MEMPSODE: Comparing Particle Swarm Optimization and Differential Evolution on a Hybrid Memetic Global Optimization Framework

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1 MEMPSODE: Comparing Particle Swarm Optimization and Dierential Evolution on a Hybrid Memetic Global Optimization Framework C. Voglis Computer Science Department University o Ioannina Ioannina, Greece voglis@cs.uoi.gr D. G. Papageorgiou Department o Materials Science and Engineering University o Ioannina dpapageo@cc.uoi.gr Drat version G.S. Piperagkas Computer Science Department University o Ioannina gpiperag@cs.uoi.gr I. E. Lagaris Computer Science Department University o Ioannina lagaris@cs.uoi.gr K. E. Parsopoulos Computer Science Department University o Ioannina kostasp@cs.uoi.gr ABSTRACT In this paper we present an experimental comparison between two well known population-based schemes, namely Particle Swarm Optimization (PSO) and Dierential Evolution (DE), that are incorporated in a memetic global optimization ramework. We use the recently published MEMP- SODE sotware [], that implements the memetic global optimization irst described in [] and incorporates Merlin optimization environment []. Since the original description o the algorithm in [] involved only a PSO variant or the exploration phase, using MEMPSODE sotware we attempt an empirical assessment o the DE. The results based on the noiseless testbed, indicate that the usage o DE may lead to superior perormance. Categories and Subject Descriptors G.. [Numerical Analysis]: Optimization global optimization, memetic algorithms, particle swarm optimization, dierential evolution, local search; G. [Mathematical Sotware] Submission deadline: March th. Corresponding author. Permission to make digital or hard copies o all or part o this work or personal or classroom use is granted without ee provided that copies are not made or distributed or proit or commercial advantage and that copies bear this notice and the ull citation on the irst page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior speciic permission and/or a ee. GECCO, July,, Philadelphia, USA. Copyright ACM 9----//...$.. Keywords Memetic global optimization, Particle swarm optimization, Dierential evolution, Benchmarking, Black-box optimization, Merlin optimization environment. INTRODUCTION Evolutionary Algorithms (EAs) and Swarm Intelligence (SI) approaches have been established as powerul optimization tools or solving optimization problems [,, ]. They are based on models that draw their inspiration rom physical systems. Based on natural selection and evolution schemes these algorithms exhibit remarkable capability o locating global solutions or optimization problems. The rise o EA and SI algorithms has sparked the development o a closely related category, namely the Memetic Algorithms (MAs). MAs constitute a class o hybrid meta heuristics that combine population based optimization algorithms with local search procedures [9]. The rationale behind their development was the necessity or powerul algorithms where the global exploration capability o EAs and SI approaches would be complemented with the eiciency and accuracy o classical local optimization techniques. In this work we examine the perormance o a memetic optimization sotware that incorporates Particle Swarm Optimization (PSO) or Dierential Evolution (DE) schemes or exploration in addition to powerul local optimization algorithms or exploitation. Our primary target is to determine the impact o the choice between UPSO and DE in the algorithmic ramework presented in []. By no means we are attempting an extensive benchmark o all MEMPSODE user deined parameters. Instead we use a deault set o parameters both on PSO and on DE and the same local search procedure.. ALGORITHM PRESENTATION The tested sotware ollows closely the PSO based memetic approaches reported in [] and extends them also to the DE

2 ramework. More speciically, the Uniied PSO (UPSO) approach [], which harnesses the strengths o standard local and global PSO variants is implemented. In both cases, direct calls to LS procedures are acilitated via the established Merlin optimization environment [], providing the ability to develop a variety o MAs. In order to present the algorithm implemented by MEMSPODE we provide summary or all key algorithms incorporated. A pseudo-code summarising the methodology is presented in Algorithm.. Uniied Particle Swarm Optimization PSO was introduced by Eberhart and Kennedy []. The main concept o the method includes a population, also called swarm, o search points, also called particles, searching or optimal solutions within the search space, simultaneously. The particles move in the search space by assuming an adaptable position shit, called velocity, at each iteration. Moreover, each particle retains in a memory the best position it has ever visited, i.e., the position with the lowest