Extending the QCR method to general mixed-integer programs

Let ( MQP ) be a general mixed integer quadratic program that consists of minimizing a quadratic function subject to linear constraints. In this paper, we present a convex reformulation of ( MQP ), i.e. we reformulate ( MQP ) into an equivalent program, with a convex objective function. Such a refor...

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Bibliographic Details
Published inMathematical programming Vol. 131; no. 1-2; pp. 381 - 401
Main Authors Billionnet, Alain, Elloumi, Sourour, Lambert, Amélie
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer-Verlag 01.02.2012
Springer
Springer Nature B.V
Subjects
Algorithms
Applied sciences
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Eigenvalues
Equivalence
Exact sciences and technology
Full Length Paper
Inequalities
Inequality
Integer programming
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical models
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Mixed integer
Numerical Analysis
Operational research and scientific management
Operational research. Management science
Optimization techniques
Quadratic programming
Semidefinite programming
Solvers
Studies
Theoretical
General integer programming
90C22 Semidefinite programming
90C11 Mixed integer programming
90C20 Quadratic programming
Mixed-integer programming
Convex reformulation
Quadratic programming
Experiments
Semi-definite programming
90C26 Nonconvex programming
Non convex programming
Semi definite programming
Convex programming
Relaxation
Inequality constraint
Relaxation method
Mathematical programming
Branch and bound method
Minimization
Mixed integer programming
Mixed method
Standards
Integer programming
Computation time
Implicit enumeration method
Equality constraint
Objective function
Quadratic function
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Summary:Let ( MQP ) be a general mixed integer quadratic program that consists of minimizing a quadratic function subject to linear constraints. In this paper, we present a convex reformulation of ( MQP ), i.e. we reformulate ( MQP ) into an equivalent program, with a convex objective function. Such a reformulation can be solved by a standard solver that uses a branch and bound algorithm. We prove that our reformulation is the best one within a convex reformulation scheme, from the continuous relaxation point of view. This reformulation, that we call MIQCR (Mixed Integer Quadratic Convex Reformulation), is based on the solution of an SDP relaxation of ( MQP ). Computational experiences are carried out with instances of ( MQP ) including one equality constraint or one inequality constraint. The results show that most of the considered instances with up to 40 variables can be solved in 1 h of CPU time by a standard solver.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-010-0381-7