Separable Relaxation for Nonconvex Quadratic Integer Programming: Integer Diagonalization Approach

We present in this paper an integer diagonalization approach for deriving new lower bounds for general quadratic integer programming problems. More specifically, we introduce a semiunimodular transformation in order to diagonalize a symmetric matrix and preserve integral property of the feasible set...

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Bibliographic Details
Published in:Journal of optimization theory and applications Vol. 146; no. 2; pp. 463 - 489
Main Authors: Zheng, X. J., Sun, X. L., Li, D.
Format: Journal Article
Language:English
Published: Boston Springer US 01.08.2010
Springer
Springer Nature B.V
Subjects:
Algorithms
Applications of Mathematics
Applied sciences
Calculus of Variations and Optimal Control; Optimization
Capital budgeting
Engineering
Exact sciences and technology
Integer programming
Integers
Integrals
Knapsack problem
Linear programming
Lower bounds
Mathematical programming
Mathematics
Mathematics and Statistics
Methods
Operational research and scientific management
Operational research. Management science
Operations Research/Decision Theory
Optimization
Preserves
Production planning
Quadratic programming
Studies
Theory of Computation
Tightness
Transformations
Variables
Convex relaxation
Semiunimodular congruence transformation
Quadratic integer programming
Separable relaxation
Lagrangian relaxation
Lower bound
Non convex programming
Congruence
Leakproofness
Quadratic programming
Integer programming
Separation method
Lagrange multiplier
Symmetric matrix
Relaxation method
ISSN:0022-3239, 1573-2878
Online Access:Get full text
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Summary:We present in this paper an integer diagonalization approach for deriving new lower bounds for general quadratic integer programming problems. More specifically, we introduce a semiunimodular transformation in order to diagonalize a symmetric matrix and preserve integral property of the feasible set at the same time. Via the semiunimodular transformation, the resulting separable quadratic integer program is a relaxation of the nonseparable quadratic integer program. We further define the integer diagonalization dual problem to identify the best semiunimodular transformation and analyze some basic properties of the set of semiunimodular transformations for a rational symmetric matrix. In particular, we present a complete characterization of the set of all semiunimodular transformations for a nonsingular 2×2 symmetric matrix. We finally discuss Lagrangian relaxation and convex relaxation schemes for the resulting separable quadratic integer programming problem and compare the tightness of different relaxation schemes.
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ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-010-9653-x