In these notes, we present a general result concerning the Lipschitz regularity of a certain type of set-valued maps often found in constrained optimization and control problems. The class of multifunctions examined in this paper is characterized by means of a set of Lipschitz continuous constraint functions defined on some Lipschitz manifold. The proof of the regularity result for this class of multifunctions is based on a quantitative version of the Implicit Function Theorem for Lipschitzian maps which provides estimates for the neighborhoods where the implicit map can be defined.
In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity.
If $F$ is a set-valued mapping from $\R^n$ into $\R^m$ with closed graph, then $y\in \R^m$ is a critical value of $F$ if for some $x$ with $y\in F(x)$, $F$ is not metrically regular at $(x,y)$. We prove that the set of critical values of a set-valued mapping whose graph is a definable (tame) set in an $o$-minimal structure containing additions and multiplications is a set of dimension not greater than $m-1$ (resp. a porous set). As a corollary of this result we get that the collection of asymptotically critical values of a semialgebraic set-valued mapping has dimension not greater than $m-1$, …
We propose a new concept of generalized differentiation of set-valued maps that captures the first order information. This concept encompasses the standard notions of Frechet differentiability, strict differentiability, calmness and Lipschitz continuity in single-valued maps, and the Aubin property and Lipschitz continuity in set-valued maps. We present calculus rules, sharpen the relationship between the Aubin property and coderivatives, and study how metric regularity and open covering can be refined to have a directional property similar to our concept of generalized differentiation. Finall…
This paper discusses the regularity of multiple-valued Dirichlet minimizing maps into the sphere. It shows that even at branched point, as long as the normalized energy is small enough, we have the energy decay estimate. Combined with the previous work by Chun-Chi Lin, we get our first estimate that m-2 dimensional Hausdorff measure of singular set is zero. Furthermore, by looking at the tangent map and using dimension reduction argument, we show that the singular set is at least of codimension 3.
The presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that t…
We study set-valued mappings defined by solution sets of parametric systems of equalities and inequalities. We prove Lipschitz-like continuity of these mappings under relaxed constant rank constraint qualification.
We provide necessary and sufficient conditions for a set-valued mapping between finite dimensional spaces to be directionally open by relating this property with directional regularity, Hölder continuity of the inverse mapping, coderivatives and variations. These generalize and refine some previously known results.
We introduce and study a notion of co-radiantness for set-valued mappings between nonnegative orthants of Euclidean spaces. We analyze them from an abstract convexity perspective. Our main results consist in representations, in terms of intersections of graphs, of the increasing co-radiant mappings that take closed normal values, by means of elementary members in the class of such mappings.
This paper concerns generalized differential characterizations of maximal monotone set-valued mappings. Using advanced tools of variational analysis, we establish coderivative criteria for maximal monotonicity of set-valued mappings, which seem to be the first infinitesimal characterizations of maximal monotonicity outside the single-valued case. We also present second-order necessary and sufficient conditions for lower-${\mathcal C}^2$ functions to be convex and strongly convex. Examples are provided to illustrate the obtained results and the imposed assumptions.
This work concerns the local convergence theory of Newton and quasi-Newton methods for convex-composite optimization: minimize f(x):=h(c(x)), where h is an infinite-valued proper convex function and c is C^2-smooth. We focus on the case where h is infinite-valued piecewise linear-quadratic and convex. Such problems include nonlinear programming, mini-max optimization, estimation of nonlinear dynamics with non-Gaussian noise as well as many modern approaches to large-scale data analysis and machine learning. Our approach embeds the optimality conditions for convex-composite optimization problem…
This paper aims at providing further studies of the notion of quasi-relative interior for convex sets introduced by Borwein and Lewis. We obtain new formulas for representing quasi-relative interiors of convex graphs of set-valued mappings and for convex epigraphs of extended-real-valued functions defined on locally convex topological vector spaces. We also show that the role, which this notion plays in infinite dimensions and the results obtained in this vein, are similar to those involving relative interior in finite-dimensional spaces.
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.
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.
We systematically study the local single-valuedness of the Bregman proximal mapping and local smoothness of the Bregman--Moreau envelope of a nonconvex function under relative prox-regularity - an extension of prox-regularity - which was originally introduced by Poliquin and Rockafellar. As Bregman distances are asymmetric in general, in accordance with Bauschke et al., it is natural to consider two variants of the Bregman proximal mapping, which, depending on the order of the arguments, are called left and right Bregman proximal mapping. We consider the left Bregman proximal mapping first. Th…
The paper is devoted to the study of the twice epi-differentiablity of extended-real-valued functions, with an emphasis on functions satisfying a certain composite representation. This will be conducted under the parabolic regularity, a second-order regularity condition that was recently utilized in [13] for second-order variational analysis of constraint systems. Besides justifying the twice epi-differentiablity of composite functions, we obtain precise formulas for their second subderivatives under the metric subregularity constraint qualification. The latter allows us to derive second-order…
Given a semialgebraic set-valued map $F \colon \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ with closed graph, we show that the map $F$ is Holder metrically subregular and that the following conditions are equivalent: (i) $F$ is an open map from its domain into its range and the range of $F$ is locally closed; (ii) the map $F$ is Holder metrically regular; (iii) the inverse map $F^{-1}$ is pseudo-Holder continuous; (iv) the inverse map $F^{-1}$ is lower pseudo-Holder continuous. An application, via Robinson's normal map formulation, leads to the following result in the context of semialgebraic…
In this paper, we consider set optimization problems where the solution concept is given by the set approach. Specifically, we deal with the lower less and the upper less set relations. First, we derive the convexity and Lipschitzianity of suitable scalarizing functionals under the assumption that the set-valued objective mapping has certain convexity and Lipschitzianity properties. Then, we obtain upper estimates of the limiting subdifferential of these functionals. These results, together with the properties of the scalarization functionals, allow us to obtain a Fermat rule for set optimizat…
A general nonlinear regularity model for a set-valued mapping $F:X\times R_+\rightrightarrows Y$, where $X$ and $Y$ are metric spaces, is considered using special iteration procedures, going back to Banach, Schauder, Lusternik and Graves. Namely, we revise the induction theorem from Khanh, J. Math. Anal. Appl., 118 (1986) and employ it to obtain basic estimates for studying regularity/openness properties. We also show that it can serve as a substitution of the Ekeland variational principle when establishing other regularity criteria. Then, we apply the induction theorem and the mentioned estim…
Several primal and dual characterizations of regularity properties of collections of sets in normed linear spaces are discussed. Relationships between regularity properties of collections of sets and those of set-valued mappings are provided.
