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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Veröffentlicht in:Mathematical programming Jg. 162; H. 1-2; S. 115 - 144
Hauptverfasser: Anthony, Martin, Boros, Endre, Crama, Yves, Gruber, Aritanan
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2017
Springer Nature B.V
Springer
Schlagworte:
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
Online-Zugang:Volltext
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Zusammenfassung: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.
Bibliographie:SourceType-Scholarly Journals-1
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scopus-id:2-s2.0-84973103255
PAI P7/36 Comex
ISSN:0025-5610
1436-4646
1436-4646
DOI:10.1007/s10107-016-1032-4