A Bayesian Optimisation Approach for Multidimensional Knapsack Problem

Abstract

This paper considers the application of Bayesian optimisation to the well-known multidimensional knapsack problem which is strongly NP-hard. For the multidimensional knapsack problem with a large number of items and knapsack constraints, a two-level formulation is presented to take advantage of the global optimisation capability of the Bayesian optimisation approach, and the efficiency of integer programming solvers on small problems. The first level makes the decisions about the optimal allocation of knapsack capacities to different item groups, while the second level solves a multidimensional knapsack problem of reduced size for each item group. To accelerate the Bayesian optimisation guided search process, various techniques are proposed including variable domain tightening, initialisation by the Genetic Algorithm, and optimisation landscape smoothing by local search. Computational experiments are carried out on the widely used benchmark instances with up to 100 items and 30 knapsack constraints. The preliminary results demonstrate the effectiveness of the proposed solution approach.