On a Stochastic Knapsack Problem

The deterministic knapsack problem is a well known and well studied NP-hard combinatorial optimization problem. It consists in filling a knapsack with items out of a given set such that the weight capacity of the knapsack is respected and the total reward maximized. For a review of references on the stochastic knapsack problem, stochastic gradient algorithms and branch-and-bound methods see [4]. In the deterministic problem, all parameters (item weights, rewards, knapsack capacity) are known (deterministic). In the stochastic counterpart, some (or all) of these parameters are assumed to be random, i.e. not known at the moment the decision has to be made. In this paper, we study the stochastic knapsack problem with expectation constraint. The item weights are assumed to be independently normally distributed. We solve the relaxed version of this problem using a stochastic gradient algorithm in order to provide upper bounds for a branch-and-bound framework. Two approaches to estimate the needed gradients are applied, one based on Integration by Parts and one using Finite Differences. Finite Differences is a robust and simple approach with efficient results despite the fact that the estimated gradients are biased, meanwhile Integration by Parts is based upon a more theoretical analysis and permits to enlarge the field of applications.