Stability in linear optimization and related topics. A personal tour

López, Marco A.

doi:10.1007/s11750-011-0213-9

Cited by 9 publications

(10 citation statements)

References 55 publications

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“…If the Fourier-Motzkin procedure for some permutation results in a clean system then there are no conic index sets by the contrapositive of Proposition A. 16. If there are no conic index sets, then applying the Fourier-Motzkin procedure to any permutation of the variables cannot result in dirty variables since a dirty variable implies the existence of a conic index set by Corollary A.11.…”

Section: A12 Clean Systems Are Permutation Independentmentioning

confidence: 90%

“…Proposition A. 16. If (SILP) contains a conic index set, then the Fourier-Motzkin elimination procedure will terminate with at least one dirty variable regardless of the variable permutation used in the elimination procedure.…”

Section: A12 Clean Systems Are Permutation Independentmentioning

confidence: 99%

“…These are linear optimization problems with a finite number of variables and possibly infinitely many constraints. For surveys covering theory, applications and algorithms for semi-infinite linear programming see Goberna and López [11], Hettich and Kortanek [12] and López [16]. In this paper we use the duality theory of semi-infinite linear programs to understand the duality of both convex and conic programs.…”

Section: Introductionmentioning

confidence: 99%

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Projection: A Unified Approach to Semi-Infinite Linear Programs and Duality in Convex Programming

2015

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Fourier-Motzkin elimination is a projection algorithm for solving finite linear programs. We extend Fourier-Motzkin elimination to semi-infinite linear programs which are linear programs with finitely many variables and infinitely many constraints. Applying projection leads to new characterizations of important properties for primal-dual pairs of semi-infinite programs such as zero duality gap, feasibility, boundedness, and solvability. Extending the Fourier-Motzkin elimination procedure to semi-infinite linear programs yields a new classification of variables that is used to determine the existence of duality gaps. In particular, the existence of what the authors term dirty variables can lead to duality gaps. Our approach has interesting applications in finite-dimensional convex optimization. For example, sufficient conditions for a zero duality gap, such as the Slater constraint qualification, are reduced to guaranteeing that there are no dirty variables. This leads to completely new proofs of such sufficient conditions for zero duality.

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Section: A12 Clean Systems Are Permutation Independentmentioning

confidence: 90%

Section: A12 Clean Systems Are Permutation Independentmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Projection: A Unified Approach to Semi-Infinite Linear Programs and Duality in Convex Programming

2015

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“…For all these reasons, several survey papers and extended reviews on LSIO have been published in the past, the last ones dated in 2005 ( [87], [88]), and 2014 ([104], exclusively focused on uncertain LSIO). Linear semi-in…nite optimization was also considered in two other surveys, the …rst one on semi-in…nite optimization, published in 2007 ( [160]), and the second one on the stability analysis of (ordinary, semi-in…nite and in…nite) linear optimization problems, published in 2012 ( [159]). It is worth mentioning another survey on non-linear semi-in…nite optimization [196] which explicitly precludes LSIO.…”

Section: Introductionmentioning

confidence: 99%

Recent contributions to linear semi-infinite optimization: an update

Goberna

López

2018

Ann Oper Res

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This paper reviews the state-of-the-art in the theory of deterministic and uncertain linear semi-in…nite optimization, presents some numerical approaches to this type of problems, and describes a selection of recent applications in a variety of …elds. Extensions to related optimization areas, as convex semi-in…nite optimization, linear in…nite optimization, and multi-objective linear semi-in…nite optimization, are also commented.

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“…For all these reasons, several survey papers and extended reviews on LSIO have been published in the past, the last ones dated in 2005 ( [74], [75]), and 2014 ( [91], exclusively focused on uncertain LSIO). Linear semi-in…nite optimization was also considered in two other surveys, the …rst one on semi-in…nite optimization, published in 2007 ( [139]), and the second one on the stability analysis of (ordinary, semi-in…nite and in…nite) linear optimization problems, published in 2012 ( [138]). It is worth mentioning another survey on non-linear semi-in…nite optimization [170] which explicitly precludes LSIO.…”

Section: Introductionmentioning

confidence: 99%