Generalizing and Unifying Gray-box Combinatorial Optimization Operators

TL;DR

This paper introduces a comprehensive framework for gray-box combinatorial optimization operators, unifying existing methods and guiding the design of new efficient operators across various problem representations.

Contribution

It presents a general framework that unifies gray-box operators, providing theoretical insights and practical algorithms for diverse combinatorial problems.

Findings

Unified proofs of efficiency for gray-box hill climbers and crossovers

Framework applicable to permutation problems like Linear Ordering and TWT

Proposed new efficient operators demonstrating improved performance

Abstract

Gray-box optimization leverages the information available about the mathematical structure of an optimization problem to design efficient search operators. Efficient hill climbers and crossover operators have been proposed in the domain of pseudo-Boolean optimization and also in some permutation problems. However, there is no general rule on how to design these efficient operators in different representation domains. This paper proposes a general framework that encompasses all known gray-box operators for combinatorial optimization problems. The framework is general enough to shed light on the design of new efficient operators for new problems and representation domains. We also unify the proofs of efficiency for gray-box hill climbers and crossovers and show that the mathematical property explaining the speed-up of gray-box crossover operators, also explains the efficient identification of improving moves in gray-box hill climbers. We illustrate the power of the new framework by proposing an efficient hill climber and crossover for two related permutation problems: the Linear Ordering Problem and the Single Machine Total Weighted Tardiness Problem.