unction value. To each particle is assigned a neighborhood, which determines the indices o its mates that will share its experience. Obviously, the neighborhood scheme aects the low o inormation among the particles. Two well known neighborhood scheme have been used extensively. The local lbest scheme were each particle is assumed to communicate only with its mates with adjacent indices and the global gbest scheme were any new inormation (best position) is immediately communicated to every single particle in each iteration. Putting our description in a mathematical ramework, let us assume the n dimensional continuous optimization problem: min (x), () x X Rn where the search space X is an orthogonal hyperbox in n : X [l, r ] [l, r ] [l n, r n]. A swarm o N particles is a set o search points: S = {x, x,..., x N }, where the i th particle is deined as: x i = (x i, x i,..., x in) X, i =,,..., N. The velocity (position shit) o x i is denoted as: v i = (v i, v i,..., v in), i =,,..., N, and its best position as: p i = (p i, p i,..., p in) X, i =,,..., N. Let g i = arg min (p j), and t denote the algorithm s iteration counter. j N i The classical PSO model can be generalized in the UPSO scheme []. Following this scheme and i we assume that G (t+) i and L (t+) i denote the velocity update o x i in the gbest and lbest PSO model, respectively: [ ( ij = χ v (t) ij + cr p (t) ij [ ( L (t+) ij = χ v (t) ij + cr p (t) ij G (t+) x(t) ij x(t) ij ) + c r ( ) + c r ( where g is the index o the overall best particle, i.e.: g = arg min (pj), j=,...,n p (t) gj x(t) ij p (t) g i j x(t) ij )] (), )] (), then the particle is updated as ollows []: U (t+) ij = u G (t+) ij + ( u) L (t+) ij, () x (t+) ij = x (t) ij + U (t+) ij, () i =,,..., N, j =,,..., n. The parameter u [, ] is called the uniication actor and it balances the inluence (trade o) o the global and local velocity update. Obviously, the lbest PSO model is retrieved or u =, while or u = the gbest PSO model is obtained. All intermediate values produce combinations with diverse convergence properties.. Dierential Evolution The Dierential Evolution (DE) algorithm was introduced by Storn and Price [] as a population based stochastic optimization algorithm or numerical optimization problems. DE is ormulated similarly to PSO. A population: P = {x, x,..., x N }, o N individuals is utilized to probe the search space, X R n. The population is randomly initialized, usually ollowing a uniorm distribution within the search space. Each individual is an n dimensional vector: x i = (x i, x i,..., x in) X, i =,,..., N, serving as a candidate solution o the problem at hand. The population is iteratively evolved by applying two operators, mutation and recombination, on each individual to produce new candidate solutions. Then, the new and the old individuals are merged and selection takes place to construct the new population consisting o the N best individuals. The procedure continues in the same manner until a termination criterion is satisied. The mutation operator produces a new vector, v i, or each individual, x i, i =,,... N, by combining some o the rest individuals o the population. There most common (but not the only) operator proposed to accomplish this task: OP: v (t+) i = x (t) g + F ( ) x (t) r x (t) r, () where t denotes the iteration counter; F (, ] is a ixed user deined parameter; g denotes the index o the best individual in the population, i.e., the one with the lowest unction value; and r j {,,..., N}, j =,,...,, are mutually dierent randomly selected indices that dier also rom the index i. Thus, in order to be able to apply all mutation operators, it must hold that N >. Ater the mutation, a recombination operator is applied producing a trial vector: u (t+) ij = u i = (u i, u i,..., u in), i =,,..., N, or each individual. This vector is deined as ollows: v (t+) ij, i R j CR or j = RI(i), x (t) ij, i Rj > CR and j RI(i), () where j =,,..., n; R j is a random variable uniormly distributed in the range [, ]; CR [, ] is a user deined crossover constant; and RI(i) {,,..., n}, is a randomly selected index. Finally, each trial vector is compared against the corresponding individual and the best between them comprise

