Convex Combinatorial Optimization

Onn, Shmuel; Rothblum, Uriel G.

doi:10.1007/s00454-004-1138-y

Cited by 48 publications

(57 citation statements)

References 54 publications

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“…We now show that a set E covering all edge-directions of the polytope conv(S) underlying a convex discrete optimization problem over a finite set S ⊂ Z n allows to solve it by solving polynomially many linear discrete optimization counterparts over S. The following theorem extends and unifies the corresponding reductions in [49] and [17] for convex combinatorial optimization and convex integer programming respectively. Recall from §1.3 that the radius of a finite set S ⊂ Z n is defined to be ρ(S) := max{|x i | : x ∈ S, i = 1, .…”

Section: Strongly Polynomial Reduction Of Convex To Linear Discrete Omentioning

confidence: 87%

“…The precise relevant definitions and statements of the theorems and corollaries mentioned here are provided in the relevant sections in the monograph body. As mentioned above, most of these results are adaptations or extensions of results from one of the papers [5,12,13,14,15,16,17,25,39,47,48,49,50,51]. The monograph gives many more applications and results that may turn out to be useful in future development of the theory of convex discrete optimization.…”

Section: Outline and Overview Of Main Results And Applicationsmentioning

confidence: 99%

“…The monograph has been written under the support of the ISF -Israel Science Foundation. The theory developed here is based on and is a culmination of several recent papers including [5,12,13,14,15,16,17,25,39,47,48,49,50,51] written in collaboration with several colleagues -Eric Babson, Jesus De Loera, Komei Fukuda, Raymond Hemmecke, Frank Hwang, Vera Rosta, Uriel Rothblum, Leonard Schulman, Bernd Sturmfels, Rekha Thomas, and Robert Weismantel. By developing and using a combination of geometric and algebraic tools, we are able to provide polynomial time algorithms for several broad classes of convex discrete optimization problems.…”

Section: In Combinatorial Optimization S ⊆ {0 1}mentioning

confidence: 99%

“…, n} possessing some combinatorial property of interest (for instance, the family of bases of a matroid over N, see §3.4.2). The convex combinatorial optimization problem then also has the following interpretation (taken in [47,49]). We are given a weighting ω : N −→ Z d of elements of the ground set by d-dimensional integer vectors.…”

Section: Convex Combinatorial Optimization and Morementioning

confidence: 99%

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Convex Discrete Optimization

Onn

2008

Encyclopedia of Optimization

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We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial optimization problems and convex integer programming problems in variable dimension. We discuss some of the many applications of this theory including to quadratic programming, matroids, bin packing and cutting-stock problems, vector partitioning and clustering, multiway transportation problems, and privacy and confidential statistical data disclosure. Highlights of our work include a strongly polynomial time algorithm for convex and linear combinatorial optimization over any family presented by a membership oracle when the underlying polytope has few edge-directions; a new theory of so-termed n-fold integer programming, yielding polynomial time solution of important and natural classes of convex and linear integer programming problems in variable dimension; and a complete complexity classification of high dimensional transportation problems, with practical applications to fundamental problems in privacy and confidential statistical data disclosure.

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Section: Strongly Polynomial Reduction Of Convex To Linear Discrete Omentioning

confidence: 87%

Section: Outline and Overview Of Main Results And Applicationsmentioning

confidence: 99%

Section: In Combinatorial Optimization S ⊆ {0 1}mentioning

confidence: 99%

Section: Convex Combinatorial Optimization and Morementioning

confidence: 99%

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Convex Discrete Optimization

Onn

2008

Encyclopedia of Optimization

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“…The first key lemma, extending results of [101] for combinatorial optimization, shows that when a suitable geometric condition holds, it is possible to efficiently reduce the convex integer maximization problem to the solution of polynomially many linear integer programming counterparts. As we will see, this condition holds naturally for a broad class of problems in variable dimension.…”

Section: Boundary Cases Of Complexitymentioning

confidence: 91%

Nonlinear Integer Programming

Li¹,

Sun²

2006

International Series in Operations Research &Amp; Management Science

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Abstract. Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic.The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms.We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems.It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.

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