In this paper, we introduce the concept of nearly convex set-valued mappings and investigate fundamental properties of these mappings. Additionally, we establish a geometric approach for generalized differentiation of nearly convex set-valued mappings and nearly convex functions. Our contributions expand the current knowledge of nearly convex sets and functions, while providing several new results pertaining to nearly convex set-valued mappings.
In the early 1960's, Moreau and Rockafellar introduced a concept of called \emph{subgradient} for convex functions, initiating the developments of theoretical and applied convex analysis. The needs of going beyond convexity motivated the pioneer works by Clarke considering generalized differentiation theory of Lipschitz continuous functions. Although Clarke generalized differentiation theory is applicable for nonconvex functions, convexity still plays a crucial role in Clarke subdifferential calculus. In the mid 1970's, Mordukhovich developed another generalized differentiation theory for nonc…
In this paper we will establish some necessary condition and sufficient condition respectively for a set-valued mapping to have the Lipschitz-like property relative to a closed set by employing regular normal cone and limiting normal cone of a restricted graph of the set-valued mapping. We will obtain a complete characterization for a set-valued mapping to have the Lipschitz-property relative to a closed and convex set by virtue of the projection of the coderivative onto a tangent cone. Furthermore, by introducing a projectional coderivative of set-valued mappings, we establish a verifiable ge…
In the contest of optimal control problems, regularity results for optima are known when addressing fiber-strictly convex Lagrangian. For infinite time horizons, or for settings with infinite dimensional dynamics, the equivalence between minima/maxima and extremals could break down. Commonly, this is due to a loss of convexity/concavity of the cost functional or to a presence of state constraints, in which further controllability assumptions are needed. For many science applications, this a trend is not required, as in energy saving problems. In the present paper, we deal with the set of a fun…
We present necessary conditions for monotonicity, in one form or another, of fixed point iterations of mappings that violate the usual nonexpansive property. We show that most reasonable notions of linear-type monotonicity of fixed point sequences imply {\em metric subregularity}. This is specialized to the alternating projections iteration where the metric subregularity property takes on a distinct geometric characterization of sets at points of intersection called {\em subtransversality}. Our more general results for fixed point iterations are specialized to establish the necessity of subtra…
In the present paper, several properties concerning generalized derivatives of multifunctions implicitly defined by set-valued inclusions are studied by techniques of variational analysis. Set-valued inclusions are problems formalizing the robust fulfilment of cone constraint systems, whose data are affected by a "crude knowledge" of uncertain elements, so they can not be casted in traditional generalized equations. The focus of this study in on the first-order behaviour of the solution mapping associated with a parameterized set-valued inclusion, starting with Lipschitzian properties and then…
In this paper, we study a first order solution method for a particular class of set optimization problems where the solution concept is given by the set approach. We consider the case in which the set-valued objective mapping is identified by a finite number of continuously differentiable selections. The corresponding set optimization problem is then equivalent to find optimistic solutions to vector optimization problems under uncertainty with a finite uncertainty set. We develop optimality conditions for these types of problems, and introduce two concepts of critical points. Furthermore, we p…
Metric regularity has emerged during last 2-3 decades as one of the central concepts of variational analysis. The roots of this concept go back to a circle of fundamental regularity ideas of classical analysis embodied in such results as the implicit function theorem, Banach open mapping theorem, theorems of Lyusternik and Graves, on the one hand, and the Sard theorem and the Thom-Smale transversality theory, on the other. The three principal themes that are in the focus of attention are: (a) regularity criteria (containing quantitative estimates for rates of regularity) including formal compa…
This article is devoted to the analysis of necessary and/or sufficient conditions for metric regularity in terms of Demyanov-Rubinov-Polyakova quasidifferentials. We obtain new necessary and sufficient conditions for the local metric regularity of a multifunction in terms of quasidifferentials of the distance function to this multifunction. We also propose a new MFCQ-type constraint qualification for a parametric system of quasidifferentiable equality and inequality constraints and prove that it ensures the metric regularity of a multifunction associated with this system. As an application, we…
