Exploring A Two-market Genetic Algorithm

The ordinary genetic algorithm may be thought of as conducting a single market in which solutions compete for success, as measured by the fitness funtion. We introduce a two-market genetic algorithm, consisting of two phases, each of which is an ordinary single-market genetic algorithm. The twomarket genetic algorithm has a natural interpretation as a method of solving constrained optimization problems. Phase 1 is optimality improvement; it works on the problem without regard to constraints. Phase 2 is feasibility improvement; it works on the existing population of solutions and drives it towards feasibility. We tested this concept on 14 standard knapsack test problems for genetic algorithms, with excellent results. The paper concludes with discussions of why the twomarket genetic algorithm is successful and of how this work can be extended. 1. Paper’s category: Genetic Algorithms. /* $Header: exploring-two-market-ga.tex,v 1.7 2002/01/22 13:59:52 Exp $ */ 1 Motivation & Experimental Setup Genetic algorithms (GAs), and more generally evolutionary computation, have much to recommend them as heuristics for unconstrained optimization. These problems are often otherwise intractable and experience has yielded broadly successful results. The case of constained optimization is more problematic for evolutionary computation and GAs in particular. In spite of considerable attention paid to the matter (see [2, 5, 6] for excellent reviews), there is no clearly best approach to encoding constrained optimization problems as GAs. Techniques are available and used, largely based on penalty functions. Still, this important class of problems remains somewhat recalcitrant. We explore a conceptually new approach to constrained optimization, at least in the GA context. We interpret a constrained optimization problem as a market between two players adapting as GAs: the objective function player and the constraint player. We explore how their markets behave (and how good the solutions are they find), compared to standard GA approaches for handling constraints. We use 14 wellstudied knapsack test problems for our benchmarking [8]. Although the knapsack is perhaps the simplest of integer constrained optimization problems, it is NPcomplete. Thus, we may hope its lessons apply to other problems of interest. 1.1 Penalty Functions & Fitness Evaluation Given a solution, ~s, to a constrained optimization problem, its absolute fitness, W (~s), in the presence of penalties for constraint violation is commonly (“standardly”) measured as: W (~s) = Z(~s)− P (~s) (1) We note a certain analogy between large-scale decomposition methods in optimization theory and our two-