Constraint Handling in Genetic Algorithms via Artificial Immune Systems Heder S. Bernardino Universidade Federal de Juiz de Fora Cidade Universit´ aria 36036 330 Juiz de Fora, MG, Brazil hedersb@gmail.com Helio J.C. Barbosa ∗ Laborat ´ orio Nacional de Computac ¸˜ ao Cient´ ıfica Av. Get´ ulio Vargas 333 25651 075 Petr´ opolis, RJ, Brazil hcbm@lncc.br Afonso C.C. Lemonge Universidade Federal de Juiz de Fora Departamento de Estruturas Cidade Universit´ aria 36036 330 Juiz de Fora, MG, Brazil lemonge@numec.ufjf.br ABSTRACT The combination of an artificial immune system (AIS) with a genetic algorithm (GA) is proposed as an alternative to tackle constrained optimization problems. The AIS is in- spired in the clonal selection principle and is embedded into a standard GA search engine in order to help move the pop- ulation into the feasible region. The procedure is applied to well known test-problems from the evolutionary compu- tation literature and compared to other alternative tech- niques. Categories and Subject Descriptors I.2.8 [Problem Solving, Control Methods, and Search]: [Heuristic methods]; J.2 [Physical Sciences and Engi- neering]: Engineering General Terms Algorithms Keywords constrained optimization, genetic algorithm, artificial im- mune systems 1. INTRODUCTION Evolutionary algorithms (EAs) can be readily applied to unconstrained optimization problems by adopting a fitness function closely related to the desired objective function. However, when the solution must satisfy a set of constraints, the EA must be equiped with an additional constraint han- dling procedure. To fix ideas and without loss of generality, only minimization problems will be considered here. The techniques for handling constraints within EAs can be direct (feasible or interior), when only feasible elements are considered, or indirect (exterior), when both feasible and infeasible elements are used during the search process. ∗ Corresponding author Copyright is held by the author/owner(s). GECCO’06, July 8–12, 2006, Seattle, Washington, USA. ACM 1-59593-186-4/06/0007. Direct techniques comprise: a) special closed genetic op- erators[29], b) special decoders[21], c) repair techniques[25, 27], and d) “death penalty”. Direct techniques are problem dependent (with the ex- ception of the “death penalty”) and actually of extremely reduced practical applicability. Indirect techniques include: a) the use of Lagrange mul- tipliers[1, 2], b) the use of fitness as well as constraint viola- tion values in a multi-objective optimization setting[31, 7], c) the use of special selection techniques[28], d) “lethaliza- tion”: any infeasible offspring is just assigned a given, very low, fitness value[32], and e) penalty techniques. Due to its simple intuitive basis and generality, penalty techniques, in spite of their shortcomings, are the most pop- ular ones. The fitness function value of an unfeasible solu- tion is increased by a penalty term which usually grows with the number of violated constraints and also with the amount of violation. One can have additive as well as multiplicative penalty functions. Usually, the performance of the technique depends strongly on one or more penalty parameters that must be set by the user for a given problem. Two-parameter penalty (Le Riche et al.[20]), multi-para- meter penalty (Homaifar et al.[23]), dynamically varying pa- rameter penalty (Joines & Houck[18]), and adaptive penalty techniques (Bean & Hadj-Alouane[5], Coit et al.[30], Bar- bosa & Lemonge[3, 4]) are available in the literature. Another technique which will be used here for numerical comparisons is that due to Runarsson & Yao[28] where a good balance between the objective and the penalty function values is sought by means of a stochastic ranking. For other constraint handling methods in evolutionary computation see [29, 26, 17, 22, 19, 12, 16, 33], references therein, and the still growing literature. However, of particular interest here is the application of ideas from artificial immune systems[10] in constrained op- timization problems. 2. CONSTRAINED OPTIMIZATION PROB- LEMS A standard constrained optimization problem in R n can be thought of as the minimization of a given objective func- tion f (x), where x ∈ R n is the vector of design/decision variables, subject to inequality constraints gp(x) ≥ 0,p =