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

On a Hyperbolic Augmented Lagrangian Artificial FishSwarm Based Method: Convergence IssuesAna Maria A.C. Rocha1;3, M. Fernanda P. Costa2, Edite M.G.P. Fernandes31 Department of Production and Systems, University of Minho, Campus deGualtar, 4710-057 Braga, Portugal arocha@dps.uminho.pt2 Department of Mathematics and Applications and Centre of Mathematics,University of Minho, Guimara˜es, Portugal mfc@math.uminho.pt3 Algoritmi R&D Centre, University of Minho, 4710-057 Braga, Portugalemgpf@dps.uminho.ptExtended Abstract1 IntroductionThe problem to be addressed in this paper has the general form:minx2f(x) subject to g(x)  0 (1)where f : Rn ! R and g : Rn ! Rp are nonlinear continuous functions and = fx 2 Rn :  1 < l  x  u < 1g. Problems with equality constraints,h(x) = 0, can be reformulated into the above form by converting into a couple ofinequality constraints h(x)    0 and  h(x)    0, where  is a small posi-tive relaxation parameter. Since we do not assume that the objective function f isconvex, the problem may have multiple optimal solutions in the feasible region. Aglobal optimal solution is to be computed. For this class of optimization problems,methods based on penalty functions are quite common in the literature. In this typeof methods, the constraint violation is combined with the objective function to de-fine a penalty function, which aims at penalizing infeasible solutions by increasingtheir fitness values proportionally to their level of constraint violation. We are in-terested in a particular class of penalty functions known as augmented Lagrangianfunctions to handle the equality and inequality constraints of the problem (1). Anaugmented Lagrangian is a more sophisticated penalty function for which a finitepenalty parameter value is sufficient to yield convergence to the solution of theconstrained problem.We aim at analyzing the theoretical and practical behavior of an augmentedLagrangian algorithm that is constructed following the most common augmentedLagrangian paradigm [1, 2, 4]. The penalty function basis of our proposal is theexterior penalty function presented in [5], the so-called 2-parameter hyperbolic1penalty function. This is a continuously differentiable function that depends ontwo penalty parameters   0 and   0, whose penalty term is:P(x; ; ) =pXi=1gi(x) +q2 (gi(x))2 + 2: (2)2 Hyperbolic Augmented Lagrangian methodologyAn augmented Lagrangian technique solves a sequence of very simple subprob-lems where the objective function penalizes all or some of the constraint violation.Our implementation of the hyperbolic augmented Lagrangian framework aims topenalize the inequality constraints of the nonlinear optimization problem (1). Thepenalty term of the hyperbolic (2) is herein reformulated so that a new penalty pa-rameter  emerges as  =and  > 0. The simple bounds are not includedin the augmented Lagrangian although the solution method to solve the subprob-lem (see (3) below) guarantees that they are satisfied. Thus, each subproblem ofthe sequence that is solved for fixed values of the penalties  and  is a boundconstrained optimization problem of the form:minx2(x; ; ; )  f(x) +pXi=1igi(x) + pXi=1gi(x) +q(gi(x))2 + 2(3)where  = (1; : : : ; p)T is the Lagrange multiplier vector associated with the in-equality constraints. Another important issue in the augmented Lagrangian contextis concerned with the selection of the method to compute an approximate solutionto the subproblem (3), within a specified ‘optimal’ tolerance. Since a global solu-tion of problem (1) is to be computed, a global optimization solution method forthe subproblem (3) is to be provided.A formal description of the hyperbolic augmented Lagrangian algorithm forsolving problem (1) is presented in Algorithm 1.We note that in the presented algorithm, the penalty parameter  is reduced atall iterations aiming to converge to 0. Although the penalty  is to be increasedas k ! 