SAWing on Symmetry

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SAWing on Symmetry Luk Schoofs, Bart Naudts and Ives Landrieu Intelligent Systems Lab, Department of Mathematics and Computer Science, University of Antwerp, RUCA, Groenenborgerlaan 171, B-2020 Antwerpen. e-mail: {lschoofs, bnaudts, landrieu}@ruca.ua.ac.be. Corresponding author: B. Naudts, [email protected].

Abstract In this paper we investigate the behavior of mutation-based evolutionary algorithms on highly symmetric binary constraint satisfaction problems. With empirical methods we study why and when these algorithms perform better under the stepwise adaptive weighting of penalties (SAWing) than under the standard penalty function. We observe that SAWing has little effect when the local optima of the symmetric problems are not very strong. However, while the use of the standard penalty function can lead to strong local optima, the SAWing mechanism can avoid this situation. The symmetric problems we consider are the standard one-dimensional Ising model and a more complex construction with the Ising model as the core component.

1 Introduction Constraint satisfaction problems (CSPs, e.g., (Tsang, 1993), (Prosser, 1996)) form a class of NP-complete decision problems which are often solved using evolutionary algorithms (e.g., (Paredis, 1994), (Michalewicz, 1996)). From the class of binary CSPs, which contain only constraints between two variables, we selected two problems: a problem corresponding to the standard Ising model of statistical physics, and a self-made construction called the torus. The main characteristic of these two problems is an extremely high amount of local optima in Hamming space due to the symmetric placement of constraints. The aim of this paper is to study the differences between the standard penalty function and the step-wise adaptation of weights (SAWing, (Eiben & van der Hauw, 1997b)) on the two symmetric CSPs.

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The context of this paper is limited by the choice of optimization algorithm: we only consider mutation-based evolutionary algorithms. The reason for this explicit restriction is that without niching, most genetic algorithms would never be able to maintain a high enough diversity in their populations to exploit the crossover operator. The loss of diversity is due to the slow exploration of the fairly flat fitness landscape of the symmetric problems. This paper is structured as follows. After a brief introduction of binary CSP, SAWing and symmetry, we study the difference between the standard penalty function and SAWing on the standard Ising model. Then, in section 3, we move from the Ising model to the torus, which is a more complex variant. Section 4 ends this paper with conclusions and hints for further work.

1.1 Binary constraint satisfaction problems A binary constraint satisfaction problem consist of the following components: • a set of variables x0 , x1 , ..., xn−1 ; • a domain Di corresponding to each variable; • a subset Cij ⊂ Di × Dj for each pair of variables (i, j ), with 0 ≤ i < j < n, which represents a constraint when it differs from the Cartesian product Di × Dj . (Note that we use a slightly different formalism from the one described in (Tsang, 1993).) The goal of a constraint satisfaction problem is to assign to the variables xi a value from their domain Di in such a way that all constraints are satisfied. Formally, we say that a constraint Cij is satisfied if and only if (xi , xj ) ∈ Cij . The couple (xi , xj ) is then called a valid assignment. The difficulty with using black-box algorithms like genetic algorithms or simulated annealing for finding solutions of a CSP is that the problem has no explicit objective function to be optimized. It is clear that one has to be constructed. The most obvious choice is the standard penalty function, which simply counts the number of constraints violated by one assignment. Formally, this means that we have to minimize the function X f : x 7→ Vij (xi , xj ), (1) 0≤i