New Sequential Lagrange Multiplier Conditions Characterizing Optimality without Constraint Qualification for Convex Programs

Jeyakumar, V.; Lee, G. M.; Dinh, N.

doi:10.1137/s1052623402417699

Cited by 113 publications

(81 citation statements)

References 14 publications

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“…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%

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: Introductionmentioning

confidence: 99%

New Farkas-type constraint qualifications in convex infinite programming

et al. 2007

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“…Given SDP problem in the form (11), the condition (17) is equivalent to the emptiness of the set L * .…”

Section: Propositionmentioning

confidence: 99%

“…In literature, a special attention is devoted to the results that do not need additional conditions on the constraints, so called constraint qualifications (CQ). For SIP such optimality conditions were proposed, for example, in [4,11,14], and for SDP in [8,17,18], and some other papers.…”

Section: Introductionmentioning

confidence: 99%

Optimality criteria without constraint qualifications for linear semidefinite problems

Kostyukova

Tchemisova

2012

J Math Sci

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We consider two closely related optimization problems: a problem of convex SemiInfinite Programming with multidimensional index set and a linear problem of Semidefinite Programming. In study of these problems we apply the approach suggested in our recent paper [14] and based on the notions of immobile indices and their immobility orders. For the linear semidefinite problem, we define the subspace of immobile indices and formulate the first order optimality conditions in terms of a basic matrix of this subspace. These conditions are explicit, do not use constraint qualifications, and have the form of criterion. An algorithm determining a basis of the subspace of immobile indices in a finite number of steps is suggested. The optimality conditions obtained are compared with other known optimality conditions.

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“…This motivates us to derive sequential optimality conditions for multiobjective fractional programming problem. Recently, work has been done in this direction for convex programming problems with cone convex constraints by Jeyakumar et al [10,12] and Bai et al [1]. Jeyakumar et al [10,12] introduced the concept of sequential Lagrange multiplier rules for convex programs with cone convex constraints using the concept of epigraph of conjugate function in terms of e-subdifferential computed at optimal solution.…”

Section: Introductionmentioning

confidence: 99%