3 the new individual in the next generation, i.e.: ( ) ( ) u (t+) x (t+) i, i u (t+) i < x (t) i, i = x (t) i, otherwise.. Local search A local solution to an optimization problem can be obtained by applying local optimization methods. The hybrid schemes implemented in MEMPSODE have a need or deterministic local search procedures that require a starting point, x, and generate a sequence o points, {x k } k=, in order to determine a minimizer within a prescribed accuracy. The generation o a new point, x k+, in the sequence is based on inormation collected or the current iterate, x k. Typically, this inormation includes the unction value at x k, as well as the irst and probably second order derivatives o (x) at x k. In all cases, the aim is to ind a new iterate with lower unction value than the current one. The Merlin optimization environment [] is an eicient and robust general purpose optimization package. It is designed to solve multi dimensional optimization problems. Merlin oers a variety o well established gradient based and gradient ree optimization algorithms. Gradient based algorithms include three methods rom the conjugate gradient amily, the method o Levenberg-Marquardt, the DFP and several variations o the BFGS algorithms (BFGS) [?]. The gradient ree algorithms include a pattern search and the nonlinear Simplex method.. Memetic Algorithm The design o MPSO in [] was based on three undamental schemes, henceorth called the memetic strategies: Scheme : LS is applied only on the overall best position, p g, o the swarm. Scheme : LS is applied on each locally best position, p i, i =,,..., N, with a prescribed ixed probability, ρ (, ]. Scheme : LS is applied both on the best position, p g, as well as on some randomly selected localy best positions, p i, i {,,..., N}. These schemes can be applied either at each iteration or whenever a speciic number o consecutive iterations has been completed. O course, many other memetic strategies can be considered. For instance, a simple one would be the application o LS on every particle. However, such an approach would be costly in terms o unction evaluations. In practice, only a small number o particles are considered as start points or LS, as pointed out in []. The memetic strategies proposed in [] were also adopted in MEMPSODE or both PSO and DE based MAs.. EXPERIMENTAL PROCEDURE We used the deault restart mechanism provided by the testbed or a maximum number o n unction evaluations. The third memetic scheme was used with probability o local search set to p i =.. Both UPSO and DE used a swarm size N = particles. In UPSO the uniication actor u was set to and the initial velocity vector was restraint by a actor o.. For the DE experiments we applied OP with deault values F =. and CR =.. () Each local seach has an upper limit o unction evaluations. Whenever derivatives were needed (eg. BFGS) we applied an O(h) inite dierences ormula where h is an adaptable step size(see []). For the local search we applied the BFGS method implemented in Merlin. The experiments have been conducted on an Intel I- processor on. GHz with GB RAM.. RESULTS Results rom experiments according to [] on the benchmark unctions given in [, ] are presented in Figures, and and in Tables. The expected running time (ERT), used in the igures and table, depends on a given target unction value, t = opt +, and is computed over all relevant trials as the number o unction evaluations executed during each trial while the best unction value did not reach t, summed over all trials and divided by the number o trials that actually reached t []. A direct comparison between UPSO and DE variants o MEMPSODE memetic algorithm can be deduced by observing the scatter plots in igure and the starred records in table. In the separable case DE variant outperorms UPSO especially as dimensionality increases. For the moderate category DE seems to outperorm only on unction (step ellipsoid) and scores marginally better in all other cases. The same behaviour is repeated or the ill-conditioned cases where DE variant is slightly better. Finally, DE variant outperorms PSO variant in all multimodal unctions but PSO variant seems to behave better or the weak structured cases. DE variant superiority is also obvious by inspecting igure. In almost all cases (except the weak structured unctions) the ECDF o DE variant lies higher than the corresponding ECDF o PSO variant and this pattern is repeated in all levels o accuracy. It is also worth mentioning that the DE variant scored the best recorded ETF or some accuracy levels in the case o ill condition unctions (-). From the corresponding lines o table we can see that or relatively low levels o accuracy the achieved ERT scores are quite competitive. As a general remark, MEMPSODE seems a very promising and competitive new algorithm that incorporates state o the art schemes o swarm intelligence algorithms with the robust and versatile Merlin optimization environment.. REFERENCES [] R. C. Eberhart and J. Kennedy. A new optimizer using particle swarm theory. In Proceedings Sixth Symposium on Micro Machine and Human Science, pages 9, Piscataway, NJ, 99. IEEE Service Center. [] A. P. Engelbrecht. Fundamentals o Computational Swarm Intelligence. Wiley,. [] S. Finck, N. Hansen, R. Ros, and A. Auger. Real-parameter black-box optimization benchmarking 9: Presentation o the noiseless unctions. Technical Report 9/, Research Center PPE, 9. Updated February. [] N. Hansen, A. Auger, S. Finck, and R. Ros. Real-parameter black-box optimization benchmarking : Experimental setup. Technical report, INRIA,.

