Calculus of tangent sets and derivatives of set-valued maps under metric subregularity conditions

Durea, Marius; Strugariu, Radu

doi:10.1007/s10898-011-9800-4

Cited by 18 publications

(5 citation statements)

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“…This property as well as its equivalent calmness (upper Lipschitzian) counterpart for inverse mappings have drawn much attention in recent publications. The reader can find various results in this direction and their applications in, e.g., [1,2,10,11,17,18,19,21,22,23,36,37] and the references therein. In what follows we concentrate on the study of metric subregularity and its higher-order extensions, while the results obtained can be reformulated in calmness terms.…”

Section: Introductionmentioning

confidence: 99%

Hölder Metric Subregularity with Applications to Proximal Point Method

Mordukhovich²

2012

SIAM J. Optim.

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This paper is mainly devoted to the study and applications of Hölder metric subregularity (or metric q-subregularity of order q ∈ (0, 1]) for general set-valued mappings between infinite-dimensional spaces. Employing advanced techniques of variational analysis and generalized differentiation, we derive neighborhood and pointbased sufficient conditions as well as necessary conditions for q-metric subregularity with evaluating the exact subregularity bound, which are new even for the conventional (firstorder) metric subregularity in both finite and infinite-dimensions. In this way we also obtain new fractional error bound results for composite polynomial systems with explicit calculating fractional exponents. Finally, metric q-subregularity is applied to conduct a quantitative convergence analysis of the classical proximal point method for finding zeros of maximal monotone operators on Hilbert spaces.

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Section: Introductionmentioning

confidence: 99%

Hölder Metric Subregularity with Applications to Proximal Point Method

Mordukhovich²

2012

SIAM J. Optim.

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“…A similar but different result could be done taking into account the special structure of this case, using directly Proposition 3.4, and some results one can find in literature concerning the calculus of Bouligand tangent cone to the counter image of a set through a differentiable mapping. Let us recall some facts from [8]. Let f : X → Y be a function and D ⊂ X be a nonempty closed set.…”

Section: Standard Arguments Yield Ymentioning

confidence: 99%

“…In fact, the above notion coincides with that of calmness of the set-valued map y ⇒ f −1 (y) ∩ D at (f (x), x) (see, for instance, [4, Section 3H]). One of the main results in [8] reads as follows.…”

Section: Standard Arguments Yield Ymentioning

confidence: 99%

Directional Pareto Efficiency: Concepts and Optimality Conditions

Chelmuş

Durea

Florea

2019

J Optim Theory Appl

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We introduce and study a notion of directional Pareto minimality with respect to a set that generalizes the classical concept of Pareto efficiency. Then we give separate necessary and sufficient conditions for the newly introduced efficiency and several situations concerning the objective mapping and the constraints are considered. In order to investigate different cases, we adapt some well-known constructions of generalized differentiation and the connections with some recent directional regularities come naturally into play. As a consequence, several techniques from the study of genuine Pareto minima are considered in our specific situation.1 means of an inverse image of a cone through another set-valued map. For the study of this general case, we introduce an adapted tangent cone, along with several directional regularity properties of the involved maps, and this approach allows us to derive necessary optimality conditions that, in turn, generalize the prototype of Fermat Theorem at an endpoint of an interval. Furthermore, we present as well optimality conditions in terms of tangent limiting cones and coderivatives. Both on primal and dual spaces we have under consideration several situations concerning the objective and constraint mappings with their specific techniques of study, among which we mention generalized constraint qualification conditions, Gerstewitz scalarization, openness vs. minimality paradigm, Clarke penalization, extremal principle. Some results are dedicated to the sufficient optimality conditions under convexity assumptions. Finally, we consider as well the situation of minimality for sets and a brief discussion of this concept reveals the similarities and the differences with respect to the known situation of Pareto efficiency.The paper is organized as follows. First of all, we introduce the notation we use and then we present the concepts of directional minimality we study in this work. The definitions of these notions along with some comparisons and examples are the subjects of the second section. The main section of the paper is the third one, and it deals with optimality conditions for the above introduced concepts, being, in turn, divided into two subsections. Firstly, we derive optimality conditions using tangent cones and to this aim we adapt a classical concept of the Bouligand tangent cone and Bouligand derivative of a set-valued map. Using some directional metric regularities, we get several assertions concerning these objects and this allows us to present necessary optimality conditions for a wide range of situations going from problems governed by set-valued mappings having generalized inequalities constraints to fully smooth constrained problems. Secondly, we deal with optimality conditions using normal limiting cones and, again, we consider several types of problems. In this process of getting necessary optimality conditions we adapt several techniques from classical vector optimization. Moreover, some generalized convex cases are considered in order to obtain sufficient opti...

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“…(See [3,4], [6,7,8], [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29], [30,31,32,33,34,35,36,37,38,39], [41,42,43,44,45,46,47,48], [50,51,52,53,54,55,56,57,58] among many other references.) Usually they deal with a specif...…”

Section: Introductionmentioning

confidence: 99%

“…We conclude with a comparison with some classical second-order notions. For more complete comparisons with previous studies about m-order notions, we refer to [21], [24], [25], [32], [38], [43], [41], [53].…”

Section: Introductionmentioning

confidence: 99%