A fish swarm intelligence algorithm based on the filter set concept to accept, at each iteration, a population of trial solutions whenever they improve constraint violation or objective function, relative to the current solutions, is proposed for constrained global continuous optimization problems. Preliminary numerical results are provided.Fundação para a Ciência e a Tecnologia (FCT

Costa, M. Fernanda P.

Fernandes, Edite Manuela da G. P.

Rocha, Ana Maria A. C.

Universidade do Minho: RepositoriUM

Proceedings of GOW 2012, pp. 113 – 116.AFSFilter: Artificial Fish Swarm Filter-Based Algorithmfor Global Optimization∗Ana Maria A.C. Rocha,1 M. Fernanda P. Costa,2 and Edite M.G.P. Fernandes11Algoritmi R&D Centre, U Minho, 4710-057 Braga, Portugal, {arocha,emgpf}@dps.uminho.pt2Mathematics R&D Centre, U Minho, Braga, Portugal, mfc@math.uminho.ptAbstract A fish swarm intelligence algorithm based on the filter set concept to accept, at each iteration, apopulation of trial solutions whenever they improve constraint violation or objective function, rela-tive to the current solutions, is proposed for constrained global continuous optimization problems.Preliminary numerical results are provided.Keywords: Global optimization, Fish Swarm Intelligence, Filter method1. IntroductionThe problem to be addressed in this paper has the formminx∈Ωf(x) , subject to gj(x) ≤ 0 , j = 1, . . . ,m (1)where at least one of the functions f, gj : Rn → R, is nonlinear and Ω = {x ∈ Rn : lk ≤xk ≤ uk, k = 1, . . . , n}. Problems with equality constraints can be reformulated in the aboveform using a small tolerance. When convexity is not assumed, problem (1) may have multipleoptimal solutions in Ω. This paper aims at proposing a stochastic method to compute a globalsolution of (1). From the class of stochastic methods, swarm intelligence algorithms haveshown to be effective in reaching a global solution. Recent studies involving the artificialfish swarm (AFS) algorithm show that highly accurate solutions may be obtained with reducedcomputational costs [6, 7]. Although penalty function methods are probably the most knownconstraint handling techniques, a penalty function depends, in general, on a penalty parameter.Unfortunately, the performance of these algorithms depends strongly on the values set to thepenalty parameter throughout the iterative procedure. Adaptive penalties [9] and augmentedLagrangian methodologies [2, 8] are just recent strategies to overcome partially this issue. Theseparate use of objective function and constraint violation with the nondominance concept frommultiobjective programming, for example in [1], avoids the use of penalty parameters. Fletcherand Leyffer [4] proposed a filter method as an alternative to penalty functions to guaranteeconvergence to optimizers in nonlinear constrained optimization. This technique incorporatesthe concept of nondominance to build a filter set that is able to accept solutions if they improveeither the objective function or the constraint violation, instead of a linear combination of thosetwo measures.∗The financial support from FEDER COMPETE (Programa Operacional Fatores de Competitividade / OperationalProgramme Thematic Factors of Competitiveness) and FCT (Fundação para a Ciência e a Tecnologia / PortugueseFoundation for Science and Technology) Project FCOMP-01-0124-FEDER-022674 is gratefully acknowledged.114 Ana Maria A.C. Rocha, M. Fernanda P. Costa, and Edite M.G.P. FernandesIn this paper, an artificial fish swarm filter-based algorithm, hereafter denoted by AFSFilter,for nonlinear constrained global optimization problems is proposed. Results from preliminarynumerical experiments are provided.2. Artificial Fish Swarm AlgorithmHere is some notation used in the paper. Constraint violation of a point x is measured by thefunctionθ(x) =m∑j=1max {0, gj(x)}+n∑k=1(max {0, xk − uk}+max {0, lk − xk}) , (2)xi ∈ Rn represents the ith point of a population of size p, and xbest is the best point in thepopulation. Pairwise comparisons in the population use the following concept: between twopoints xi and xj , xi is better than xj if the following condition holds:θ(xi) < θ(xj) or(θ(xi) = θ(xj) and f(xi) < f(xj)). (3)In the AFS algorithm, the initial population of p points is randomly generated inside theset Ω. A crucial quantity of the algorithm is the ‘visual scope’ of a point, say xi. This isdefined as the closed neighborhood with center xi and radius equal to a positive quantityv = ς maxk∈{1,...,n}(uk − lk), where ς is a positive visual parameter. Let ni be the number ofpoints in its ‘visual scope’ (ni < p). If the condition ni/p ≤ κ holds, where κ ∈ (0, 1] is thecrowd parameter, the ‘visual scope’ of xi is said to be not crowded. Depending on the relativeposition of the points in the population, one of the following three situations occurs.1. When ni = 0, the ‘visual scope’ is empty, and the point xi, with no other points in itsneighborhood to follow, has a random behavior. Here, a point is randomly generated inthe search space, xr, and a movement is tried along the direction defined by d = xr − xi.2. When the ‘visual scope’ is crowded, the point has some difficulty in following any particularpoint, and starts by following a searching behavior. A point inside the ‘visual scope’ israndomly generated, xs, and a movement towards it is carried out if xs is better than xi(see condition (3)); otherwise, xi moves according to a random behavior.3. When the ‘visual scope’ is not crowded, the point firstly tries the chasing behavior movingtowards the best point inside the ‘visual scope’, denoted by xmin, if this is better thanxi, thus being the direction of movement d = xmin − xi. Otherwise, the point tries tofollow the swarming behavior moving towards the central point, c, of the ‘visual scope’.However, if c is not better than xi, the point tries to follow a searching behavior; andif that randomly generated point xs is not better than xi, the point follows a randombehavior.The algorithm also implements an elitism procedure in the sense that the best point of thepopulation is not moved and is maintained throughout the iterative process. For each currentpoint xi of the population, the trial point yi is generated according to a direction d and a stepsize α ∈ (0, 1]yi = xi + αd , i = 1, . . . , p and i 6= best. (4)The procedure that decides if the trial solution is to be accepted and replaces the current pointis a filter method combined with a backtracking line search, as described in the next section.The algorithm terminates with a successful run if the stopping conditions∣∣∣f(xbest)− f∗∣∣∣ ≤ ǫ1 |f∗|+ ǫ2 and θ(xbest) ≤ ǫ2AFSFilter: Artificial Fish Swarm Filter-Based Algorithm 115are satisfied, for small positive tolerances ǫ1, ǫ2; otherwise xi ← yi for all i 6= best and theprocedure is repeated until a maximum number of iterations is reached, where f∗ is the bestglobal solution of the problem available in the literature.3. The Implemented Filter MethodologyThis section briefly describes the filter methodology that aims at deciding which trial solutionis to be accepted in the sequence of Eq. (4). The herein proposed AFSFilter method usesthe filter set concept, as outlined in [3, 4], with the ability of exploring both feasible andinfeasible regions, and building a filter set that is able to accept a trial point if it improveseither the objective function or the constraint violation, relative to the current point. Filter-based algorithms treat the optimization problem as a biobjective problem aiming to minimizeboth the objective function and the nonnegative constraint violation function (2).After a search direction d has been computed, A decreasing sequence of step sizes {αj} withlimj αj = 0 is tried, until a set of acceptance conditions are satisfied. This j denotes theiteration counter for the inner loop. A trial step size αj might be accepted if the correspondingtrial point yi = xi +αj d is acceptable by the filter. We only require an improvement in θ or inf , relative to the current point xi, to consider the trial point yi, in Eq. (4), to be acceptable,as shown:θ(yi) < θ(xi) or f(yi) < f(xi). (5)However, when xi is (almost) feasible, the trial point yi has to satisfy only the condition ofsimple reduction on f :f(yi) < f(xi) (6)to be acceptable. To prevent cycling between points that improve either θ or f , at each iteration,the algorithm maintains a filter F which is defined as a finite set of entries (θ(xj), f(xj)) thatcorrespond to a collection of infeasible solutions xj such that no filter entry is dominated by anyof the others in the filter. During the backtracking line search procedure, the yi is acceptableonly if (θ(yi), f(yi)) /∈ F . (Only solutions that are not dominated by any entry in the filtermight be accepted.)The filter is initialized with entries (θ, f) that satisfy θ ≥ θmax, where θmax > 0 is the upperbound on θ. Furthermore, the filter is augmented whenever yi is accepted because condition(5) is satisfied. When it is not possible to find a point yi with a step size αj > αmin > 0that satisfies one of the conditions (5) or (6), a restoration phase is invoked. In this phase,the algorithm performs a coordinate random local search around the best point, with length10−3maxk{uk − lk}, to find a point inside [l, u] that is acceptable to the filter. If no such pointis found, the