Dedekind order completion of C(X) by Hausdorff continuous functions

The concept of Hausdorff continuous interval valued functions, developed within the theory of Hausdorff approximations and originaly defined for interval valued functions of one real variable is extended to interval valued functions defined on a topological space X. The main result is that the set of all finite Hausdorff continuous functions on any topological space X is Dedekind order complete. Hence it contains the Dedekind order completion of the set C(X) of all continuous real functions defined on X as well as the Dedekind order completion of the set C_b(X) of all bounded continuous functions on X. Under some general assumptions about the topological space X the Dedekind order completions of both C(X) and C_b(X) are characterised as subsets of the set of all Hausdorff continuous functions. This solves a long outstanding open problem about the Dedekind order completion of C(X). In addition, it has major applications to the regularity of solutions of large classes of nonlinear PDEs

The Radius of Metric Subregularity

There is a basic paradigm, called here the radius of well-posedness, which quantifies the "distance" from a given well-posed problem to the set of ill-posed problems of the same kind. In variational analysis, well-posedness is often understood as a regularity property, which is usually employed to measure the effect of perturbations and approximations of a problem on its solutions. In this paper we focus on evaluating the radius of the property of metric subregularity which, in contrast to its siblings, metric regularity, strong regularity and strong subregularity, exhibits a more complicated behavior under various perturbations. We consider three kinds of perturbations: by Lipschitz continuous functions, by semismooth functions, and by smooth functions, obtaining different expressions/bounds for the radius of subregularity, which involve generalized derivatives of set-valued mappings. We also obtain different expressions when using either Frobenius or Euclidean norm to measure the radius. As an application, we evaluate the radius of subregularity of a general constraint system. Examples illustrate the theoretical findings.Comment: 20 page

Stability and Error Analysis for Optimization and Generalized Equations

Stability and error analysis remain challenging for problems that lack regularity properties near solutions, are subject to large perturbations, and might be infinite dimensional. We consider nonconvex optimization and generalized equations defined on metric spaces and develop bounds on solution errors using the truncated Hausdorff distance applied to graphs and epigraphs of the underlying set-valued mappings and functions. In the process, we extend the calculus of such distances to cover compositions and other constructions that arise in nonconvex problems. The results are applied to constrained problems with feasible sets that might have empty interiors, solution of KKT systems, and optimality conditions for difference-of-convex functions and composite functions

Set-valued convex compositions

We study the composition of two set-valued functions defined on locally convex topological linear spaces. We assume that these functions map into certain complete lattices of sets that have been used to establish a conjugation theory for set-valued functions in the literature. Our main result is a formula for the conjugate of the composition in terms of the conjugates of the ingredient functions. As a special case, when the composition is proper and has further regularity, our formula yields a dual representation for the composition. The proof of the main result uses Lagrange duality and minimax theory in a nontrivial way.Comment: 18 page

Failure of Metric Regularity for Major Classes of Variational Systems

The paper is devoted to the study of metric regularity, which is a remarkable property of set-valued mappings playing an important role in many aspects of nonlinear analysis and its applications. We pay the main attention to metric regularity of the so- called parametric variational systems that contain, in particular, various classes of parameterized/perturbed variational and hemivariational inequalities, complementarity systems, sets of optimal solutions and corresponding Lagrange multipliers in problems of parametric optimization and equilibria, etc. Based on the advanced machinery of generalized differentiation1 we surprisingly reveal that metric regularity fails for certain major classes of parametric variational systems, which admit conventional descriptions via subdifferentials of convex as well as prox-regular extended-real-valued functions

Rectifiability of stationary varifolds branching set with multiplicity at most 2

This thesis deals with regularity and rectifiability properties on the branching set of stationary varifolds that can be represented as the graph of a two-valued function. In the first chapter I briefly show the Simon and Wickramasekera’s work in which they introduce a frequency function monotonicity formula for two-valued C1,α functions with stationary graph that leads to an estimate of the Hausdorff dimension of the branching set. In the second chapter I build upon Simon and Wickramasekera’s work and introduce several relaxed frequency functions in order to get an estimate of the Minkowski’s content of the branching set. I then use their result to prove the local (n − 2)-rectifiablility of the branching set

Transversality, regularity and error bounds in variational analysis and optimisation

Transversality properties of collections of sets, regularity properties of set-valued mappings, and error bounds of extended-real-valued functions lie at the core of variational analysis because of their importance for stability analysis, constraint qualifications, qualification conditions in coderivative and subdifferential calculus, and convergence analysis of numerical algorithms. The thesis is devoted to investigation of several research questions related to the aforementioned properties. We develop a general framework for quantitative analysis of nonlinear transversality properties by establishing primal and dual characterizations of the properties in both convex and nonconvex settings. The H¨older case is given special attention. Quantitative relations between transversality properties and the corresponding regularity properties of set-valued mappings as well as nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space are also discussed. We study a new property so called semitransversality of collections of set-valued mappings on metric (in particular, normed) spaces. The property is a generalization of the semitransversality of collections of sets and the negation of the corresponding stationarity, a weaker property than the extremality of collections of set-valued mappings. Primal and dual characterizations of the property as well as quantitative relations between the property and semiregularity of set-valued mappings are formulated. As a consequence, we establish dual necessary and sufficient conditions for stationarity of collections of set-valued mappings as well as optimality conditions for efficient solutions with respect to variable ordering structures in multiobjective optimization. We examine a comprehensive (i.e. not assuming the mapping to have any particular structure) view on the regularity theory of set-valued mappings and clarify the relationships between the existing primal and dual quantitative sufficient and necessary conditions including their hierarchy. The typical sequence of regularity assertions, often hidden in the proofs, and the roles of the assumptions involved in the assertions, in particular, on the underlying space: general metric, normed, Banach or Asplund are exposed. As a consequence, we formulate primal and dual conditions for the stability properties of solution mappings to inclusions. We propose a unifying general framework of quantitative primal and dual sufficient and necessary error bound conditions covering linear and nonlinear, local and global settings. The function is not assumed to possess any particular structure apart from the standard assumptions of lower semicontinuity in the case of sufficient conditions and (in some cases) convexity in the case of necessary conditions. We expose the roles of the assumptions involved in the error bound assertions, in particular, on the underlying space: general metric, normed, Banach or Asplund. As a consequence, the error bound theory is applied to characterize subregularity of set-valued mappings, and calmness of the solution mapping in convex semi-infinite optimization problems.Doctor of Philosoph