The SECQ, Linear Regularity, and the Strong CHIP for an Infinite System of Closed Convex Sets in Normed Linear Spaces

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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.

Section: 2)mentioning

confidence: 99%

Section: 2)mentioning

confidence: 99%

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

Fang¹,

Li²,

Ng³

2010

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“…In particular, (1.3) reduces to the SECQ introduced in [35] if g i = δ Ci for some family of closed convex sets {C i : i ∈ I}. We show that by suitably choosing the family {g i : i ∈ I}, the conical EHP reduces to the CCCQ defined in [7,26].…”

Section: Introductionmentioning

confidence: 99%

Constraint Qualifications for Convex Inequality Systems with Applications in Constrained Optimization

Li¹,

Ng²,

Pong

2008

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Abstract. For an inequality system defined by an infinite family of proper convex functions, we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions and study relationships between these new constraint qualifications and other well known constraint qualifications including the basic constraint qualification studied by Hiriart-Urrutty and Lemarechal, Li, Nahak and Singer. Extensions of known results to more general settings are presented, and applications to particular important problems, such as conic programming and approximation theory, are also studied.

“…(see, e.g., [7], [40], and [41]). Let = ff t (x) 0; t 2 T ; x 2 Cg be consistent and let v 2 X and 2 R: Then the asymptotic Farkas'Lemma (Theorem 4.1 in [13]) establishes that…”

Section: Introduction Many Optimization Problems Are Formulated In Tmentioning

confidence: 99%

“…The stability theory of the feasible set in semiin…nite programming has been reviewed in [18], where the connection between lower semicontinuity of F and constraint quali…cations (e.g., Slater-type and interior-type conditions) are discussed. There exists a wide literature on constraint quali…cations in convex (and extended convex) in…nite dimensional optimization, where they provide optimality conditions and duality theorems (see, e.g., [6], [15], [28], [29], [36], [38], [39], [40], and [41]). …”

Section: Introduction Many Optimization Problems Are Formulated In Tmentioning

confidence: 99%

On the Stability of the Feasible Set in Optimization Problems

Dinh¹,

Goberna²,

López³

2010

Abstract. This paper provides stability theorems for the feasible set of optimization problems posed in locally convex topological vector spaces. The problems considered in this paper have an arbitrary number of inequality constraints and one constraint set. Di¤erent models are discussed, depending on the properties of the constraint functions (linear or not, convex or not, but at least lower semicontinuous) and one closed constraint set (but not necessarily convex). The parameter space is formed by systems of the same type as the nominal one (with the same space of variables and the same number of constraints), where the constraint set can be perturbed or not, equipped with the metric of the uniform convergence on the positive multiples of a …xed barrelled neighborhood of zero. In …nite dimensions, this topology describes the unifom convergence on compact sets and, in the particular case that the constraints are linear, the uniform convergence of the vector coe¢ cients. The paper examines, in a uni…ed way, the lower and upper semicontinuity, and the closedness, of the feasible set mapping, the stable consistency of the constraint system with respect to arbitrary and right-hand side perturbations, Tuy and Robinson regularities, and other desirable stability properties of the feasible set.