Abstract :
The behavior of the Crossover and Mutation Operators on candidate solutions to any discrete and constrained optimization problem is investigated in some detail, affording new understanding in the construction of Evolutionary and Genetic Algorithms for the effective resolution of problems of this kind. The important case in which some or all of the constraints are linear exemplifies the potential usefulness of these insights, by guiding the design and application of operators guaranteed to preserve feasibility. The computational challenge posed by numerous optimization problems has lead to the exploration of algorithms based on adaption observed in biological systems, as an alternative means of illation. Ideally suited to difficult conditions, in accordance with their inspiration, these methods have been successfully exploited in many problems that have defied adequate resolution by more traditional means. Despite their undeniable attraction, the applicability of Genetic Algorithms has been limited by the lack of general techniques to manage systems of constraints, a feature of many optimization tasks, and this is compounded when some proportion of the decision variables are discrete. In response, the fundamental behavior of the genetic operators with respect to any system of constraints has been investigated; this discussion summarizes and extends the results of earlier work. [Math. Comp. Modelling, 1996, 23(5), 87–111]. By investigating theoretically the behavior of the crossover operator on viable solutions of an entirely general system of constraints, the limitations of restricting attention to solely to feasible points become apparent. A relaxation of the crossover operator is also proposed to avoid calamitous performance degradation otherwise experienced in solving highly constrained problems, and its effective utilization suggests a modification in the overall algorithm in which the population is permitted to transiently increase in size.
