The quadratic knapsack problem—a survey

• Published in: Discrete Applied Mathematics Vol. 155; no. 5; pp. 623 - 648
• Main Author: Pisinger, David
• Format: Journal Article
• Language: English
• Published: Lausanne Elsevier B.V 15.03.2007; Amsterdam Elsevier; New York, NY
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Approximation algorithms; Computer science; control theory; systems; Exact sciences and technology; Mathematical programming; Operational research and scientific management; Operational research. Management science; Quadratic knapsack problem; Semidefinite programming; Theoretical computing; Upper bounds; Valid inequalities; Approximation algorithms; Semidefinite programming; Valid inequalities; Upper bounds; Quadratic knapsack problem; Polytope; Lagrangian; Computer theory; Decomposition method; Semi definite programming; Approximation algorithm; Experimental study; Constrained optimization; Knapsack problem; Relaxation; Upper bound; Objective function; Combinatorics; Quadratic function
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2006.08.007

The binary quadratic knapsack problem maximizes a quadratic objective function subject to a linear capacity constraint. Due to its simple structure and challenging difficulty it has been studied intensively during the last two decades. The present paper gives a survey of upper bounds presented in the literature, and show the relative tightness of several of the bounds. Techniques for deriving the bounds include relaxation from upper planes, linearization, reformulation, Lagrangian relaxation, Lagrangian decomposition, and semidefinite programming. A short overview of heuristics, reduction techniques, branch-and-bound algorithms and approximation results is given, followed by an overview of valid inequalities for the quadratic knapsack polytope. The paper is concluded by an experimental study where the upper bounds presented are compared with respect to strength and computational effort.