In this paper we investigate the Lipschitz-like property of the solution mapping of parametric variational inequalities over perturbed polyhedral convex sets. By establishing some lower and upper estimates for the cod...
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In this paper we investigate the Lipschitz-like property of the solution mapping of parametric variational inequalities over perturbed polyhedral convex sets. By establishing some lower and upper estimates for the coderivatives of the solution mapping, among other things, we prove that the solution mapping could not be Lipschitz-like around points where the positive linear independence condition is invalid. Our analysis is based heavily on the Mordukhovich criterion (Mordukhovich in Variational Analysis and Generalized Differentiation. vol. I: Basic Theory, vol. II: Applications. Springer, Berlin, 2006) of the Lipschitz-like property for set-valued mappings between Banach spaces and recent advances in variational analysis. The obtained result complements the corresponding ones of Nam (Nonlinear Anal 73:2271-2282, 2010) and Qui (Nonlinear Anal 74:1674-1689, 2011).
We study general constrained multiobjective optimization problems with objectives being closed multifunctions in Banach spaces. In terms of the coderivatives and normal cones, we provide generalized Lagrange multiplie...
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We study general constrained multiobjective optimization problems with objectives being closed multifunctions in Banach spaces. In terms of the coderivatives and normal cones, we provide generalized Lagrange multiplier rules as necessary optimality conditions of the above problems. In an Asplund space setting, sharper results are presented.
This paper investigates solution stability of parametric variational inequalities over Euclidean balls in finite dimensional spaces. We provide exact formulas for computing required coderivatives of the normal cone ma...
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This paper investigates solution stability of parametric variational inequalities over Euclidean balls in finite dimensional spaces. We provide exact formulas for computing required coderivatives of the normal cone mappings to Euclidean balls via the initial data. On the basis of these formulas, we establish necessary and sufficient conditions for Lipschitzian stability of the solution maps of the aforementioned variational inequalities.
The present paper shows how thelinear independence constraint qualification(LICQ) can be combined with some conditions put on the first-order and second-order derivatives of the objective function and the constraint f...
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The present paper shows how thelinear independence constraint qualification(LICQ) can be combined with some conditions put on the first-order and second-order derivatives of the objective function and the constraint functions to ensure the Robinson stability and the Lipschitz-like property of the stationary point set map of a generalC(2)-smooth parametric constrained optimization problem. So, a part of the results in two preceding papers of the authors [J. Optim. Theory Appl. 180 (2019), 91-116 (Part 1);117-139 (Part 2)], which were obtained for a problem with just one inequality constraint, now has an adequate extension for problems having finitely many equality and inequality constraints. Our main tool is an estimate of B. S. Mordukhovich and R. T. Rockafellar [SIAM J. Optim. 22 (2012), 953-986;Theorem 3.3] for a second-order partial subdifferential of a composite function. The obtained results are illustrated by three examples.
The normal subdifferential of a set-valued mapping with values in a partially ordered Banach space has been recently introduced in Bao and Mordukhovich (Control Cyber 36:531-562, 2007), by using the Mordukhovich coder...
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The normal subdifferential of a set-valued mapping with values in a partially ordered Banach space has been recently introduced in Bao and Mordukhovich (Control Cyber 36:531-562, 2007), by using the Mordukhovich coderivative of the associated epigraphical multifunction, which has proven to be useful in deriving necessary conditions for super efficient points of vector optimization problems. In this paper, we establish new formulae for computing and/or estimating the normal subdifferential of the efficient point multifunctions of parametric vector optimization problems. These formulae will be presented in a broad class of conventional vgector optimization problems with the presence of geometric, operator, equilibrium, and (finite and infinite) functional constraints.
Metric subregularity and regularity of multifunctions are fundamental notions in variational analysis and optimization. Using the concept of strong slope, in this paper we first establish a criterion for metric subreg...
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Metric subregularity and regularity of multifunctions are fundamental notions in variational analysis and optimization. Using the concept of strong slope, in this paper we first establish a criterion for metric subregularity of multifunctions between metric spaces. Next, we use a combination of abstract coderivatives and contingent derivatives to derive verifiable first order conditions ensuring metric subregularity of multifunctions between Banach spaces. Then using second order approximations of convex multifunctions, we establish a second order condition for metric subregularity of mixed smooth-convex constraint systems, which generalizes a result established recently by Gfrerer
One of the main difficulties in nonsmooth analysis is to devise calculus rules. It is our purpose here to show that a certain cooperative behavior between functions (resp. sets, resp. multifunctions) yields calculus r...
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One of the main difficulties in nonsmooth analysis is to devise calculus rules. It is our purpose here to show that a certain cooperative behavior between functions (resp. sets, resp. multifunctions) yields calculus rules for subdifferentials (resp. normal cones, resp. coderivatives). In previous contributions, the qualification conditions ensuring calculus rules were given in a non symmetric way. The new conditions can be combined easily and encompass various criteria. We also address the important question of the extension of calculus rules from the Lipschitz case to the general case.
Solution stability of a class of linear generalized equations in finite dimensional Euclidean spaces is investigated by means of generalized differentiation. Exact formulas for the Frechet and the Mordukhovich coderiv...
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Solution stability of a class of linear generalized equations in finite dimensional Euclidean spaces is investigated by means of generalized differentiation. Exact formulas for the Frechet and the Mordukhovich coderivatives of the normal cone mappings of perturbed Euclidean balls are obtained. Necessary and sufficient conditions for the local Lipschitz-like property of the solution maps of such linear generalized equations are derived from these coderivative formulas. Since the trust-region subproblems in nonlinear programming can be regarded as linear generalized equations, these conditions lead to new results on stability of the parametric trust-region subproblems.
In this paper, we consider a multiobjective optimal control problem where the preference relation in the objective space is defined in terms of a pointed convex cone containing the origin, which defines generalized Pa...
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In this paper, we consider a multiobjective optimal control problem where the preference relation in the objective space is defined in terms of a pointed convex cone containing the origin, which defines generalized Pareto optimality. For this problem, we introduce the set-valued return function V and provide a unique characterization for V in terms of contingent derivative and coderivative for set-valued maps, which extends two previously introduced notions of generalized solution to the Hamilton-Jacobi equation for single objective optimal control problems.
Using variational analysis, we study the vector optimization problems with objectives being closed multifunctions on Banach spaces or in Asplund spaces. In terms of the coderivatives and normal cones, we present Ferma...
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Using variational analysis, we study the vector optimization problems with objectives being closed multifunctions on Banach spaces or in Asplund spaces. In terms of the coderivatives and normal cones, we present Fermat's rules as necessary or sufficient conditions for a super efficient solution of the above problems. (C) 2008 Elsevier Ltd. All rights reserved.
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