An effective branch-and-bound algorithm for convex quadratic integer programming

We present a branch-and-bound algorithm for minimizing a convex quadratic objective function over integer variables subject to convex constraints. In a given node of the enumeration tree, corresponding to the fixing of a subset of the variables, a lower bound is given by the continuous minimum of th...

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Bibliographic Details
Published in:Mathematical programming Vol. 135; no. 1-2; pp. 369 - 395
Main Authors: Buchheim, Christoph, Caprara, Alberto, Lodi, Andrea
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer-Verlag 01.10.2012
Springer
Springer Nature B.V
Subjects:
Algorithms
Applied sciences
Approximation
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Ellipsoids
Enumeration
Exact sciences and technology
Full Length Paper
Integer programming
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical models
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Operational research and scientific management
Operational research. Management science
Optimization
Preprocessing
Theoretical
Trees
Variables
90C10
90C57
Computational results
90C25
90C90
Convex quadratic minimization
Branch-and-bound algorithm
Closest vector problem
Lower bound
Tree(graph)
Enumeration
Branch and bound method
Linear time
Quadratic programming
Convex programming
Lattice
Integer programming
Linear process
Implicit enumeration method
Convex function
Objective function
Ellipsoid
Quadratic function
Mathematical programming
ISSN:0025-5610, 1436-4646
Online Access:Get full text
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Summary:We present a branch-and-bound algorithm for minimizing a convex quadratic objective function over integer variables subject to convex constraints. In a given node of the enumeration tree, corresponding to the fixing of a subset of the variables, a lower bound is given by the continuous minimum of the restricted objective function. We improve this bound by exploiting the integrality of the variables using suitably-defined lattice-free ellipsoids. Experiments show that our approach is very fast on both unconstrained problems and problems with box constraints. The main reason is that all expensive calculations can be done in a preprocessing phase, while a single node in the enumeration tree can be processed in linear time in the problem dimension.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-011-0475-x