Regularity properties of constrained set-valued mappings (original) (raw)
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Uniform Regularity of Set-Valued Mappings and Stability of Implicit Multifunctions
2020
We establish primal and dual quantitative sufficient and necessary conditions for uniform regularity of set-valued mappings in metric and normed/Banach/Asplund spaces as well as complete quantitative characterizations of the properties in the convex setting. As a consequence, we obtain primal and dual conditions for the conventional metric regularity and subregularity properties as well as stability properties of solution mappings to generalized equations.
Applied Mathematics and Optimization, 1991
In this paper we establish some implicit function theorems for a class of locally Lipschitz set-valued maps and then apply them to investigate some questions concerning the stability of optimization problems with inclusion constraints. In consequence we have an extension of some of the corresponding results of Robinson, Aubin, and others. Implicit Function Theorems for Set-Valued Maps 37 radius c > 0 by Bz(z, c), and its interior by Bz(z, c). The symbols B z and S z stand for the unit ball and the unit sphere in Z, respectively, i.e., Bz = Bz(0, 1), S z = {z ~ Z/N z II = 1}. By B* and S* we mean the unit ball and the unit sphere in Z*. The subscript will be deleted if no confusion is possible. The distance from a point z e Z to a subset A c Z is denoted by d(z, A) and the excess of A over a subset B c Z is denoted by e(A, B), that is d(z, A) = inf{[] z-al[/a ~ A} and e(A, B) = sup{a(a, B)/a ~ A}. We admit that d(z, A) = + ~ if A = ~ (the empty set). The abbreviation "int" is used to denote the interior of a set. The Clarke tangent cone to a dosed subset E c Z at a point z o ~ E, denoted by Tr.(Zo), is defined in [5] and is characterized in [14] as the set of all z ~ Z satisfying the condition: for every q > 0 there exist r > 0, s > 0 such that [z' + tBz(z, q)] ~ E ~ for all t e (0, r) and z' E Bz(zo, s) n E. Note that Tr.(Zo) is a closed convex cone. Its dual Nr(zo) = {z* e Z*/(z*, z) < 0 for all z e Tr(zo)} is the Clarke normal cone to E at z o. It is well known that Tr(zo) coincides with the tangent cone of E at Zo in the sense of convex analysis if E is convex. Throughout the forthcoming, unless otherwise specified, X and Y are Hilbert spaces, P is a normed space, H is a set-valued map from X to Y, and F is a setvalued map from X x P to Y. Denote by gr H and dom H the graph and the domain of H, respectively, that is, gr n = {(x, y) ~ X x Y/y ~ n(x)}, dora n = {x ~ x/n(x) ~ ~5}. We say that H is convex (resp. closed) if gr H is convex (resp. closed) in the product space X x Y. We say that H is locally Lipschitz at x 0 e X if there is U ~ N(xo) such that, for some k > 0 and for all Xl, x2 e U, the following inclusion holds: n(xx) ~ n(x2) + k II xx-x2 IIB.
On Subregularity Properties of Set-Valued Mappings
Set-Valued and Variational Analysis, 2013
In this work we classify the at-point regularities of set-valued mappings into two categories and then we analyze their relationship through several implications and examples. After this theoretical tour, we use the subregularity properties to deduce implicit theorems for set-valued maps. Finally, we present some applications to the study of multicriteria optimization problems.
Lipschitz properties of nonsmooth functions and set-valued mappings via generalized differentiation
Nonlinear Analysis: Theory, Methods & Applications, 2013
In this paper, we revisit the Mordukhovich's subdifferential criterion for Lipschitz continuity of nonsmooth functions and 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, that play an important role in many aspects of nonsmooth analysis and optimization.
First order optimality condition for constrained set-valued optimization
Pacific Journal of Optimization
A constrained optimization problem with set-valued data is considered. Different kind of solutions are defined for such a problem. We recall weak minimizer, efficient minimizer and proper minimizer. The latter are defined in a way that embrace also the case when the ordering cone is not pointed. Moreover we present the new concept of isolated minimizer for set-valued optimization. These notions are investigated and appear when establishing first-order necessary and sufficient optimality conditions derived in terms of a Dini type derivative for set-valued maps. The case of convex (along rays) data is considered when studying sufficient optimality conditions for weak minimizers. Key words: Vector optimization, Set-valued optimization, First-order optimality conditions.
Characterizations of Uniform Regularity of Set-Valued Mappings
arXiv: Optimization and Control, 2020
We establish primal and dual quantitative sufficient and necessary conditions of uniform regularity of set-valued mappings in metric and normed (in particular, Banach/Asplund) spaces as well as complete quantitative characterizations of the properties in the convex setting. As a consequence, we obtain primal and dual characterizations of the conventional metric regularity and subregularity properties as well as stability properties of solution mappings to generalized equations.
Optimality conditions for a nonconvex set-valued optimization problem
Computers & Mathematics with Applications, 2008
In this paper we study necessary and sufficient optimality conditions for a set-valued optimization problem. Convexity of the multifunction and the domain is not required. A definition of K-approximating multifunction is introduced. This multifunction is the differentiability notion applied to the problem. A characterization of weak minimizers is obtained for invex and generalized K-convexlike multifunctions using the Lagrange multiplier rule.
A study of variational inequalities for set-valued mappings
Journal of Inequalities and Applications, 1999
In this paper, Ky Fan's KKM mapping principle is used to establish the existence of solutions for simultaneous variational inequalities. By applying our earlier results together with Fan-Glicksberg fixed point theorem, we prove some existence results for implicit variational inequalities and implicit quasi-variational inequalities for set-valued mappings which are either monotone or upper semi-continuous.
Upper semicontinuity of closed-convex-valued multifunctions
Mathematical Methods of Operations Research, 2003
In this paper we study the (Berge) upper semicontinuity of a generic multifunction assigning to each parameter, in a metric space, a closed convex subset of the n-dimensional Euclidean space. A relevant particular case arises when we consider the feasible set mapping associated with a parametric family of convex semi-infinite programming problems. Related to such a generic multifunction, we introduce the concept of e-reinforced mapping, which will allow us to establish a su‰cient condition for the aimed property. This condition turns out to be also necessary in the case that the boundary of the image set at the nominal value of the parameter contains no half-lines. On the other hand, it is well-known that every closed convex set in the Euclidean space can be viewed as the solution set of a linear semi-infinite inequality system and, so, a parametric family of linear semi-infinite inequality systems can always be associated with the original multifunction. In this case, a di¤erent necessary condition is provided in terms of the coe‰cients of these linear systems. This condition tries to measure the relative variation of the right hand side with respect to the left hand side of the constraints of the systems in a neighborhood of the nominal parameter.