Sequential convexification in reverse convex and disjunctive programming

Balas, Egon; Tama, Joseph M.; Tind, Jørgen

doi:10.1007/bf01587096

Cited by 19 publications

(5 citation statements)

References 8 publications

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“…The foundations of disjunctive programming were laid in a July 1974 technical report, published 24 years later as an invited paper ) with a foreword. For additional work on disjunctive programming in the seventies and eighties see Balas (1979Balas ( , 1985; Balas and Jeroslow (1980); Balas, Tama, and Tind (1989); Blair (1976Blair ( , 1980; Jeroslow (1977Jeroslow ( , 1987Jeroslow ( , 1989; Sherali and Shetty (1980). In particular, Balas (1979) contains a detailed account of the origins of the disjunctive approach and the relationship of disjunctive cuts to Gomory's mixed integer cut, intersection cuts and others.…”

Section: Disjunctive Programmingmentioning

confidence: 99%

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Projection, Lifting and Extended Formulation in Integer and Combinatorial Optimization

Balas

2005

Ann Oper Res

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Abstract. This is an overview of the significance and main uses of projection, lifting and extended formulation in integer and combinatorial optimization. Its first two sections deal with those basic properties of projection that make it such an effective and useful bridge between problem formulations in different spaces, i.e. different sets of variables. They discuss topics like projection and restriction, the integrality-preserving property of projection, the dimension of projected polyhedra, conditions for facets of a polyhedron to project into facets of its projections, and so on. The next two sections describe the use of projection for comparing the strength of different formulations of the same problem, and for proving the integrality of polyhedra by using extended formulations or lifting. Section 5 deals with disjunctive programming, or optimization over unions of polyhedra, whose most important incarnation are mixed 0-1 programs and their partial relaxations. It discusses the compact representation of the convex hull of a union of polyhedra through extended formulation, the connection between the projection of the latter and the polar of the convex hull, as well as the sequential convexification of facial disjunctive programs, among them mixed 0-1 programs, with the related concept of disjunctive rank. Section 6 reviews lift-and-project cuts, the construction of cut generating linear programs, and techniques for lifting and for strengthening disjunctive cuts. Section 7 discusses the recently discovered possibility of solving the higher dimensional cut generating linear program without explicitly constructing it, by a sequence of properly chosen pivots in the simplex tableau of the linear programming relaxation. Finally, section 8 deals with different ways of combining cuts with branch and bound, and briefly discusses computational experience with lift-and-project cuts.

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Section: Disjunctive Programmingmentioning

confidence: 99%

“…A necessary and sufficient condition is given in Balas, Tama, and Tind (1989). The most important class of facial disjunctive programs are mixed 0-1 programs, and for that case Theorem 5.5. asserts that if we denote…”

Section: Theorem 55 (Balas 1998) Letmentioning

confidence: 99%

Projection, Lifting and Extended Formulation in Integer and Combinatorial Optimization

Balas

2005

Ann Oper Res

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“…While faciality is a sufficient condition for the theorem to hold, it is not necessary. A necessary condition was given in [BTT89].…”

Section: Where P (K) Is the Set Of Those (A#) E !'+I For Which There Exist Vectors U V E R++pmentioning

confidence: 99%

A lift-and-project cutting plane algorithm for mixed 0–1 programs

Balas

Ceria

Cornuéjols

1993

Mathematical Programming

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this document has bien oapprov-ed for public release and sale; its distribution is unhimited.This research was supported in part by the National Science Foundation, Grant #DDM-8901495 and the Office of Naval Research through Contract N00014-85-K-0198. Management Science Research Group Graduate School of Industrial AdministrationCarnegie Mellon University Pittsburgh, PA 15213 AbstractWe propose a cutting plane algorithm for mixed 0-1 programs based on a family of polyhedra which strengthen the usual LP relaxation. We show how to generate a facet of a polyhedron in this family which is most violated by the current fractional point. This cut is found through the solution of a linear program that has about twice the size of the usual LP relaxation. A lifting step is used to reduce the size of the LP's needed to generate the cuts. An additional strengthening step suggested by Baas and Jeroslow is then applied. We report our computational experience with a preliminary version of the algorithm. This approach is related to the work of Balas on disjunctive programming, the matrix cut relaxations of Lovisz and Schrijver and the hierarchy of relaxations of Sherali and Adams.

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“…(6) (7) (8) Even in the simplest cases when all data are linear: /(z, y) = c1z + d1y, VJ(Y) = d2y, D = {(z, y): Atz + BtY ~ Dt. z E R~} while O(z) = {y: A2z + B2Y 5 02, y E R~}, the feedback relation between upper and lower levels creates nonconvexities which can not be easily handled by standards methods of nonlinear programming.…”

Section: (Multilevel Programming)mentioning

confidence: 99%

“…R such that is called the quasiconjugate of f(x). When sup{f(z) : z E R"} = +oo, as it often occurs, we simply have f 8 …”

Section: Duality In Global Optimizationmentioning

confidence: 99%