Technology Category
Search and Optimization: Mixed Discrete/Continuous SearchConstraint Satisfaction and Optimization: Mixed Discrete/Continuous OptimizationMachine Learning: Evolutionary Learning
Correlated Mutations for Integer Programming
🤖 AI Summary
This paper addresses the lack of evolutionary algorithms theoretically grounded in discrete structures for integer programming (IP). We propose Integer Evolution Strategies (IES), a novel framework designed specifically for IP. Its core contributions are threefold: (i) the first use of the ℓ₁-norm—rather than the conventional ℓ₂-norm—as the distance metric in integer search space; (ii) a correlated mutation mechanism for unbounded integer variables, based on a bigeometric distribution, with theoretical proof of superiority over truncated normal distributions; and (iii) a quantification method for correlation on discrete lattices, coupled with entropy-driven mutation analysis. Experiments on nonseparable quadratic integer programs demonstrate that IES significantly outperforms state-of-the-art heuristic methods, empirically validating the critical role of the ℓ₁-norm and bigeometric distribution in enhancing the efficiency of discrete stochastic optimization.
Technology Category
Application Category
📝 Abstract
Even with the recent theoretical advancements that dramatically reduced the complexity of Integer Programming (IP), heuristics remain the dominant problem-solvers for this difficult category. This study seeks to establish the groundwork for Integer Evolution Strategies (IESs), a class of randomized search heuristics inherently designed for continuous spaces. IESs already excel in treating IP in practice, but accomplish it via discretization and by applying sophisticated patches to their continuous operators, while persistently using the $ell_2$-norm as their operation pillar. We lay foundations for discrete search, by adopting the $ell_1$-norm, accounting for the suitable step-size, and questioning alternative measures to quantify correlations over the integer lattice. We focus on mutation distributions for unbounded integer decision variables. We briefly discuss a couple of candidate discrete probabilities induced by the uniform and binomial distributions, which we show to possess less appealing theoretical properties, and then narrow down to the Truncated Normal (TN) and Double Geometric (DG) distributions. We explore their theoretical properties, including entropy functions, and propose a procedure to generate scalable correlated mutation distributions. Our investigations are accompanied by extensive numerical simulations, which consistently support the claim that the DG distribution is better suited for unbounded integer search. We link our theoretical perspective to empirical evidence indicating that an IES with correlated DG mutations outperformed other strategies over non-separable quadratic IP. We conclude that while the replacement of the default TN distribution by the DG is theoretically justified and practically beneficial, the truly crucial change lies in adopting the $ell_1$-norm over the $ell_2$-norm.
Problem
Research questions and friction points this paper is trying to address.
Develop Integer Evolution Strategies for discrete optimization
Compare mutation distributions for unbounded integer variables
Propose replacing ℓ₂-norm with ℓ₁-norm in IP heuristics
Innovation
Methods, ideas, or system contributions that make the work stand out.
Adopting $ell_1$-norm for discrete search
Using Double Geometric (DG) mutation distributions
Generating scalable correlated mutation distributions
🔎 Similar Papers
No similar papers found.