4 Sphere Ellipsoid separable Rastrigin separable Skew Rastrigin-Bueche separ pso-bgs de-bgs target=e- Linear slope target=e- Attractive sector target=e- Step-ellipsoid target=e- Rosenbrock original target=e- 9 Rosenbrock rotated target=e- Ellipsoid target=e- Discus target=e- Bent cigar target=e- Sharp ridge target=e- Sum o dierent powers target=e- Rastrigin target=e- Weierstrass target=e- Schaer F, condition target=e- Schaer F, condition target=e- 9 Griewank-Rosenbrock FF target=e- Schweel x*sin(x) target=e- Gallagher peaks target=e- Gallagher peaks target=e- Katsuuras target=e- Lunacek bi-rastrigin pso-bgs de-bgs target=e- target=e- target=e- target=e- Figure : Expected running time (ERT in number o -evaluations) divided by dimension or target unction value as log values versus dimension. Dierent symbols correspond to dierent algorithms given in the legend o and. Light symbols give the maximum number o unction evaluations rom the longest trial divided by dimension. Horizontal lines give linear scaling, slanted dotted lines give quadratic scaling. Black stars indicate statistically better result compared to all other algorithms with p <. and Bonerroni correction number o dimensions (six). Legend: : pso-bgs, : de-bgs.

5 Sphere Sphere Linear slope Linear slope 9 Rosenbrock rotated 9 Rosenbrock rotated Sharp ridge Sharp ridge Schaer F, cond. Schaer F, condition Gallagher peaks Gallagher peaks Ellipsoid separable Ellipsoid separable Attractive sector Attractive sector Ellipsoid Ellipsoid Sum o di. powers Sum o dierent powers Schaer F, cond. Schaer F, condition Gallagher peaks Gallagher peaks Rastrigin separable Rastrigin separable Step-ellipsoid Step-ellipsoid Discus Discus Rastrigin Rastrigin 9 Griewank-Rosenbrock 9 Griewank-Rosenbrock FF Katsuuras Katsuuras Skew Rastrigin-Bueche Skew Rastrigin-Bueche separ Rosenbrock original Rosenbrock original Bent cigar Bent cigar Weierstrass Weierstrass Schweel x*sin(x) Schweel x*sin(x) Lunacek bi-rastrigin Lunacek bi-rastrigin Figure : Expected running time (ERT in log o number o unction evaluations) o pso-bgs (x-axis) versus de-bgs (y-axis) or target values [, ] in each dimension on unctions. Markers on the upper or right edge indicate that the target value was never reached. Markers represent dimension: :+, :, :, :, :, :.

6 all unctions o trials D -D +: / -: / -: / -: /9 o trials : 9/ -: /9 -: / -: 9/ separable cts moderate cts ill-conditioned cts multi-modal cts weak structure cts o trials o trials o trials o trials o trials. log o FEvals / DIM log o FEvals / DIM log o FEvals / DIM log o FEvals / DIM log o FEvals / DIM log o FEvals / DIM log o FEvals ratio +: / -: / -: / -: / log o FEvals ratio +: / -: / -: / -: / log o FEvals ratio +: / -: / -: / -: / log o FEvals ratio +: / -: / -: / -: / log o FEvals ratio +: / -: / -: / -: / log o FEvals ratio o trials o trials o trials o trials o trials. log o FEvals / DIM log o FEvals / DIM log o FEvals / DIM log o FEvals / DIM log o FEvals / DIM log o FEvals / DIM log o FEvals ratio +: / -: / -: / -: / log o FEvals ratio +: / -: / -: / -: / log o FEvals ratio +: / -: / -: / -: / log o FEvals ratio +: / -: / -: / -: / log o FEvals ratio +: / -: / -: / -: / log o FEvals ratio Figure : Empirical cumulative distributions (ECDF) o run lengths and speed-up ratios in -D (let) and - D (right). Let sub-columns: ECDF o the number o unction evaluations divided by dimension D (FEvals/D) to reach a target value opt + with = k, where k {,,, } is given by the irst value in the legend, or pso-bgs ( ) and de-bgs ( ). Light beige lines show the ECDF o FEvals or target value = o all algorithms benchmarked during BBOB-9. Right sub-columns: ECDF o FEval ratios o pso-bgs divided by de-bgs, all trial pairs or each unction. Pairs where both trials ailed are disregarded, pairs where one trial ailed are visible in the limits being > or <. The legends indicate the number o unctions that were solved in at least one trial (pso-bgs irst).

