Sensitivity theorems in integer linear programming

Cook, William J.; Gerards, A.M.H.; Schrijver, Alexander; Tardos, Éva

doi:10.1007/bf01582230

Cited by 195 publications

(121 citation statements)

References 18 publications

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“…The subadditive dual has been used primarily for sensitivity analysis in integer programming (e.g. [5]). It has apparently not been used in the context of nogood-based search.…”

Section: Nondecreasing Subadditive Function H and That H(ax) ≥ H(b)mentioning

confidence: 99%

Duality in Optimization and Constraint Satisfaction

Hooker

2006

Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems

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Abstract. We show that various duals that occur in optimization and constraint satisfaction can be classified as inference duals, relaxation duals, or both. We discuss linear programming, surrogate, Lagrangean, superadditive, and constraint duals, as well as duals defined by resolution and filtering algorithms. Inference duals give rise to nogood-based search methods and sensitivity analysis, while relaxation duals provide bounds. This analysis shows that duals may be more closely related than they appear, as are surrogate and Lagrangean duals. It also reveals common structure between solution methods, such as Benders decomposition and Davis-Putnam-Loveland methods with clause learning. It provides a framework for devising new duals and solution methods, such as generalizations of mini-bucket elimination. Two Kinds of DualsDuality is perennial theme in optimization and constraint satisfaction. Wellknown optimization duals include the linear programming (LP), Lagrangean, surrogate, and superadditive duals. The constraint satisfaction literature discusses constraint duals as well as search methods that are closely related to duality.These many duals can be viewed as falling into two classes: inference duals and relaxation duals [12]. The two classes represent quite different concepts of duality. This is perhaps not obvious at first because the traditional optimization duals just mentioned can be interpreted as both inference and relaxation duals.Classifying duals as inference or relaxation duals reveals relationships that might not otherwise be noticed. For instance, the surrogate and Lagrangean duals do not seem closely related, but by viewing them as inference duals rather than relaxation duals, one sees that they are identical except for a slight alteration in the type of inference on which they are based.A general analysis of duality can also unify some existing solution methods and suggest new ones. Inference duals underlie a number of nogood-based search methods and techniques for sensitivity analysis. For instance, Benders decomposition and Davis-Putnam-Loveland methods with clause learning, which appear unrelated, are nogood-based search methods that result from two particular inference duals. Since any inference method defines an inference dual, one can

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“…The subadditive dual has been used primarily for sensitivity analysis in integer programming (e.g. [5]). It has apparently not been used in the context of nogood-based search.…”

Section: Nondecreasing Subadditive Function H and That H(ax) ≥ H(b)mentioning

confidence: 99%

Duality in Optimization and Constraint Satisfaction

Hooker

2006

Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems

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“…Cook, Gerards, Schrijver, and Tardos [3] extended this result by proving that for each matrix A E zmxn, there exists a t E N, such that (6) for each b E zm we have that…”

Section: From ( 1) and (2) We Getmentioning

confidence: 99%

“…(ii) As C. Blair observed, (6) is equivalent to the result, due to Blair and Jeroslow [1], that "each integer programming value function is a Gomory function." For a discussion see Cook, Gerards, Schrijver, and Tardos [3].…”

Section: From ( 1) and (2) We Getmentioning

confidence: 99%

On cutting planes and matrices

Gerards¹

1991

DIMACS Series in Discrete Mathematics And Theoretical Computer Science

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ABSTRACT. Continuing the work of Chvatal and Gomory, Schrijver proved that any rational polyhedron {x I Ax ~ b} has finite Chvatal rank. This was extended by Cook, Gerards, Schrijver, and Tardos, who proved that in fact this Chvatal rank can be bounded from above by a number only depending on A, hence independent of b . The aim of this note is to show that the latter result can be proved quite easily from the result of Chvatal and Schrijver.

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“…In fact, the $\mathrm{L}$ -convex function minimization problem can be solved in polynomial-time by combining submodular set function minimization algorithms and the proximity property [12] [8,9,17] in developing efficient algorithms for resource allocation problems. Different types of theorems on proximity have also been investigated: proximity between integral and real optimal solutions in [1,2,7,9,10] and proximity for anumber of resource allocation problems with min-max tyPe objective functions in [5]. This paper addresses proximity properties of $\mathrm{L}_{2^{-}}/\mathrm{M}_{2}$ -convex functions.…”

Section: Introductionmentioning

confidence: 99%

Proximity theorems of discrete convex functions

Murota

Tamura

2004

Mathematical Programming

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Aproximity theorem is astatement that, given an optimization problem and its relaxation, an optimal solution to the original problem exists in acertain neighborhood of asolution to the relaxation. -convexity(in their variants). $\mathrm{L}_{2}$ -convex functions and $\mathrm{M}_{2^{-}}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{x}$ functions constitute larger classes of discrete convex functions that are relevant to the polymatroid intersection problem, where an $\mathrm{L}_{2}$ -convex function is, by definition, the infimal convolution of two $\mathrm{L}$ -convex functions and an $\mathrm{M}_{2}$ -convex function is the sum of two M-convex functions. The $\mathrm{M}_{2}$ -convex function minimization problem is equivalent to the M-convex submodular flow problem [20] which is an extension of the submodular flow problem [3].

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Sensitivity theorems in integer linear programming

Cited by 195 publications

References 18 publications

Duality in Optimization and Constraint Satisfaction

Duality in Optimization and Constraint Satisfaction

On cutting planes and matrices

Proximity theorems of discrete convex functions

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