Stability and Well-Posedness in Linear Semi-Infinite Programming

Cánovas, M. J.; López, Marco A.; Parra, J.; Todorov, Maxim I.

doi:10.1137/s1052623497319869

Cited by 55 publications

(48 citation statements)

References 12 publications

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“…Thus, (9) implies that min j∈J {ᾱ j +α j } ≤ max j∈J φ Z j (−u, −v − ). Letting → 0 and noting that, for each j ∈ J, φ Z j is a continuous convex function and min…”

Section: And Hencementioning

confidence: 99%

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Radius of robust feasibility formulas for classes of convex programs with uncertain polynomial constraints

Goberna¹,

Jeyakumar²,

Li³

et al. 2016

Operations Research Letters

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The radius of robust feasibility of a convex program with uncertain constraints gives a value for the maximal 'size' of an uncertainty set under which robust feasibility can be guaranteed. This paper provides an upper bound for the radius for convex programs with uncertain convex polynomial constraints and exact formulas for convex programs with SOS-convex polynomial constraints (or convex quadratic constraints) under affine data uncertainty. These exact formulas allow the radius to be computed by commonly available software.

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“…Thus, (9) implies that min j∈J {ᾱ j +α j } ≤ max j∈J φ Z j (−u, −v − ). Letting → 0 and noting that, for each j ∈ J, φ Z j is a continuous convex function and min…”

Section: And Hencementioning

confidence: 99%

“…It was inspired by the notion of consistency radius used in linear semi-infinite programming in order to guarantee the feasibility of the nominal problem under perturbations preserving the number of constraints [7,8,9]. This notion extends the concept of radius of robust feasibility introduced in [13] for robust linear programs.…”

Section: Introductionmentioning

confidence: 99%

Radius of robust feasibility formulas for classes of convex programs with uncertain polynomial constraints

Goberna¹,

Jeyakumar²,

Li³

et al. 2016

Operations Research Letters

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“…To the best of our knowledge, this approach is new in the literature on semi-infinite and infinite programming despite many publications related to various stability properties and applications of linear infinite inequality systems, most of which concern the case of finite-dimensional spaces X of decisions variables (i.e., in the semi-infinite programming framework); see, e.g., [1,19] for comprehensive overviews on this field and also [5] confined to the parameter space of continuous perturbations P = C(T) when the index set T is a compact Hausdorff space. We refer the reader to [12] for the study of qualitative stability (formalized through certain semicontinuity properties of feasible solution and optimal solution mappings) in the framework of X = JRn, an arbitrary index set T, and arbitrary perturbations. In the same semi-infinite context, for a quantitative perspective (through ,-Lipschitzian properties), the reader is addressed to [7], and to [6] for the case of continuous perturbations.…”

Section: Introductionmentioning

confidence: 99%

Variational Analysis in Semi-Infinite and Infinite Programming, I: Stability of Linear Inequality Systems of Feasible Solutions

Cánovas¹,

López²,

Mordukhovich³

et al. 2010

SIAM J. Optim.

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Abstract. This paper concerns applications of advanced techniques of variational analysis and generalized differentiation to parametric problems of semi-infinite and infinite programming, where decision variables run over finite-dimensional and infinite-dimensional spaces, respectively. Part I is primarily devoted to the study of robust Lipschitzian stability of feasible solutions maps for such problems described by parameterized systems of infinitely many linear inequalities in Banach spaces of decision variables indexed by an arbitrary set T. The parameter space of admissible perturbations under consideration is formed by all bounded functions on T equipped with the standard supremum norm. Unless the index set Tis finite, this space is intrinsically infinite-dimensional (nonreflexive and nonseparable) of the l 00 -type. By using advanced tools of variational analysis and exploiting specific features of linear infinite systems, we establish complete characterizations of robust Lipschitzian · -st-abi1ity--entirely-v1a-tnelrinit1ah:lata·with-c-omputirrg·the exact·bnund-of bipschitzian-moduli;··-A· -· -crucial part of our analysis addresses the precise computation of the coderivative of the feasible set mapping and its norm. The results obtained are new in both semi-infinite and infinite frameworks.

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“…The fulfilment of both the Slater condition and the boundedness (and non-emptiness) of the set of optimal solutions yield high stability for optimization problems in different frameworks (see, for instance, [18,Thm. 1] and [5,Thm. 4.2] in relation to the Lipschitz continuity of the optimal value).…”

mentioning

confidence: 99%

“…T are analyzed in [5] in the linear case, and in [8] in the convex case. More details about stability of linear semi-infinite problems and their constraint systems in this general context (no continuity assumption) are gathered in [9,Chapters 6 and 10].…”

mentioning

confidence: 99%

Metric Regularity in Convex Semi-Infinite Optimization under Canonical Perturbations

Cánovas¹,

Klatte

López³

et al. 2007

SIAM J. Optim.

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This paper is concerned with the Lipschitzian behavior of the optimal set of convex semi-infinite optimization problems under continuous perturbations of the right hand side of the constraints and linear perturbations of the objective function. In this framework we provide a sufficient condition for the metric regularity of the inverse of the optimal set mapping. This condition consists of the Slater constraint qualification, together with a certain additional requirement in the Karush-Kuhn-Tucker conditions. For linear problems this sufficient condition turns out to be also necessary for the metric regularity, and it is equivalent to some well-known stability concepts. Abstract. This paper is concerned with the Lipschitzian behavior of the optimal set of convex semi-infinite optimization problems under continuous perturbations of the right hand side of the constraints and linear perturbations of the objective function. In this framework we provide a sufficient condition for the metric regularity of the inverse of the optimal set mapping. This condition consists of the Slater constraint qualification, together with a certain additional requirement in the Karush-Kuhn-Tucker conditions. For linear problems this sufficient condition turns out to be also necessary for the metric regularity, and it is equivalent to some well-known stability concepts.

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Stability and Well-Posedness in Linear Semi-Infinite Programming

Cited by 55 publications

References 12 publications

Radius of robust feasibility formulas for classes of convex programs with uncertain polynomial constraints

Radius of robust feasibility formulas for classes of convex programs with uncertain polynomial constraints

Variational Analysis in Semi-Infinite and Infinite Programming, I: Stability of Linear Inequality Systems of Feasible Solutions

Metric Regularity in Convex Semi-Infinite Optimization under Canonical Perturbations

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