7 -D -D e+ e- e- e- e- #succ e+ e- e- e- e- #succ / / : pso.9(.).... / : pso.().(.).(.).(.).(.) / : de.(.).... / : de.().9(.).9(.).9(.).9(.) / / / : pso.().().().() () / : pso.(.).(.).9(.).(.).(.) / : de.().().().() () / : de.(.).(.).(.).9(.).(.) / / / : pso.e / : pso 9.() () (9) () () / : de.().().().().() : de () / (9) () () 9(9) / 9 9 /.e 9/ : pso.e / : pso () (9) () (99) (9) / : de.() () 9() () () : de () / 9() (9) (9) () / / / : pso () () () () () / : pso.(.).(.).9(.).9(.).9(.) / : de () () () () () / : de.() 9.() 9.() 9.() 9.() / / 9 9 / : pso 9.() () (9) 9() (9) / : pso.(9).().9().() (9) / : de.() () () 9() 9() / : de.().(.9).(.).().() / 9 / 9 99 / : pso.e / : pso9(9) 9(9) (9) () () / : de.9() () 9.() 9.() () : de () /.e / 9 / 9 9 / : pso.().().().() 9.() / : pso.().(.).(.).(.).(.) / : de.().().().().() / : de.().(.).(.).(.).(.) / 9 9 / 9 9 / : pso.(.).().().9().9() / : pso.().(.9).(.).(.).(.) / : de.(.).9(.9).(.).(.).(.) / : de.().().().().() / 9 9 / 9 / : pso.().(.9).9(.).(.).(.) / : pso 9() () () () () / : de.(.).9(.).(.).9(.).(.9) : de.().().().().() / / / 9 / : pso.(.).(.).(.).9(.).(.) : pso.(.).(.).(.).(.).(.) / / : de.(.).(.).(.).(.).(.) : de.(.).(.).(.).(.).(.) / / 9 / / : pso.().().().() () / : pso.().().().(.).(.9) / : de.(.).().().(.) : de.().9().9().(.).(.) /.() / / 9 / : pso.99(.).(.).(.) : pso.(.).(.) ().e /.(.) ().e / : de.(.).(.).(.).(.) 9(9) : de.(.).(.) /.(.) 9().e / 9 / 9 / : pso.().(.).9(.).(.) () / : pso.(.).9(.).(.).(.) 9(9) / : de.().(.).9(.).(.).() / : de.(.).9(.).(.).(.) () / /.e.e.e.e / : pso () 9() () () () / : pso.e / : de.().().().().() / : de ().e / 9 9 /.9e.e.e / : pso.9() (9) () 9() 9() / : pso ().e / : de.() (9) () () () / : de 9().e / / / : pso.(.) (9) () 9.() 9(99) / : pso 9().e / : de.() () ().9() () / : de (9) (9) 9().e / / 9 9.e.e / : pso () () () ().e / : pso (9).e / : de.() () 9() () () / : de 9() (9).e / 9.e.e.e / 9.e.e.e.e / : pso 9(9) ().().() 9(9) / : pso().().e / : de () ().().() () / : de ().().e / / : pso.9(.9).().().().() /.e.e.e.e / : pso.().e / : de.() ().().(9).() / : de.().e / 9 / : pso.().().().() 9() / 9 / : pso.().(.9).(.9).() 9(9) / : de.().().().() () / : de () () () () () / 9 / : pso.().().().() () / 9 9.e / : de.() () () () () / : pso.() () 9.().() 9() / : de.() () () ().e /. 9 / : pso.().() ().e /..9e.e.e / : de.().().(.9).(.9) 9() : pso.() ().e / / : de.() ().e /.e 9.e.e.e / : pso.().(.).(.).(.).(.) /.e.e.e.e.e / : pso.e / : de.().9(.).(.).(.).(.) / : de.().e / Table : ERT in number o unction evaluations divided by the best ERT measured during BBOB-9 given in the respective irst row and the hal inter-%ile in brackets or dierent values. #succ is the number o trials that reached the inal target opt+. :pso is pso-bgs and :de is de-bgs. Bold entries are statistically signiicantly better compared to the other algorithm, with p =. or p = k where k {,,,... } is the number ollowing the symbol, with Bonerroni correction o. A indicates the same tested against the best BBOB-9.

