The quadratic knapsack problem—a survey

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 th...

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Bibliographic Details
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
Online Access:Get full text
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Summary: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.
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2006.08.007