Numerische Methoden der Approximation und semi-infiniten Optimierung

Hettich, R.; Zencke, Peter

doi:10.1007/978-3-322-93108-5

Cited by 184 publications

(57 citation statements)

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“…For continuously differentiable data f and g t and under the Slater condition, property (ii) of Proposition 9 (recall that it is a consequence of condition (10)) is known as a sufficient condition for x to be a (locally) strongly unique minimizer of P (c,b), see Theorem 3.1.16 in [14]. In the linear case, condition (10) turns out to be equivalent even to persistence of strong unicity under small parameter changes (see §4 for details).…”

Section: A Sufficient Condition For the Metric Regularity Of Gmentioning

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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Section: A Sufficient Condition For the Metric Regularity Of Gmentioning

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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“…In particular, a local minimum )2 for ~ is always a point of Fritz John type (cf. [6]). However, it need not be a Kuhn-Tucker point, unless some constraint qualification is satisfied.…”

Section: M= {X Er"l H~(x) =0 Gj(x)>~o I E I J E J}mentioning

confidence: 99%

Critical sets in parametric optimization

Jongen

Jonker

Twilt

1986

Mathematical Programming

106

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We deal with one-parameter families of optimization problems in finite dimensions. The constraints are both of equality and inequality type. The concept of a 'generalized critical point' (g.c. point) is introduced. In particular, every local minimum, Kuhn-Tucker point, and point of Fritz John type is a g.c. point. Under fairly weak (even generic) conditions we study the set Z consisting of all g.c. points. Due to the parameter, the set X is pieced together from one-dimensional manifolds. The points of Z can be divided into five (characteristic) types. The subset of 'nondegenerate critical points' (first type) is open and dense in 2 (nondegenerate means: strict complementarity, nondegeneracy of the corresponding quadratic form and linear independence of the gradients of binding constraints). A nondegenerate critical point is completely characterized by means of four indices. The change of these indices along X is presented. Finally, the Kuhn-Tucker subset of 2; is studied in more detail, in particular in connection with the (failure of the) Mangasarian-Fromowitz constraint qualification.

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“…Uniqueness and strong uniqueness are no longer equivalent in LSIO, where strong uniqueness plays a crucial role in numerical analysis (together with some regularity conditions it implies superlinear convergence of multiple exchange methods, see [16] - [15]) as well as in sensitivity analysis (it characterizes those problems for which the optimal value is a linear function of the costs on some neighborhood of c; as shown in [7]). The literature about uniqueness and strong uniqueness in LSIO up to 1995 was surveyed in [9].…”

Section: Introductionmentioning

confidence: 99%

On stable uniqueness in linear semi-infinite optimization

2011

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Abstract. This paper is intended to provide conditions for the stability of the strong uniqueness of the optimal solution of a given linear semi-in…nite optimization (LSIO) problem, in the sense of the maintaining of the strong uniqueness property under su¢ ciently small perturbations of all the data. We consider LSIO problems such that the family of gradients of all the constraints is unbounded, extending earlier results of Nürnberger for continuous LSIO problems, and of Helbig and Todorov for LSIO problems with bounded set of gradients. To do this we characterize the absolutely (a¢ nely) stable problems, i.e., those LSIO problems whose feasible set (its a¢ ne hull, respectively) remains constant under su¢ ciently small perturbations.

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Numerische Methoden der Approximation und semi-infiniten Optimierung

Cited by 184 publications

References 0 publications

Metric Regularity in Convex Semi-Infinite Optimization under Canonical Perturbations

Metric Regularity in Convex Semi-Infinite Optimization under Canonical Perturbations

Critical sets in parametric optimization

On stable uniqueness in linear semi-infinite optimization

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