Constraint Handling Approaches in Visual Studio .NET

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Constraint Handling Approaches
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While a number of ES variations have been developed to cope with constraints, this section discusses only some of these approaches Tahk and Sun [830] converted the constrained problem to an unconstrained problem using the augmented Lagrangian approach given in Section A62 A coevolutionary approach is used to nd the saddle point, (x , , ), of the Lagrangian given in equag h tion (A27) Two populations are used, each with di erent objectives, both evolved in parallel Assuming a minimization problem, the one population minimizes the tness function, (1246) f (x) = max L(x, g , h )
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where L(x, g , h ) is de ned in equation (A27) The second population maximizes the tness function, (1247) f ( g , h ) = min L(x, g , h )
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Both populations use an ES as search algorithm Kramer et al [488] developed a biased mutation operator to lead the search to more promising, feasible areas The mean of the Gaussian distribution, from which mutational step sizes are sampled, is biased to shift the center of the mutation distribution as illustrated in Figure 123 Let i (t) = ( i1 (t), , inx (t)) be the bias coe cient vector, with ij (0) U ( 1, 1), for all j = 1, , nx The bias vector, i (t), is then de ned as ij (t) = ij (t) ij (t) Mutational step sizes are calculated as xij (t) = ij (t)Nj (0, 1) + ij (t) = Nj ( ij (t), ij (t)) Bias coe cients are self-adapted using ij (t) = ij (t) + N (0, 1) (1249) (1248)
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12 Evolution Strategies
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Figure 123 Biased Mutation for Evolution Strategies with = 01 (suggested in [488]) A very simple approach to handle constraints is to change the selection operator to select the next population as follows: Until individuals have been selected, First select the best feasible individuals If all feasible solutions have been selected, select those that violate the fewest constraints As last resort, when individuals are infeasible, and they violate the same number of constraints, select the most t individuals When a selection operator is applied to select one of two individuals, the following rules can be applied: If both are feasible, select the one with the best tness If one is feasible, and the other infeasible, select the feasible solution If both are infeasible, select the one that violates the fewest constraints If constraint violation is the same, select the most t individual
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Multi-Objective Optimization
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One of the rst, simple ES for solving multi-objective (MOO) problems was developed by Knowles and Corne [469] The Pareto archived evolution strategy (PAES)
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consists of three parts: (1) a candidate solution generator, (2) the candidate solution acceptance function, and (3) the nondominated-solutions archive The candidate solution generator is an (1 + 1)-ES, where the individual that survives to the next generation is based on dominance If the parent dominates the o spring, the latter is rejected, and the parent survives If the o spring dominates the parent, the o spring survives to the next generation If neither the parent nor the o spring is dominating, the o spring is compared with the nondominated solutions in the archive, as summarized in Algorithm 124 The archive maintains a set of nondominated solutions to the MOO The size of the archive is restricted When an o spring dominates the current solutions in the archive, it is included in the archive When the o spring is dominated by any of the solutions in the archive, the o spring is not included in the archive When the o spring and the solutions in the archive are nondominating, the o spring is accepted and included in the archive based on the degree of crowding in the corresponding area of objective space To keep track of crowding, a grid is de ned over objective space, and for each cell of the grid a counter is maintained to keep track of the number of nondominated solutions for that part of the Pareto front When an o spring is accepted into the archive, and the archive has reached its capacity, the o spring replaces one of the solutions in the highest populated grid cell (provided that the grid cell corresponding to the o spring has a lower frequency count) When the parent and its o spring are nondominating, the one with the lower frequency count is accepted in the archive The PAES is summarized in Algorithm 123 Costa and Oliveira [159] developed a di erent approach to ensure that nondominated solutions survive to next generations, and to produce diverse solutions with respect to objective space Fitnesses of individuals are based on a Pareto ranking, where individuals are grouped into a number of Pareto fronts An individual s tness is determined based on the Pareto front in which the individual resides At each generation, the Pareto ranking process proceeds as follows All of the nondominated solutions from the o spring, or + parents and o spring, form the rst Pareto front These individuals are then removed, and the nondominating solutions from the remainder of the individuals form the second Pareto front This process of forming Pareto fronts continues until all (or + ) individuals are assigned to a Pareto front Individuals of the rst Pareto front is assigned a tness of 1/nc , where nc is the niche count The niche count is the number of individuals in this front that lies within a distance of share from the individual (distance is measured with respect to objective space) The threshold, share , is referred to as the niche radius Individuals of the next Pareto front is assigned a tness of (1 + fworst )/nc , where fworst is the worst tness from the previous front This process of tness assignment continues until all individuals have been assigned a tness The tness sharing approach described above promotes diversity of nondominated solutions The process of creating Pareto fronts ensures that dominated individuals are excluded from future generations as follows: Depending on the ES used, individuals
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