt orthe relative error in the best objective function value,with respect to fglobal is less than 0.01%.Two multi-modal functions are selected: Gold-stein and Price (GP) with fglobal = 3, and Hump (H)with fglobal = 0. The first has one global solutiononly and three local solutions. The second one hastwo global optima. The other 3 tested functions areuni-modal: Rosenbrock2 (R) with fglobal = 0, Mc-Cormick (M) with fglobal = −1.9133, and Spherical(S) with fglobal = 0.In the experiments, we used m = 20 (initialpopulation), LSIt = 10, MaxIt = 50 (exceptfor Rosenbrock2, where we chose 500 due to con-vergence problems) and δ=1E-03 (except for Rosen-brock2, where we used 1E-02). The results that are re--2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2-1.5-1-0.500.511.52Goldstein PriceFigure 1: Original EM algorithm: points in the initialand final populations (GP)ported in Table 1 include the name of the test function,Tf , the number of points in the final (last iteration)population (when the shrinking strategy is used), mf ,the number of iterations of the EM algorithm, ItEM ,and the total number of function evaluations, fe.The tested codes are denoted by: orig (originalEM algorithm); orig-shri (original EM algorithmwith shrinking strategy); ps (EM pattern search algo-rithm); ps-shri (EM pattern search algorithm withshrinking strategy). All random quantities were ob-tained with the seed number set to 0.We noticed that the population shrinking processwas never activated when solving problems Mc-Cormick and Spherical. The deviation ratios werenever under 30%.From Table 1 one can see that the implementa-tions based on the pattern search algorithm requiresmaller number of EM iterations, although in somecases with larger number of function evaluations. Theaccuracy of the solutions is also good. The results pro-duced by the shrinking process show a reduction onthe iterations or on the function evaluations. In par-ticular, with the function Rosenbrock2, the two codesbased on shrinking significantly outperform the oth-ers in terms of convergence rate and accuracy of thesolution. Codes orig and ps reach the maximumnumber of iterations (500) without meeting the defaultrelative error criterion.We use the functions Goldstein and Price andHump to illustrate the population in the first and finaliterations of the 4 tested codes. Figures 1 - 8 show thelocation of the randomly generated points in the ini-tial population, represented by O, and the best pointrepresented by H; the location of the points in the fi-nal population, represented by ¤, and the best pointrepresented by ¥. The location of the known global-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2-1.5-1-0.500.511.52Goldstein PriceFigure 2: Original EM with shrinking strategy: pointsin the initial and final populations (GP)-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2-1.5-1-0.500.511.52Goldstein PriceFigure 3: EM pattern search algorithm: points in theinitial and final populations (GP)-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-2-1.5-1-0.500.511.52Goldstein PriceFigure 4: EM pattern search algorithm with shrink-ing strategy: points in the initial and final populations(GP)solution is represented by ×.First the function Goldstein and Price. Figures 1and 2 correspond to the original EM algorithm with-out and with the shrinking strategy, respectively. Inthe second case, the population of 20 points is reducedto 10 points after the first iteration and reduced to 5points in iteration 6. The algorithm takes 11 iterationsto reach the solution (see Table 1). Figures 3 and 4correspond to the EM pattern search algorithm with-out and with the shrinking strategy, respectively. Here,the population is reduced to 10 points after the firstiteration and reduced again to 5 points in iteration 5.The remaining Figures 5 - 8 contain similar illus-trations for the function Hump. With the original EMalgorithm, the shrinking strategy reduced the popula-tion in iteration 3 to 10 points which are maintaineduntil the final iteration. In the EM pattern search al-gorithm, the population is reduced to 10 points in thethird iteration and the iterative process terminates justafter the reduction. Overall the reduction of the popu-lation did not affect the rate of convergence and accu-racy of the results.6 ConclusionsWe have presented a modification to theElectromagnetism-like algorithm given in [3] forsolving the global optimization problem (1). Theoriginal Hook and Jeeves pattern search method isproposed as the Local procedure and a populationshrinking process is incorporated in the algorithmso that the population can be reduced over theiterative process, and fewer evaluations of the ob-jective functions are required, without affecting theconvergence rate. The numerical experiments andcomparisons that were carried out show that themodified EM algorithm is efficient, although a moresignificant reduction on the function evaluations wasexpected. More numerical experiments have to bedone. Another important challenge is to extend thismodified EM algorithm, incorporating the shrinkingstrategy, to problems with large numbers of globaland non-global optima in the region.To reduce the need for large sets of points inthe initial population we intend to apply the number-theoretic method, a deterministic process that pro-duces a set of uniformly scattered points in the fea-sible region [6]. It seems that this new process has theability to explore the search space uniformly and re-quires in practice smaller number of function evalua-tions.-5 -4 -3 -2 -1 0 1 2 3 4 5-5-4-3-2-1012345HumpFigure 5: Original EM algorithm: points in the initialand final populations (H)-5 -4 -3 -2 -1 0 1 2 3 4 5-5-4-3-2-1012345HumpFigure 6: Original EM with shrinking strategy: pointsin the initial and final populations (H)-5 -4 -3 -2 -1 0 1 2 3 4 5-5-4-3-2-1012345HumpFigure 7: EM pattern search algorithm: points in theinitial and final populations (H)-5 -4 -3 -2 -1 0 1 2 3 4 5-5-4-3-2-1012345HumpFigure 8: EM pattern search algorithm with shrinkingstrategy: points in the initial and final populations (H)References:[1] C. Audet and J.E. Dennis, Analysis of general-ized pattern search, SIAM J. Optim. 13, 2003, pp.889–903.[2] S.I. Birbil, Stochastic Global OptimizationTechniques, PhD Thesis, North Carolina StateUniversity 2002.[3] S.I. Birbil SI and S. Fang, An Electromagnetism-like mechanism for global optimization, J. ofGlobal Optimization 25, 2003, pp. 263–282.[4] S.I. Birbil, S. Fang and R. Sheu, On the conver-gence of a population-based global optimizationalgorithm, J. of Global Optimization 30, 2004,pp. 301-318.[5] R. Hooke and T.A. Jeeves, Direct search solutionof numerical and statistical problems, J. Assoc.Comput. Mach. 8, 1961, pp. 212–229.[6] Y-C. Jiao, C. Dang, Y. Leung and Y. Hao, Amodification to the new version of the Pricesalgorithm for continuous global optimizationproblems, J. of Global Optimization 36, 2006,pp. 609-626.[7] T.G. Kolda, R.M. Lewis and V. Torczon, Opti-mization by direct search: new perspectives onsome classical and modern methods, SIAM Re-view 45, 2003, pp. 385–482.[8] R.M. Lewis and V. Torczon, Pattern search al-gorithms for bound constrained minimization,SIAM J. Optim. 9, 1999, pp. 1082–1099.[9] R.M. Lewis and V. Torczon, A globally conver-gent augmented Lagrangian pattern search algo-rithm for optimization with general constraintsand simple bounds, SIAM J. Optim. 12, 2002,pp. 1075–1089.[10] V. Torczon, On the convergence of pattern searchalgorithms, SIAM J. Optim. 7, 1997, pp. 1–25.

A new electromagnetism-like algorithm with a population shrinking strategy

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A new electromagnetism-like algorithm with a population shrinking strategy

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