8 [] N. Hansen, S. Finck, R. Ros, and A. Auger. Real-parameter black-box optimization benchmarking 9: Noiseless unctions deinitions. Technical Report RR-9, INRIA, 9. Updated February. [] W. E. Hart. Adaptive Global Optimization with Local Search. PhD thesis, University o Caliornia, San Diego,USA, 99. [] J. Kennedy and R. C. Eberhart. Particle swarm optimization. In Proc. IEEE Int. Con. Neural Networks, volume IV, pages 9 9, Piscataway, NJ, 99. IEEE Service Center. [] J. Kennedy and R. C. Eberhart. Swarm Intelligence. Morgan Kaumann Publishers,. [9] P. Moscato. Memetic algorithms: A short introduction. In D. Corne, M. Dorigo, and F. Glover, editors, New Ideas in Optimization, pages 9. McGraw Hill, London, 999. [] D. Papageorgiou, I. Demetropoulos, and I. Lagaris. MERLIN-... A new version o the Merlin optimization environment. Computer Physics Communications, 9():,. [] K. E. Parsopoulos and M. N. Vrahatis. UPSO: A uniied particle swarm optimization scheme. In Lecture Series on Computer and Computational Sciences, Vol., Proceedings o the International Conerence o Computational Methods in Sciences and Engineering (ICCMSE ), pages. VSP International Science Publishers, Zeist, The Netherlands,. [] K. E. Parsopoulos and M. N. Vrahatis. Particle Swarm Optimization and Intelligence: Advances and Applications. Inormation Science Publishing (IGI Global),. [] Y. G. Petalas, K. E. Parsopoulos, and M. N. Vrahatis. Memetic particle swarm optimization. Annals o Operations Research, ():99,. [] R. Storn and K. Price. Dierential evolution a simple and eicient heuristic or global optimization over continuous spaces. J. Global Optimization, : 9, 99. [] C. Voglis, P. Hadjidoukas, I. Lagaris, and D. Papageorgiou. A numerical dierentiation library exploiting parallel architectures. Computer Physics Communications, ():, 9. [] C. Voglis, K. Parsopoulos, D. Papageorgiou, I. Lagaris, and M. Vrahatis. Mempsode: A global optimization sotware based on hybridization o population-based algorithms and local searches. Computer Physics Communications, ():9,. Algorithm : Pseudocode o the implemented memetic algorithm Input: Objective unction, : S R n R; algorithm PSO/DE: algo ; swarm size: N; memetic strategy: memetic, maximum unction evaluations: maxev; uniication actor: UF; use mutation: mut; probability or local search: ρ Output: Best detected solution: x, (x ). // Initialization or i =,,..., N do Initialize position x i and velocity u i Set p i x i // Initialize best position i (x i()) // Evaluate particle p i i // Best position local i // Best position is minimum is set to alse end // Update Best Indices Calculate global best index g and local best index g // Main Iteration Loop 9 Set t while termination criterion do // Update Swarm i algo = pso then or i =,,..., N do Calculate local best velocity update or particle u l i using g Calculate global best velocity update or particle u g i using g i mut = then // Uniied PSO with mutation R N(µ, σ) i rand(). then u i RUFu l i + ( UF)ug i // Uniied PSO + Mutate local term 9 else u i UFu l i + R( UF)ug i // Uniied PSO + Mutate global term end else u i UFu l i + ( UF)ug i // Uniied PSO end x i = x i + u i // Update particle s position end else i algo = de then or i =,,..., N do 9 x i p i // Replicate best positions p to swarm array x end or i =,,..., N do Calculate u i using a strategy rom Eqs. () (??) i rand() C then x i u i else x i p i end end 9 end // Evaluate Swarm or i =,,..., N do i (x i) // Evaluate particle end // Update Best Positions or i =,,..., N do i i < (p i) then p i x i p i i end end 9 Calculate global best index g and local best index g Apply one o the schemes using ρ Calculate global best index g and local best index g // I all best positions are local minima, restart i N i= locali = N then Keep global best particle and reinitialize the swarm end end