On Lipschitz semicontinuity properties of variational systems with application to parametric optimization

In this paper two properties of recognized interest in variational analysis, known as Lipschitz lower semicontinuity and calmness, are studied with reference to a general class of variational systems, i.e. to solution mappings to parameterized generalized equations. In consideration of the metric nature of such properties, some related sufficient conditions are established, which are expressed via nondegeneracy conditions on derivative-like objects appropriate for a metric space analysis. For certain classes of generalized equations in Asplund spaces, it is shown how such conditions can be formulated by using the Fr\'echet coderivative of the field and the derivative of the base. Applications to the stability analysis of parametric constrained optimization problems are proposed

On a class of convex sets with convex images and its application to nonconvex optimization

In the present paper, conditions under which the images of uniformly convex sets through $C^{1,1}$ regular mappings between Banach spaces remain convex are established. These conditions are expressed by a certain quantitative relation betweeen the modulus of convexity of a given set and the global regularity behaviour of the mapping on it. Such a result enables one to extend to a wide subclass of convex sets the Polyak's convexity principle, which was originally concerned with images of small balls around points of Hilbert spaces. In particular, the crucial phenomenon of the preservation of convexity under regular $C^{1,1}$ transformations is shown to include the class of $r$-convex sets, where the value of $r$ depends on the regularity behaviour of the involved transformation. Two consequences related to nonconvex optimization are discussed: the first one is a sufficient condition for the global solution existence for infinite-dimensional constrained extremum problems; the second one provides a zero-order Lagrangian type characterization of optimality in nonlinear mathematical programming.Comment: This paper has been withdrawn by the author due to errors found in a proo

A strong metric subregularity analysis of nonsmooth mappings via steepest displacement rate

In this paper, a systematic study of the strong metric subregularity property of mappings is carried out by means of a variational tool, called steepest displacement rate. With the aid of this tool, a simple characterization of strong metric subregularity for multifunctions acting in metric spaces is formulated. The resulting criterion is shown to be useful for establishing stability properties of the strong metric subregularity in the presence of perturbations, as well as for deriving various conditions, enabling to detect such a property in the case of nonsmooth mappings. Some of these conditions, involving several nonsmooth analysis constructions, are then applied in studying the isolated calmness property of the solution mapping to parameterized generalized equations

An implicit multifunction theorem for the hemiregularity of mappings with application to constrained optimization

The present paper contains some investigations about a uniform variant of the notion of metric hemiregularity, the latter being a less explored property obtained by weakening metric regularity. The introduction of such a quantitative stability property for set-valued mappings is motivated by applications to the penalization of constrained optimization problems, through the notion of problem calmness. As a main result, an implicit multifunction theorem for parameterized inclusion problems is established, which measures the uniform hemiregularity of the related solution mapping in terms of problem data. A consequence on the exactness of penalty functions is discussed

Convexity of the images of small balls through perturbed convex multifunctions

In the present paper, the following convexity principle is proved: any closed convex multifunction, which is metrically regular in a certain uniform sense near a given point, carries small balls centered at that point to convex sets, even if it is perturbed by adding C^{1,1} smooth mappings with controlled Lipschizian behaviour. This result, which is valid for mappings defined on a subclass of uniformly convex Banach spaces, can be regarded as a set-valued generalization of the Polyak convexity principle. The latter, indeed, can be derived as a special case of the former. Such an extension of that principle enables one to build large classes of nonconvex multifunctions preserving the convexity of small balls. Some applications of this phenomenon to the theory of set-valued optimization are proposed and discussed

Solution analysis for a class of set-inclusive generalized equations: a convex analysis approach

In the present paper, classical tools of convex analysis are used to study the solution set to a certain class of set-inclusive generalized equations. A condition for the solution existence and global error bounds is established, in the case the set-valued term appearing in the generalized equation is concave. A functional characterization of the contingent cone to the solution set is provided via directional derivatives. Specializations of these results are also considered when outer prederivatives can be employed

An extension of the Polyak convexity principle with application to nonconvex optimization

The main problem considered in the present paper is to single out classes of convex sets, whose convexity property is preserved under nonlinear smooth transformations. Extending an approach due to B.T. Polyak, the present study focusses on the class of uniformly convex subsets of Banach spaces. As a main result, a quantitative condition linking the modulus of convexity of such kind of set, the regularity behaviour around a point of a nonlinear mapping and the Lipschitz continuity of its derivative is established, which ensures the images of uniformly convex sets to remain uniformly convex. Applications of the resulting convexity principle to the existence of solutions, their characterization and to the Lagrangian duality theory in constrained nonconvex optimization are then discussed

On the Polyak convexity principle and its application to variational analysis

According to a result due to B.T. Polyak, a mapping between Hilbert spaces, which is $C^{1,1}$ around a regular point, carries a ball centered at that point to a convex set, provided that the radius of the ball is small enough. The present paper considers the extension of such result to mappings defined on a certain subclass of uniformly convex Banach spaces. This enables one to extend to such setting a variational principle for constrained optimization problems, already observed in finite dimension, that establishes a convex behaviour for proper localizations of them. Further variational consequences are explored.Comment: 13 page

On a strong covering property of multivalued mappings

In this paper, a strong variant for multivalued mappings of the well-known property of openness at a linear rate is studied. Among other examples, a simply characterized class of closed convex processes between Banach spaces, which satisfies such a covering behaviour, is singled out. Equivalent reformulations of this property and its stability under Lipschitz perturbations are investigated in a metric space setting. Applications to the solvability of set-valued inclusions and to the exact penalization of optimization problems with set-inclusion constraints are discussed

On finite-dimensional set-inclusive constraint systems: local analysis and related optimality conditions

In the present paper, some aspects of the finite-dimensional theory of set-inclusive generalized equations are studied. Set-inclusive generalized equations are problems arising in several contexts of optimization and variational analysis, involving multi-valued mappings and cones. The aim of this paper is to propose an approach to the local analysis of the solution set to such kind of generalized equations. In particular, a study of the contingent cone to the solution set is carry out by means of first-order approximations of set-valued mappings, which are expressed by prederivatives. Such an approach emphasizes the role of the metric increase property for set-valued mappings, as a condition triggering crucial error bound estimates for the tangential description of solution sets. Some of the results obtained through this kind of analysis are then exploited for formulating necessary optimality conditions, which are valid for problems with constraints formalized by set-inclusive generalized equations.Comment: Draft version to be continue