1, to avoid an ill-conditioned subproblem, its value is not updated when-ever complementarity, as well as feasibility, improve. Another important issueaddressed in this paper is the estimation of the multiplier vector. A safeguardedfirst-order Lagrange multiplier formula is used, so that multiplier vector bounded-ness is guaranteed.2.1 Artificial Fish Swarm AlgorithmTo solve subproblem (3) the artificial fish swarm (AFS) algorithm is used. This isa stochastic population based method. At each iteration, a population of NP solu-tions, herein denoted by xi, i = 1; : : : ; NP , are used to generate a trial population2Algorithm 1 Hyperbolic augmented Lagrangian algorithm1: Given initial values and parameters;2: Set k = 0;3: while feasibility and complementarity are not achieved do4: Find an approximate global minimizer x(k) of subproblem (3), within an‘optimal’ tolerance, using Algorithm 2;5: Estimate the multiplier vector ;6: Reduce the penalty ;7: if feasibility and complementarity have been improved then8: Maintain the value of penalty  ;9: else10: Increase the value of penalty  ;11: end if12: Reduce ‘optimal’ tolerance;13: k = k + 1;14: end whileyi, i = 1 : : : ; NP . Initially, the population is randomly generated in the entiresearch space . Each fish/point movement is defined according to the number ofpoints inside its visual. The visual is the closed neighborhood centered at xi witha radius v > 0. The three possible situations that may occur are:i) the visual is empty;ii) the visual is crowded;iii) the visual is not crowded.When the visual is empty, a random behavior is performed, in which the trial yiis randomly generated inside the visual of xi. When the visual is crowded, and apoint randomly selected from the visual has a better fitness than xi, the searchingbehavior is implemented, i.e., yi is randomly generated along the direction fromxi to that selected point. Otherwise, the random behavior is performed. When thevisual is not crowded, and the best point inside the visual has a better fitness thanxi, the chasing behavior is performed. This means that yi is randomly generatedalong the direction from xi to that best point. However, if that best point is notbetter than xi, the swarming behavior is tried instead. Here, the central point ofthe visual is computed and if it has better fitness than xi, yi is computed randomlyalong the direction from xi to the central; otherwise, the searching behavior isimplemented instead [3].When selecting between the current xi and the trial yi, the solution with bestobjective function value (fitness) wins. The pseudo code of this algorithm is shownbelow in Algorithm 2.3Algorithm 2 Artificial fish swarm algorithm1: Generate a population of NP points: xi, i = 1; : : : ; NP ;2: while an approximate minimizer of subproblem (3) is not reached do3: for all xi do4: Compute visual;5: Perform accordingly chasing, swarming, searching or random behavior tofind yi;6: Select between yi and xi;7: end for8: end while2.2 Convergence to Global MinimizersThe global convergence of the hyperbolic augmented Lagrangian algorithm hasbeen analyzed. The analysis has some similarities with that of the global aug-mented Lagrangian method presented in [2] and relies on the fact that, at eachiteration k, an approximate global minimizer of the subproblem (3) is obtained,x(k), within a certain ‘optimal’ tolerance. Based on the following assumptions a global minimizer z of the problem (1) exists; the sequence fx(k)g generated by the Algorithm 1 is well defined and thereexists a set of indices N  N so that limk2N x(k) = x; The functions f : Rn ! R and g : Rn ! Rp are continuous in a neighbor-hood of x; for all i = 1; : : : ; p, the optimal multipliers i 2 [0; max]; the sequence of penalty parameters f (k)g tends to1; the sequence of penalty parameters f(k)g tends to zero;we prove that every limit point x of the sequence fx(k)g generated by the Al-gorithm 1 is feasible and is an approximate global minimizer of the problem (1),within a sufficiently small ‘optimal’ tolerance.A stochastic algorithm for solving an optimization problem of the type (3) issaid to exhibit global convergence when the sequence of solutions generated bythe algorithm converges in some stochastic manner to the global optimum. SinceAFS algorithm generates random sequences of solutions, convergence here meansstochastic convergence that relies on a probability measure and is different fromthe convergence concept of classical