Farkas-Type Results With Conjugate Functions

Boţ, Radu Ioan; Wanka, Gert

doi:10.1137/030602332

Cited by 44 publications

(25 citation statements)

References 12 publications

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“…Semi-infinite versions of Lemma 4, with f t : R n → R convex for all t ∈ T , are [16], Theorem 4.1 (where C = R n and f : R n → R) and [3], Theorem 5.6. Observe that if f is either linear or continuous at some point of A then, by Theorem 1, we can replace (4.4) with…”

Section: Lemma 4 Let σ Be Fm and α ∈ R Then F (X) ≥ α Is Consequencmentioning

confidence: 99%

“…Farkas-type results for convex systems (characterizing families of inequalities which are consequences of a consistent convex system σ) are fundamental in convex optimization and in other fields as game theory, set containment problems, etc. Since the literature on Farkas lemma, and its extensions, is very wide (see, e.g., the survey in [15]), we just mention here some works giving Farkas-type results for the kind of systems considered in the paper: [3,11,16,22] for semi-infinite systems, [9,14,17,21] for infinite systems, and [8,18,19] for cone convex systems.…”

Section: Introductionmentioning

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: Lemma 4 Let σ Be Fm and α ∈ R Then F (X) ≥ α Is Consequencmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

New Farkas-type constraint qualifications in convex infinite programming

et al. 2007

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“…This problem has been studied extensively under various degrees of restrictions imposed on f t , t ∈ T or on the underlying space and many problems in optimization and approximation theory such as linear semi-infinite optimization and the best approximation with restricted ranges can be recast into the form (1.1), see for example [8,16,17,23,24,33,35,36,38,40,41,42]. Another important and classical example of (1.1) is the following so-called conic programming problem, which recently received much attention (cf.…”

Section: 2)mentioning

confidence: 99%

Constraint Qualifications for Extended Farkas's Lemmas and Lagrangian Dualities in Convex Infinite Programming

Fang¹,

Li²,

Ng³

2010

SIAM J. Optim.

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Abstract. For an inequality system defined by a possibly infinite family of proper functions (not necessarily lower semicontinuous), we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions. Under the new constraint qualifications, we obtain characterizations of those reverse-convex inequalities which are consequence of the constrained system, and we provide necessary and/or sufficient conditions for a stable Farkas lemma to hold. Similarly, we provide characterizations for constrained minimization problems to have the strong or strong stable Lagrangian dualities. Several known results in the conic programming problem are extended and improved.

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“…The proof follows from (3.4). It is well known that the Farkas lemma has a great deal of applications in optimization (see [3], [5], [7], and references therein). Observe that Theorem 3.1 yields directly the strong Lagrange duality and stable strong Lagrange duality for the problem…”

Section: Preliminariesmentioning

confidence: 99%

“…To each version of the stable Farkas lemma corresponds a stable (strong) duality theorem showing that strong duality still holds whenever perturbing the objective function of the primal problem with a linear continuous functional (see, e.g., [2], [14]). Some of the recent versions of the Farkas lemma (see, e.g., [3], [5], [6], [7], [10], [15]) are so general that the following question arises in a natural way: is it possible to approach the fundamentals of mathematics from suitable generalized versions of the Farkas lemma?…”

mentioning

confidence: 99%

From the Farkas Lemma to the Hahn--Banach Theorem

Dinh¹,

Goberna²,

López³

et al. 2014

SIAM J. Optim.

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Abstract. This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f (x) ≥ 0 which are consequences of a composite convex inequality (S • g)(x) ≤ 0 on a closed convex subset of a given locally convex topological vector space X, where f is a proper lower semicontinuous convex function defined on X, S is an extended sublinear function, and g is a vector-valued S-convex function. In parallel, associated versions of a stable Farkas lemma, considering arbitrary linear perturbations of f , are also given. These new versions of the Farkas lemma, and their corresponding stable forms, are established under the weakest constraint qualification conditions (the so-called closedness conditions), and they are actually equivalent to each other, as well as equivalent to an extended version of the so-called Hahn-Banach-Lagrange theorem, and its stable version, correspondingly. It is shown that any of them implies analytic and algebraic versions of the Hahn-Banach theorem and the Mazur-Orlicz theorem for extended sublinear functions.

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Farkas-Type Results With Conjugate Functions

Cited by 44 publications

References 12 publications

New Farkas-type constraint qualifications in convex infinite programming

New Farkas-type constraint qualifications in convex infinite programming

Constraint Qualifications for Extended Farkas's Lemmas and Lagrangian Dualities in Convex Infinite Programming

From the Farkas Lemma to the Hahn--Banach Theorem

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