Variants of an adaptive penalty scheme for steady-state genetic algorithms in engineering optimization

Abstract

Purpose – The purpose of this paper is to propose variants of an adaptive penalty scheme for steady-state genetic algorithms applied to constrained engineering optimization problems. Design/methodology/approach – For each constraint a penalty parameter is adaptively computed along the evolution according to information extracted from the current population such as the existence of feasible individuals and the level of violation of each constraint. The adaptive penalty method (APM), as originally proposed, computes the constraint violations of the initial population, and updates the penalty coefficient of each constraint after a given number of new individuals are inserted in the population. A second variant, called sporadic APM with constraint violation accumulation, works by accumulating the constraint violations during a given insertion of new offspring into the population, updating the penalty coefficients, and fixing the penalty coefficients for the next generations. The APM with monotonic penalty coefficients is the third variation, where the penalty coefficients are calculated as in the original method, but no penalty coefficient is allowed to have its value reduced along the evolutionary process. Finally, the penalty coefficients are defined by using a weighted average between the current value of a coefficient and the new value predicted by the method. This variant is called the APM with damping. Findings – The paper checks new variants of an APM for evolutionary algorithms; variants of an APM, for a steady-state genetic algorithm based on an APM for a generational genetic algorithm, largely used in the literature previously proposed by two co-authors of this manuscript; good performance of the proposed APM in comparison with other techniques found in the literature; innovative and general strategies to handle constraints in the field of evolutionary computation. Research limitations/implications – The proposed algorithm has no limitations and can be applied in a large number of evolutionary algorithms used to solve constrained optimization problems. Practical implications – The proposed algorithm can be used to solve real world problems in engineering as can be viewed in the references, presented in this manuscript, that use the original (APM) strategy. The performance of these variants is examined using benchmark problems of mechanical and structural engineering frequently discussed in the literature. Originality/value – It is the first extended analysis of the variants of the APM submitted for possible publication in the literature, applied to real world engineering optimization problems.

A shifted hyperbolic augmented Lagrangian-based artificial fish two-swarm algorithm with guaranteed convergence for constrained global optimization

“…A comparison of the results with those of the adaptive penalty scheme used within a steadystate genetic algorithm (APS-GA) (Lemonge, Barbosa, and Bernardino 2015), the dynamic use of differential evolution variants combined with the adaptive penalty method (DUVDE+APM) (Silva, Barbosa, and Lemonge 2011), the filter simulated annealing algorithm (FSA), available in Hedar and Fukushima (2006), the hybrid evolutionary algorithm with an adaptive constraint handling technique (HEA-ACT) (Y. , the harmony search metaheuristic (HSm) algorithm (Lee and Geem 2005) and the hybrid version of the electromagnetism-like algorithm (Hybrid EM) (Rocha and Fernandes 2009) is carried out. Table 3 shows the values of the variables and of the objective function of the best run, the average objective function value, f av , and the average number of function evaluations, n.f.e.…”

“…This function aims at penalizing infeasible solutions by increasing their fitness values proportionally to their level of constraint violation. In Ali, Golalikhani, and Zhuang (2014); Ali and Zhu (2013); Barbosa and Lemonge (2008); Coello (2002); Lemonge, Barbosa, and Bernardino (2015); Mezura-Montes and Coello (2011); Silva, Barbosa, and Lemonge (2011), penalty methods and stochastic approaches are used to generate a population of points, at each iteration, aiming to explore the search space and to converge to a global optimal solution.…”

Extensions of Firefly Algorithm for Nonsmooth Nonconvex Constrained Optimization Problems

“…In population-based methods, the widely used approach to deal with constrained optimization problems is based on exterior penalty methods [18,19,20,21]. In this type of approach, the constrained problem is replaced by a sequence of unconstrained subproblems, defined by penalty functions.…”

Improved NSGA-II Algorithm for Optimization of Constrained Functions

“…However, this method has a reasonable penalty factor setting problem: the penalty factor is too large, although it can make the evolutionary population converge quickly to the feasible domain of the problem, but neglects the value of the population to be of no value. By using the feasible solution, it is difficult to find the optimal solution of the problem; on the contrary, if the penalty factor is set too small, then the evolutionary population will stay in the unfeasible region [2] [3], and the other way to deal with the constraints is that it is difficult to get the feasible solution of the problem. The problem of optimization is transformed into a multi-objective optimization problem without constraints, which is solved by using the existing unconstrained optimization methods.…”