This paper considers a distributionally robust version of a quadratic knapsack problem. In this model, a subsets of items is selected to maximizes the total profit while requiring that a set of knapsack constraints be satisfied with high probability. In contrast to the stochastic programming version of this problem, we assume that only part of the information on random data is known, i.e., the first and second moment of the random variables, their joint support, and possibly an independence assumption. As for the binary constraints, special interest is given to the corresponding semidefinite programming (SDP) relaxation. While in the case that the model only has a single knapsack constraint we present an SDP reformulation for this relaxation, the case of multiple knapsack constraints is more challenging. Instead, two tractable methods are presented for providing 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 value of the proposed distributionally robust approach.


  1. stochastic programming
  2. chance constraints
  3. distributionally robust optimization
  4. semidefinite programming
  5. knapsack problem

MSC codes

  1. 90C15
  2. 90C22
  3. 90C59

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Information & Authors


Published In

cover image SIAM Journal on Optimization
SIAM Journal on Optimization
Pages: 1485 - 1506
ISSN (online): 1095-7189


Submitted: 12 June 2013
Accepted: 3 July 2014
Published online: 11 September 2014


  1. stochastic programming
  2. chance constraints
  3. distributionally robust optimization
  4. semidefinite programming
  5. knapsack problem

MSC codes

  1. 90C15
  2. 90C22
  3. 90C59



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