Stochastic Quadratic Knapsack with Recourse

This paper is dedicated to a study of different extensions of the classical knapsack problem to the case when different elements of the problem formulation are subject to a degree of uncertainty described by random variables. This brings the knapsack problem into the realm of stochastic programming. In this paper, we propose a model of two-stage quadratic knapsack with recourse in which we introduce a probability constraint on the capacity of the knapsack on the first stage. As far as we know, this is the first time such a constraint has been used in a two-stage model. The solution techniques are based on the semidefinite relaxations. This allows for solving large instances, for which exact methods cannot be used.