Locally Farkas–Minkowski Systems in Convex Semi-Infinite Programming

Fajardo, Maria Dolores; López, Marco A.

doi:10.1023/a:1021700702376

Cited by 37 publications

(14 citation statements)

References 5 publications

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“…It was extended in [24] to the framework of linear semi-infinite systems in the Euclidean space, and deeply studied in [11], Chapter 5. The consequences of its extension to convex semi-infinite systems were analyzed in [10].…”

Section: Lemmamentioning

confidence: 99%

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New Farkas-type constraint qualifications in convex infinite programming

et al. 2007

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Abstract. This paper provides KKT and saddle point optimality conditions, duality theorems and stability theorems for consistent convex optimization problems posed in locally convex topological vector spaces. The feasible sets of these optimization problems are formed by those elements of a given closed convex set which satisfy a (possibly infinite) convex system. Moreover, all the involved functions are assumed to be convex, lower semicontinuous and proper (but not necessarily real-valued). The key result in the paper is the characterization of those reverse-convex inequalities which are consequence of the constraints system. As a byproduct of this new versions of Farkas' lemma we also characterize the containment of convex sets in reverse-convex sets. The main results in the paper are obtained under a suitable Farkas-type constraint qualifications and/or a certain closedness assumption.Mathematics Subject Classification. 90C25, 90C34, 90C46, 90C48.

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Section: Lemmamentioning

confidence: 99%

“…An example of infinite convex system (similar to [10], Ex. 2.1) that also illustrates this fact is the following.…”

Section: Proposition 3 Let Z ∈ Amentioning

confidence: 99%

New Farkas-type constraint qualifications in convex infinite programming

et al. 2007

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“…Observe that D(F; x) is nothing else that the normal cone to F at x; also represented by N (F; x): As a consequence of Lemma 2.1 in [4], the LFM property should be investigated only at the boundary feasible points (if is LFM, one has A(x) 6 = f0 n g for every x 2 bd(F ); i.e., is tight): Also, it is shown in Theorem 7.10 in [3] that is LFM if and only if L is LFM.…”

Section: Locally Farkas-minkowski Systemsmentioning

confidence: 97%

“…In [4] the relationship between this constraint quali…cation and the upper semicontinuity (in the sense of Berge) of the so-called active and sup-active mappings is analyzed, as well as the ful…lment of the Valadier formula for the supremum function under some conditions involving the LFM property. Section 2 of this paper studies the relative interior and the relative boundary of F for a general consistent convex system.…”

Section: Introductionmentioning

confidence: 99%

Some results about the facial geometry of convex semi-infinite systems

Fajardo

López

2006

Optimization

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“…In [6] the relationship between this constraint quali cation and the upper semicontinuity (in the sense of Berge) of the socalled active and sup-active mappings is analyzed, as well as the ful lment of the Valadier formula for the supremum function under some conditions involving the LFM property.…”

Section: Introductionmentioning

confidence: 99%

Linear representations and quasipolyhedrality of a finite-valued convex function

2008

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The possibility of representing the epigraph of a nite-valued convex function by means of a (locally) Farkas-Minkowski linear semi-in nite inequalities system is studied in this paper. Moreover, we prove that the so-called locally polyhedral representations characterize the function, giving rise to the concept of quasipolyhedral function. Conditions for its conjugate to be also quasipolyhedral are obtained, as well as the characterization of its subdifferential and " subdifferential in terms of a speci c sequence of ordinary polyhedral functions.

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Locally Farkas–Minkowski Systems in Convex Semi-Infinite Programming

Cited by 37 publications

References 5 publications

New Farkas-type constraint qualifications in convex infinite programming

New Farkas-type constraint qualifications in convex infinite programming

Some results about the facial geometry of convex semi-infinite systems

Linear representations and quasipolyhedrality of a finite-valued convex function

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