Distributionally Robust Stochastic Knapsack Problem
Résumé
This paper considers a distributionally robust version of a quadratic knapsack prob-lem. In this model, a subsets of items is selected to maximizes the total profit while requiring thata set of knapsack constraints be satisfied with high probability. In contrast to the stochastic pro-gramming version of this problem, we assume that only part of the information on random datais known, i.e., the first and second moment of the random variables, their joint support, and pos-sibly an independence assumption. As for the binary constraints, special interest is given to thecorresponding semidefinite programming (SDP) relaxation. While in the case that the model onlyhas a single knapsack constraint we present an SDP reformulation for this relaxation, the case ofmultiple knapsack constraints is more challenging. Instead, two tractable methods are presented forproviding upper and lower bounds (with its associated conservative solution) on the SDP relaxation.An extensive computational study is given to illustrate the tightness of these bounds and the valueof the proposed distributionally robust approach.