The Boolean quadric polytope: some characteristics, facets and relatives
Mathematical Programming: Series A and B
A hierarchy of relaxation between the continuous and convex hull representations
SIAM Journal on Discrete Mathematics
Discrete Applied Mathematics
Solving quadratic (0,1)-problems by semidefinite programs and cutting planes
Mathematical Programming: Series A and B
Fast Approximation Algorithms for the Knapsack and Sum of Subset Problems
Journal of the ACM (JACM)
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
A Spectral Bundle Method for Semidefinite Programming
SIAM Journal on Optimization
Combining Semidefinite and Polyhedral Relaxations for Integer Programs
Proceedings of the 4th International IPCO Conference on Integer Programming and Combinatorial Optimization
The Dynamic and Stochastic Knapsack Problem
Operations Research
A Comparison of the Sherali-Adams, Lovász-Schrijver, and Lasserre Relaxations for 0--1 Programming
Mathematics of Operations Research
Approximating the Stochastic Knapsack Problem: The Benefit of Adaptivity
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
The quadratic knapsack problem-a survey
Discrete Applied Mathematics
Binary Accelerated Particle Swarm Algorithm (BAPSA) for discrete optimization problems
Journal of Global Optimization
Approximability of the two-stage stochastic knapsack problem with discretely distributed weights
Discrete Applied Mathematics
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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. Two different model formulations are proposed, based on the introduction of probability constraints. The first one is a static quadratic knapsack with a probability constraint on the capacity of the knapsack. The second one is a two-stage quadratic knapsack model, with recourse, where we introduce a probability constraint on the capacity of the knapsack in the second 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. Numerical experiments on a set of randomly generated instances are discussed below.