A study of adaptive penalty functions for constrained genetic algorithm-based optimization

Abstract

Constrained optimization via the Genetic Algorithm (GA) is often a challenging endeavor, as the GA is most directly suited to unconstrained optimization. Traditionally, external penalty functions have been used to convert a constrained optimization problem into an unconstrained problem for GA-based optimization. This approach requires the somewhat arbitrary selection of penalty draw-down coefficients. In this paper, several potential approaches are presented that utilize adaptive penalty functions that change the value of the draw-down coefficients during a run of the genetic algorithm. A simple one-dimensional constrained problem and a more complex twodimensional constrained problem were solved using the adaptive penalty strategies. Then, a stiffened composite panel was optimized for minimum weight, subject to several constraints using the adaptive penalty methods to provide insight to how the approaches perform on an engineering problem. Based on these problem solutions, conclusions were drawn regarding the efficacy of adaptive penalty functions for constrained optimization. Nomenclature c penalty draw-down coefficient f(x) fitness function g(x) constraint function n generation number N con number of constraints P(x) penalty function x design variable vector ϕ(x) objective function σ standard deviation of fitness σ 2 variance of fitness

A New Adaptive Penalty Method for Constrained Genetic Algorithm and Its Application to Water Distribution Systems

“…Choosing the penalty coefficient values for a penalty function is often arbitrary, [23]. A small coefficient will impose a smaller penalty than a large coefficient for the same magnitude of constraint violation.…”

“…Crossley and Williams [23] defined three basic forms of draw-down coefficient strategies: constant penalty coefficient, generation number-based strategies (increasing the value of c with successive generations) and population fitness-based strategies (using the standard deviation and the variance of the population's fitness values).…”

“…The goal of an adaptive penalty function is to change the value of the penalty coefficient during the search allowing exploration of infeasible regions to find optimal building blocks, while preserving the feasibility of the final solution, [23].…”

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Characterization of particle dispersion and volume fraction in alumina-filled epoxy nanocomposites using photo-stimulated luminescence spectroscopy

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