analysis. There are different types of stochas-tic convergence for a sequence of random solutions. Convergence with probabilityone (or almost sure convergence), convergence in probability, convergence in meansquare, and convergence in distribution are the most known. Until now, conver-gence of the fish swarm iterative process in mean square has been guaranteed [4].43 Discussion of Preliminary Numerical ResultsWe consider the below example to analyze the behavior of the algorithm whendealing with both equality and inequality constraints:min f(x)  sin3(2x1) sin(2x2)x31(x1+x2)subject to x21   x2 + 2:272  01  x1 + (x2   4)2 = 00  xi  10; i = 1; 2:The global optimum is f =  0:047764467932642 located at (1.1550767073652,3.6062021998978), where both constraints are active. This example was solved 10times by the presented algorithm with NP = 50, and the best obtained solution(‘fbest point’) is  0:047762. During the best of the 10 runs, the hyperbolic aug-mented Lagrangian algorithm completed 10 iterations, after 9 166 function eval-uations (‘n.func.eval.’). Algorithm 2 was allowed to run for 20 iterations, at eachouter iteration. For the augmented Lagrangian shown in (3), the equality was trans-formed into an inequality using  = 0:00001.0 20 40 60 80 100 120 140 160 180−1012AFS algorithm total iterationsConstraints  051015inequalityequality1 2 3 4 5 6 7 8 9 1005001000k  (iteration from the augmented Lagrangian algorithm)Penalty parameters  0510penalty parameter τpenalty parameter µτ µFigure 1: Constraints behavior (left) and penalty parameters behavior (right).From Figure 1 - plot on the left - we may observe that the feasible region isreached within k = 7 outer iterations (equivalent to 140 AFS algorithm total iter-ations), with a sum of violations (‘violation’) equal to 2.545298E-05. The plot onthe right shows the values of the penalty parameters  and  at every outer iteration.Their convergence behavior is as expected. Figure 2 aims to depict the initial pop-ulation at the beginning of the iterative process and the population after completingsix iterations. In this figure, we show the values of ‘n.func.eval.’, ‘fbest point’, ‘viola-tion’,  and  attained at iterations k = 0 and k = 6. It is also shown that the initialpopulation is nicely spread over the set [0; 10]2 and the minimizer is attracting thepoints of the population.50 1 2 3 4 5 6 7 8 9 10012345678910k=0n.func.eval.=50     fbest point=  −18.039778    violation=8.421006E−001τ=1.000000E+000    µ=1.000000E+001x1x 2minimizer1 1.1 1.2 1.3 1.4 1.5 1.6 1.73.544.55k=6n.func.eval.=6160     fbest point=   −0.046723    violation=7.869158E−004τ=6.400000E+001    µ=1.562500E−01x1x 2minimizerFigure 2: Minimizer and population at k = 0 (left) and k = 6 (right).AcknowledgmentsThis work was financed by FEDER funds through COMPETE-Programa Opera-cional Fatores de Competitividade and by portuguese funds through FCT-Fundac¸a˜opara a Cieˆncia e a Tecnologia, within projects FCOMP-01-0124-FEDER-022674and PEst-C/MAT/UI0013/2011.References[1] D.P. Bertsekas: Nonlinear Programming, Athena Scientific, Belmont, 2nd Edi-tion, 1999.[2] E.G. Birgin, C.A. Floudas, J.M. Martinez, Global minimization using an Aug-mented Lagrangian method with variable lower-level constraints, Mathemati-cal Programming, Ser. A, 125(1), pp. 139–162 (2010).[3] A.M.A.C. Rocha, E.M.G.P. Fernandes, T.F.M.C. Martins, Novel fish swarmheuristics for bound constrained global optimization problems. Lecture Notesin Computer Science, Vol 6784, Part III, B Murgante et al. (eds.) pp. 185–199(2011).[4] A.M.A.C. Rocha, T.F.M.C. Martins, E.M.G.P. Fernandes, An augmented La-grangian fish swarm based method for global optimization, Journal of Compu-tational and Applied Mathematics, 235(16), pp. 4611–4620 (2011).[5] A.E. Xavier, Hyperbolic penalty: a new method for nonlinear programmingwith inequalities, International Transactions in Operational Research 8, pp.659–671 (2001).6

On a hyperbolic augmented lagrangian artificial fish swarm based method: convergence issues

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On a hyperbolic augmented lagrangian artificial fish swarm based method: convergence issues

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