On the stability of closed-convex-valued mappings and the associated boundaries

Goberna, Miguel Á.; López, Marco A.; Todorov, Maxim I.

doi:10.1016/j.jmaa.2005.01.003

Cited by 17 publications

(9 citation statements)

References 4 publications

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“…The next result characterizes the usc property of F provided the decision space X is locally compact, in which case X is …nite dimensional and, so, isomorphic to R n ([34, §15.7 (1)]). The su¢ cient part is totally general and its proof is a straightforward consequence of statement (iii) in [22,Lemma 2]. For j = 6; 7; this result generalizes the corresponding ones in [21] and [44] because we consider here a …xed set constraint C 0 R n and extended constraint functions f t ; t 2 T: Theorem 7.4.…”

Section: Upper Semicontinuity In Rmentioning

confidence: 67%

On the Stability of the Feasible Set in Optimization Problems

Dinh¹,

Goberna²,

López³

2010

SIAM J. Optim.

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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.

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Section: Upper Semicontinuity In Rmentioning

confidence: 67%

On the Stability of the Feasible Set in Optimization Problems

Dinh¹,

Goberna²,

López³

2010

SIAM J. Optim.

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“…In particular, taking A = bd F we get the direct statement of Proposition 1 in [5], whose corresponding converse statement establishes that, if F = conv bd F is lsc and closed at y 0 ∈ dom F, then bd F is also lsc at y 0 . The next two results are counterparts of this converse statement for rbd F and extr F (instead of bd F).…”

Section: Lower Semicontinuitymentioning

confidence: 83%

“…The following result (Lemma 2 in [5]), which establishes the relationships between M and M ρ , will be useful in the next sections.…”

Section: Preliminariesmentioning

confidence: 99%

“…Throughout this paper we consider given an arbitrary convex-valued mapping F : Y ⇒ R n , where the domain Y is a locally metrizable space (i.e., Y is equipped with the topology induced by an extended distance on Y , δ, taking values on R + ∪{+∞}), and its boundary mapping, relative boundary mapping, and extreme points set mapping, bd F, rbd F, and extr F. The relationships between F and bd F, assuming that F = conv bd F, have been studied in [5]. In the same vein, this paper considers the relationships between the stability properties of F, rbd F, and extr F, assuming that F = conv rbd F and F = conv extr F, respectively.…”

Section: Introduction the Main Objective Of The Paper Is To Analyze mentioning

confidence: 99%

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On the Stability of Convex-valued Mappings and Their Relative Boundary and Extreme Points Set Mappings

Goberna¹,

Todorov²,

Serio³

2006

SIAM J. Optim.

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This paper deals with the transmission of the main stability properties (lower and upper semicontinuity in Berge sense, and closedness) from a given closed-convex-valued mapping to its corresponding relative boundary and extreme point set mappings, and vice versa. The domain of the mappings considered in this paper are locally metrizable spaces and the images range on Euclidean spaces. Important examples of the class of mappings considered in this paper are the feasible set mapping and the optimal set mapping of convex optimization problems, for which the space of parameters is the result of perturbing a given nominal problem.

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“…The statement of the following well-known result (see, e.g., The next result is Lemma 2(iii) in [13].…”

Section: Preliminariesmentioning

confidence: 92%

Stability of the Feasible Set Mapping of Linear Systems with an Exact Constraint Set

2007

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This paper deals with the stability of the feasible set mapping of linear systems of an arbitrary number (possibly infinite) of equations and inequalities such that the variable x ranges on a certain fixed constraint set X ⊂ R n (X could represent the solution set of a given constraint system, e.g., the positive cone of R n in the case of sign constraints). More in detail, the paper provides necessary as well as sufficient conditions for the lower and upper semicontinuity (in Berge sense), and the closedness, of the set-valued mapping which associates, with any admissible perturbation of the given (nominal) system its feasible set. The parameter space is formed by all the systems having the same structure (i.e., the same number of variables, equations and inequalities) as the nominal one, and the perturbations are measured by means of the pseudometric of the uniform convergence. Mathematics Subject Classification (2000): Primary: 49K40, 90C34; Secondary: 15A39, 90C05.

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On the stability of closed-convex-valued mappings and the associated boundaries

Cited by 17 publications

References 4 publications

On the Stability of the Feasible Set in Optimization Problems

On the Stability of the Feasible Set in Optimization Problems

On the Stability of Convex-valued Mappings and Their Relative Boundary and Extreme Points Set Mappings

Stability of the Feasible Set Mapping of Linear Systems with an Exact Constraint Set

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