The Influence of Local Search over Genetic Algorithms with Balanced Representations

We continue the study of Genetic Algorithms (GA) on combinatorial optimization problems where the candidate solutions need to satisfy a balancedness constraint. It has been observed that the reduction of the search space size granted by ad-hoc crossover and mutation operators does not usually translate to a substantial improvement of the GA performances. There is still no clear explanation of this phenomenon, although it is suspected that a balanced representation might yield a more irregular fitness landscape, where it could be more difficult for GA to converge to a global optimum. In this paper, we investigate this issue by adding a local search step to a GA with balanced operators, and use it to evolve highly nonlinear balanced Boolean functions. In particular, we organize our experiments around two research questions, namely if local search (1) improves the convergence speed of GA, and (2) decreases the population diversity. Surprisingly, while our results answer affirmatively the first question, they also show that adding local search actually \emph{increases} the diversity among the individuals in the population. We link these findings to some recent results on fitness landscape analysis for problems on Boolean functions.

Tip the Balance: Improving Exploration of Balanced Crossover Operators by Adaptive Bias

The use of balanced crossover operators in Genetic Algorithms (GA) ensures that the binary strings generated as offsprings have the same Hamming weight of the parents, a constraint which is sought in certain discrete optimization problems. Although this method reduces the size of the search space, the resulting fitness landscape often becomes more difficult for the GA to explore and to discover optimal solutions. This issue has been studied in this paper by applying an adaptive bias strategy to a counter-based crossover operator that introduces unbalancedness in the offspring with a certain probability, which is decreased throughout the evolutionary process. Experiments show that improving the exploration of the search space with this adaptive bias strategy is beneficial for the GA performances in terms of the number of optimal solutions found for the balanced nonlinear Boolean functions problem.

Balanced Crossover Operators in Genetic Algorithms

In several combinatorial optimization problems arising in cryptography and design theory, the admissible solutions must often satisfy a balancedness constraint, such as being represented by bitstrings with a fixed number of ones. For this reason, several works in the literature tackling these optimization problems with Genetic Algorithms (GA) introduced new balanced crossover operators which ensure that the offspring has the same balancedness characteristics of the parents. However, the use of such operators has never been thoroughly motivated, except for some generic considerations about search space reduction. In this paper, we undertake a rigorous statistical investigation on the effect of balanced and unbalanced crossover operators against three optimization problems from the area of cryptography and coding theory: nonlinear balanced Boolean functions, binary Orthogonal Arrays (OA) and bent functions. In particular, we consider three different balanced crossover operators, two of which have never been published before, and compare their performances with classic one-point crossover. The statistical comparison shows that for the problems of nonlinear balanced Boolean functions and binary OA the use of balanced crossover operators gives GA a definite advantage over one-point crossover. For the case of bent functions, the situation is reversed, with the unbalanced crossover providing the best performances.