We consider parametric families of constrained problems in mathematical programming and conduct a local sensitivity analysis for multivalued solution maps. coderivatives of set-valued mappings are our basic tool to an...
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We consider parametric families of constrained problems in mathematical programming and conduct a local sensitivity analysis for multivalued solution maps. coderivatives of set-valued mappings are our basic tool to analyze the parametric sensitivity of either stationary points or stationary point-multiplier pairs associated with parameterized optimization problems. An implicit mapping theorem for coderivatives is one key to this analysis for either of these objects, and in addition, a partial coderivative rule is essential for the analysis of stationary points. We develop general results along both of these lines and apply them to study the parametric sensitivity of stationary points alone, as well as stationary point-multiplier pairs. Estimates are computed for the coderivative of the stationary point multifunction associated with a general parametric optimization model, and these estimates are refined and augmented by estimates for the coderivative of the stationary point-multiplier multifunction in the case when the constraints are representable in a special composite form. When combined with existing coderivative formulas, our estimates are entirely computable in terms of the original data of the problem.
In this paper, we provide a comprehensive study of coderivative formulas for normal cone mappings. This allows us to derive necessary and sufficient conditions for the Lipschitzian stability of parametric variational ...
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In this paper, we provide a comprehensive study of coderivative formulas for normal cone mappings. This allows us to derive necessary and sufficient conditions for the Lipschitzian stability of parametric variational inequalities in reflexive Banach spaces. Our development not only gives an answer to the open questions raised in Yao and Yen (2009) [11], but also establishes generalizations and complements of the results given in Henrion et al. (2010) [4] and Yao and Yen (2009) [11,12]. (C) 2010 Elsevier Ltd. All rights reserved.
This paper concerns sensitivity analysis for general parametric constrained problems of multiobjective optimization in infinite-dimensional spaces by using advanced tools of modem variational analysis and generalized ...
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This paper concerns sensitivity analysis for general parametric constrained problems of multiobjective optimization in infinite-dimensional spaces by using advanced tools of modem variational analysis and generalized differentiation. We pay the main attention to computing and estimating coderivatives of frontier and efficient solution maps in parametric multiobjective problems with respect to generalized order optimality that include a vast majority of conventional multiobjective problems in the presence of geometric, operator, functional, and equilibrium constraints. The obtained results are new in both finite-dimensional and infinite-dimensional spaces.
Fuzzy calculus for coderivatives of multifunctions is studied. The generalized differentiation of set-valued mappings between Banach spaces is explored. Appropriative derivative-like concepts are derived to provide an...
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Fuzzy calculus for coderivatives of multifunctions is studied. The generalized differentiation of set-valued mappings between Banach spaces is explored. Appropriative derivative-like concepts are derived to provide an effective study of local behavior of multifunctions with successive applications. One of the advantages of the coderivative is a rich calculus supported by effective characterizations of the Lipschitzian. Comprehensive results for Frechet coderivatives are obtained.
Our basic object in this paper is to establish calculus rules for coderivatives of multivalued mappings between Banach spaces. We consider the coderivative which is associated to some geometrical approximate subdiffer...
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Our basic object in this paper is to establish calculus rules for coderivatives of multivalued mappings between Banach spaces. We consider the coderivative which is associated to some geometrical approximate subdifferential for functions.
The paper deals with the minimization of an integral functional over an L-p space subject to various types of constraints. For such optimization problems, new necessary optimality conditions are derived, based on seve...
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The paper deals with the minimization of an integral functional over an L-p space subject to various types of constraints. For such optimization problems, new necessary optimality conditions are derived, based on several concepts of nonsmooth analysis. In particular, we employ the generalized differential calculus of Mordukhovich and the fuzzy calculus of proximal subgradients. The results are specialized to nonsmooth two-stage and multistage stochastic programs.
This paper concerns the study of solution maps to parameterized variational inequalities over generalized polyhedra in reflexive Banach spaces. It has been recognized that generalized polyhedral sets are significantly...
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This paper concerns the study of solution maps to parameterized variational inequalities over generalized polyhedra in reflexive Banach spaces. It has been recognized that generalized polyhedral sets are significantly different from the usual convex polyhedra in infinite dimensions and play an important role in various applications to optimization, particularly to generalized linear programming. Our main goal is to fully characterize robust Lipschitzian stability of the aforementioned solution maps entirely via their initial data. This is done on the basis of the coderivative criterion in variational analysis via efficient calculations of the coderivative and related objects for the systems under consideration. The case of generalized polyhedra is essentially more involved in comparison with usual convex polyhedral sets and requires developing elaborated techniques and new proofs of variational analysis. (C) 2010 Elsevier Ltd. All rights reserved.
In this paper we establish new generalized differentiation rules in general Banach spaces regarding normal cones to set images under functions, coderivatives of compositions of set-valued mappings, as well as calculus...
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In this paper we establish new generalized differentiation rules in general Banach spaces regarding normal cones to set images under functions, coderivatives of compositions of set-valued mappings, as well as calculus results for normal compactness of sets and their images. In addition to the metric regularity of mappings, our results involve tangential distances of sets for which we also provide a fairly complete study by exploring its variations, basic properties, as well as relations to similar notions. Some related results are also established. (C) 2011 Elsevier Ltd. All rights reserved.
The derivation of multiplier-based optimality conditions for elliptic mathematical programs with equilibrium constraints (MPEC) is essential for the characterization of solutions and development of numerical methods. ...
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The derivation of multiplier-based optimality conditions for elliptic mathematical programs with equilibrium constraints (MPEC) is essential for the characterization of solutions and development of numerical methods. Though much can be said for broad classes of elliptic MPECs in both polyhedric and non-polyhedric settings, the calculation becomes significantly more complicated when additional constraints are imposed on the control. In this paper we develop three derivation methods for constrained MPEC problems: via concepts from variational analysis, via penalization of the control constraints, and via penalization of the lower-level problem with the subsequent regularization of the resulting nonsmoothness. The developed methods and obtained results are then compared and contrasted.
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