algorithm maintains the current point to the next iteration.4. Preliminary ResultsTable 1 contains the numerical results of our preliminary experiments with the AFSFiltermethod. Three well-known engineering design problems are used in the comparison with theresults obtained by the Filter Simulated Annealing (SA) method proposed in [5]. The welded‘beam’ design problem has four design variables and seven inequality constraints, the ten-sion/compression ‘spring’ design problem has three continuous variables and four inequalityconstraints and the cylindrical ‘vessel’ design problem has four design variables (two of themare multiples of 0.0625) and four inequality constraints [5]. The size of the population is set top = 5n and the algorithm was allowed to run for a maximum of 200 iterations. A comparisonwith the pattern search hybrid GA from MatLabTM (with the tournament selection option tohandle constraints) is also provided. A set of four small but difficult problems, with n = 2,116 Ana Maria A.C. Rocha, M. Fernanda P. Costa, and Edite M.G.P. Fernandesselected from [2] and a technical report from the same authors1, is also tested and the resultsare reported in the table for comparison with GA. The size of the population is set to p = 20and a maximum of 50 iterations is allowed.Each problem was run 30 times and the results reported in the table are: ‘f best’, the bestsolution obtained during all the runs and ‘avg.n.f.e.’, the average number of function evaluations(from the 30 runs). Other parameter values are set as follows: ǫ1 = 10−6, ǫ2 = 10−8, θmax = 104,αmin = 10−3 and ς is set to one and is reduced every p iterations until it reaches 0.1. We mayconclude that the proposed AFSFilter method is effective in reaching a global optimal solutionwith reasonable computational costs. New developments with further experimentation willfollow.Table 1. Comparison of AFSFilter with Filter SA in [5] and Hybrid GA in MatLab.Prob. AFSFilter Filter SA in [5] Hybrid GA from MatLabf∗ fbest avg.n.f.e. fbest avg.n.f.e. fbest avg.n.f.e.beam 2.38081 2.3866641 65 687 2.381065 56 243 2.5526753 168 119spring 0.012664 0.0126653 35 929 0.0126653 49 531 0.0126663 3 480vessel 5854.738 5868.974 45 283 5868.765 108 883 5859.977 21 289Example 1 in [2] -1.0000000 -0.9983634 7 906 n.a. n.a. -0.9999910 4 027Example 3 in [2] 1.0000000 1.0000013 10 849 n.a. n.a. 0.9999987 5 366Example 5 in [2] -2.0000000 -2.0000330 8 180 n.a. n.a. -2.0000043 51 067Problem 6† -9.4772944 -9.4772842 9 458 n.a. n.a. -3.2883714‡ 5 108† Technical report MCDO-051015 (2005) in http://www.ime.usp.br/∼egbirgin/, pre-print to [2]‡ This is a local optimal solutionReferences[1] Ali, M.M., Golalikhani, M. (2010). An electromagnetism-like method for nonlinearly constrained globaloptimization. Computers and Mathematics with Applications, 60, 2279–2285[2] Andreani, R., Birgin, E.G., Martinez, J.M., Schuverdt, M.L. (2007). On augmented Lagrangian methodswith general lower-level constraints. SIAM Journal on Optimization, 18(4), 1286–1309[3] Audet, C., Dennis, Jr., J.E. (2004). A pattern search filter method for nonlinear programming withoutderivatives. SIAM Journal on Optimization, 14(4) 980–1010[4] Fletcher, R., Leyffer, S. (2002). Nonlinear programming without a penalty function. Mathematical Program-ming, 91, 239–269[5] Hedar, A.-R., Fukushima, M. (2006). Derivative-free filter simulated annealing method for constrained con-tinuous global optimization. Journal of Global Optimization, 35, 521–549[6] Gao, X.Z., Wu, Y., Zenger, K., Huang, X. (2010). A knowledge-based artificial fish-swarm algorithm. 13thIEEE International Conference on Computational Science and Engineering, 327–332[7] Rocha, A.M.A.C., Fernandes, E.M.G.P., Martins, T.F.M.C. (2011). Novel fish swarm heuristics for boundconstrained global optimzation problems. ICCSA 2011, Part III, Lecture Notes in Computer Science 6784,B. Murgante et al. (Eds.) 185–199[8] Rocha, A.M.A.C., Martins, T.F.M.C., Fernandes, E.M.G.P. (2011). An augmented Lagrangian fish swarmbased method for global optimization. Journal of Computational and Applied Mathematics, 235(16) 4611–4620[9] Silva, E.K., Barbosa, H.J.C., Lemonge, A.C.C. (2011). An adaptive constraint handling technique for dif-ferential evolution with dynamic use of variants in engineering optimization. Optimization and Engineering,12, 31–541Technical report MCDO-051015 (2005) in http://www.ime.usp.br/∼egbirgin/.

AFSFilter: artificial fish swarm filter-based algorithm for global optimization

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AFSFilter: artificial fish swarm filter-based algorithm for global optimization

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