Quadratic reformulations of nonlinear binary optimization problems

Very large nonlinear unconstrained binary optimization problems arise in a broad array of applications. Several exact or heuristic techniques have proved quite successful for solving many of these problems when the objective function is a quadratic polynomial. However, no similarly efficient methods...

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Vydané v:	Mathematical programming Ročník 162; číslo 1-2; s. 115 - 144
Hlavní autori:	Anthony, Martin, Boros, Endre, Crama, Yves, Gruber, Aritanan
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2017 Springer Nature B.V Springer
Predmet:	Arrays Binary system Boolean Business & economic sciences Calculus of Variations and Optimal Control; Optimization Codes Combinatorics Computer vision Cost engineering Full Length Paper Linear programming Mathematical analysis Mathematical and Computational Physics Mathematical functions Mathematical Methods in Physics Mathematical models Mathematics Mathematics and Statistics Mathematics of Computing Mathématiques Méthodes quantitatives en économie & gestion nonlinear binary optimization Nonlinearity Numerical Analysis Optimization Optimization algorithms Physical, chemical, mathematical & earth Sciences Physique, chimie, mathématiques & sciences de la terre Polynomials pseudo-Boolean functions quadratic binary optimization Quantitative methods in economics & management reformulation methods Sciences économiques & de gestion Studies Theoretical Upper bounds Variables 90C20 90C10 90C30 Pseudo-Boolean functions Reformulation methods Quadratic binary optimization Nonlinear binary optimization 90C09
ISSN:	0025-5610, 1436-4646, 1436-4646
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Popis
Shrnutí:	Very large nonlinear unconstrained binary optimization problems arise in a broad array of applications. Several exact or heuristic techniques have proved quite successful for solving many of these problems when the objective function is a quadratic polynomial. However, no similarly efficient methods are available for the higher degree case. Since high degree objectives are becoming increasingly important in certain application areas, such as computer vision, various techniques have been recently developed to reduce the general case to the quadratic one, at the cost of increasing the number of variables by introducing additional auxiliary variables. In this paper we initiate a systematic study of these quadratization approaches. We provide tight lower and upper bounds on the number of auxiliary variables needed in the worst-case for general objective functions, for bounded-degree functions, and for a restricted class of quadratizations. Our upper bounds are constructive, thus yielding new quadratization procedures. Finally, we completely characterize all “minimal” quadratizations of negative monomials.
Bibliografia:	SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 ObjectType-Article-1 ObjectType-Feature-2 content type line 23 scopus-id:2-s2.0-84973103255 PAI P7/36 Comex
ISSN:	0025-5610 1436-4646 1436-4646
DOI:	10.1007/s10107-016-1032-4