Using a Mixed Integer Quadratic Programming Solver for the Unconstrained Quadratic 0-1 Problem

In this paper, we consider problem (P) of minimizing a quadratic function q(x)= xtQx + ctx of binary variables. Our main idea is to use the recent Mixed Integer Quadratic Programming (MIQP) solvers. But, for this, we have to first convexify the objective function q(x). A classical trick is to raise...

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Vydáno v:Mathematical programming Ročník 109; číslo 1; s. 55 - 68
Hlavní autoři: Billionnet, Alain, Elloumi, Sourour
Médium: Journal Article
Jazyk:angličtina
Vydáno: Heidelberg Springer 01.01.2007
Springer Nature B.V
Témata:
Applied sciences
Eigenvalues
Exact sciences and technology
Integer programming
Mathematical programming
Methods
Operational research and scientific management
Operational research. Management science
Optimization
Quadratic programming
Studies
Semidefinite relaxation
Quadratic 0-1 optimization
Eigenvalue
Minimization
Mixed integer programming
Semi definite programming
Vector method
Quadratic programming
Modeling
Convex programming
Experiments
Optimization
Semidefinite positive relaxation
Exact solution
Max-cut
Integer programming
Diagonal matrix
Convex function
Eigenvalue problem
Objective function
Convex quadratic relaxation
Quadratic function
Mathematical programming
ISSN:0025-5610, 1436-4646
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Shrnutí:In this paper, we consider problem (P) of minimizing a quadratic function q(x)= xtQx + ctx of binary variables. Our main idea is to use the recent Mixed Integer Quadratic Programming (MIQP) solvers. But, for this, we have to first convexify the objective function q(x). A classical trick is to raise up the diagonal entries of Q by a vector u until (Q + diag(u)) is positive semidefinite. Then, using the fact that xi2 = xi, we can obtain an equivalent convex objective function, which can then be handled by an MIQP solver. Hence, computing a suitable vector u constitutes a preprocessing phase in this exact solution method. We devise two different preprocessing methods. The first one is straightforward and consists in computing the smallest eigenvalue of Q. In the second method, vector u is obtained once a classical SDP relaxation of (P) is solved. [PUBLICATION ABSTRACT]
Bibliografie:SourceType-Scholarly Journals-1
ObjectType-Feature-1
content type line 14
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-005-0637-9