Robust solutions for stochastic and distributionally robust chance-constrained binary knapsack problems

Abstract: We consider chance-constrained binary knapsack problems, where the weights of items are independent random variables with the means and standard deviations known. The chance constraint can be reformulated as a second-order cone constraint under some assumptions for the probability distribution of the weights. The problem becomes a second-order cone-constrained binary knapsack problem, which is equivalent to a robust binary knapsack problem with an ellipsoidal uncertainty set. We demonstrate that optimal solutions to robust binary knapsack problems with inner and outer polyhedral approximations of the ellipsoidal uncertainty set can provide both upper and lower bounds on the optimal value of the second-order cone-constrained binary knapsack problem, and they can be obtained by solving ordinary binary knapsack problems repeatedly. Moreover, we prove that the solution providing the upper bound converges to the optimal solution to the second-order cone-constrained binary knapsack problem as the approximation of the uncertainty set becomes more accurate. Based on this, a pseudo-polynomial time algorithm is obtained under the assumption that the coefficients of the problem are integer-valued. We also propose an exact algorithm, which iteratively improves the accuracy of the approximation until an exact optimal solution is obtained. The exact algorithm also runs in pseudo-polynomial time. Computational results are presented to show the quality of the upper and lower bounds and efficiency of the exact algorithm compared to CPLEX and a previous study.