Optimizing a Binary Integer Program by Identifying Its Optimal Core Problem - A New Optimization Concept Applied to the Multidimensional Knapsack Problem

Abstract

AbstractThe core concept for solving a binary integer program (BIP) is about dividing the BIP’s variables into core and adjunct ones. We fix the adjunct variables to either 0 or 1; consequently, we reduce the problem to the core variables only, forming a core problem (CP). An optimal solution to a CP is not optimal to the original BIP unless adjunct variables are fixed to their optimal values. Consequently, an optimal CP is a CP whose associated adjunct variables are fixed to their optimal values. This paper presents a new optimization concept that solves a BIP by searching for its optimal CP. We use a hybrid algorithm of local search and linear programming to move from a CP to a better one until we find the optimal CP. We use our algorithm to solve 180 multidimensional knapsack (MKP) instances to validate this new optimization concept. Results show that it is a promising approach to investigate because we were able to find the optimal solutions of 149 instances, of which some had 500 variables, by solving several CPs having 30 variables only.KeywordsBinary integer programCore problemMultidimensional knapsackLocal search