Several notions of constraint qualifications are generalized from the setting of convex inequality systems to that of convex generalized equations. This is done and investigated in terms of the coderivatives and the n...
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Several notions of constraint qualifications are generalized from the setting of convex inequality systems to that of convex generalized equations. This is done and investigated in terms of the coderivatives and the normal cones, and thereby we provide some characterizations for convex generalized equations to have the metric subregularity. As applications, we establish formulas of the modulus of calmness and provide several characterizations of the calmness. Extending the classical concept of extreme boundary, we introduce a notion of recession cores of closed convex sets. Using this concept, we establish global metric subregularity ( i. e., error bound) results for generalized equations.
In this paper, we revisit the Mordukhovich subdifferential criterion for Lipschitz continuity of nonsmooth functions and the coderivative criterion for the Aubin/Lipschitz-like property of set-valued mappings in finit...
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In this paper, we revisit the Mordukhovich subdifferential criterion for Lipschitz continuity of nonsmooth functions and the coderivative criterion for the Aubin/Lipschitz-like property of set-valued mappings in finite dimensions. The criteria are useful and beautiful results in modern variational analysis showing the state of the art of the field. As an application, we establish necessary and sufficient conditions for Lipschitz continuity of the minimal time function and the scalarization function, which play an important role in many aspects of nonsmooth analysis and optimization. (C) 2013 Elsevier Ltd. All rights reserved.
We consider a multiobjective bilevel optimization problem with vector-valued upper- and lower-level objective functions. Such problems have attracted a lot of interest in recent years. However, so far, scalarization h...
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We consider a multiobjective bilevel optimization problem with vector-valued upper- and lower-level objective functions. Such problems have attracted a lot of interest in recent years. However, so far, scalarization has appeared to be the main approach used to deal with the lower-level problem. Here, we utilize the concept of frontier map that extends the notion of optimal value function to our parametric multiobjective lower-level problem. Based on this, we build a tractable constraint qualification that we use to derive necessary optimality conditions for the problem. Subsequently, we show that our resulting necessary optimality conditions represent a natural extension from standard optimistic bilevel programs with scalar objective functions.
We give a new necessary and sufficient condition for convexity of a set-valued map F between Banach spaces. It is established for a closed map F having nonconvex values. The main tool in this paper is the coderivative...
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We give a new necessary and sufficient condition for convexity of a set-valued map F between Banach spaces. It is established for a closed map F having nonconvex values. The main tool in this paper is the coderivative of F which is constructed with the help of an abstract subdifferential notion of Penot [16]. A detailed discussion is devoted to special cases when the contingent, the Frechet and the Clarke-Rockafellar subdifferentials are used as this abstract subdifferential.
The aim of this work is twofold. First, we use the advanced tools of modern variational analysis and generalized differentiation to study the Lipschitz-like property of an implicit multifunction. More explicitly, new ...
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The aim of this work is twofold. First, we use the advanced tools of modern variational analysis and generalized differentiation to study the Lipschitz-like property of an implicit multifunction. More explicitly, new sufficient conditions in terms of the Frechet coderivative and the normal/Mordukhovich coderivative of parametric multifunctions for this implicit multifunction to have the Lipschitz-like property at a given point are established. Then we derive sufficient conditions ensuring the Lipschitz-like property of an efficient solution map in parametric vector optimization problems by employing the above implicit multifunction results. (C) 2011 Elsevier Ltd. All rights reserved.
Using variational analysis, we study vector optimization problems with objectives being closed multifunctions on Banach spaces or in Asplund spaces. In particular, in terms of the coderivatives, we present Fermat'...
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Using variational analysis, we study vector optimization problems with objectives being closed multifunctions on Banach spaces or in Asplund spaces. In particular, in terms of the coderivatives, we present Fermat's rules as necessary conditions for an optimal solution of the above problems. As applications, we also provide some necessary conditions (in terms of Clarke's normal cones or the limiting normal cones) for Pareto efficient points.
We discuss three different characterizations of continuity properties for general multifunctions S : R-d double right arrow R-n. Each of these characterizations is given by the same simple nonsingularity condition, bu...
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We discuss three different characterizations of continuity properties for general multifunctions S : R-d double right arrow R-n. Each of these characterizations is given by the same simple nonsingularity condition, but stated in terms of three different generalized derivatives. Two of these characterizations are known, but the third is new to this paper. We discuss how all three have immediate analogues as generalized inverse mapping theorems, and we apply our new characterization to develop a fundamental and very broad sensitivity theorem for solutions to parameterized optimization problems.
We study sharp minima for multiobjective optimization problems. In terms of the Mordukhovich coderivative and the normal cone, we present sufficient and or necessary conditions for existence of such sharp minima, some...
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We study sharp minima for multiobjective optimization problems. In terms of the Mordukhovich coderivative and the normal cone, we present sufficient and or necessary conditions for existence of such sharp minima, some of which are new even in the single objective setting.
In this paper, we follow Kuroiwa's set approach in set optimization, which proposes to compare values of a set-valued objective map F with respect to various set order relations. We introduce a Hausdorff-type dist...
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In this paper, we follow Kuroiwa's set approach in set optimization, which proposes to compare values of a set-valued objective map F with respect to various set order relations. We introduce a Hausdorff-type distance relative to an ordering cone between two sets in a Banach space and use it to define a directional derivative for F. We show that the distance has nice properties regarding set order relations and the directional derivative enjoys most properties of the one of a scalar single-valued function. These properties allow us to derive necessary and/or sufficient conditions for various types of maximizers and minimizers of F.
In Part 1 of this paper, we have estimated the Frechet coderivative and the Mordukhovich coderivative of the stationary point set map of a smooth parametric optimization problem with one smooth functional constraint u...
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In Part 1 of this paper, we have estimated the Frechet coderivative and the Mordukhovich coderivative of the stationary point set map of a smooth parametric optimization problem with one smooth functional constraint under total perturbations. From these estimates, necessary and sufficient conditions for the local Lipschitz-like property of the map have been obtained. In this part, we establish sufficient conditions for the Robinson stability of the stationary point set map. This allows us to revisit and extend several stability theorems in indefinite quadratic programming. A comparison of our results with the ones which can be obtained via another approach is also given.
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