Charactarizations of Linear Suboptimality for Mathematical Programs with Equilibrium Constraints

The paper is devoted to the study of a new notion of linear suboptimality in constrained mathematical programming. This concept is different from conventional notions of solutions to optimization-related problems, while seems to be natural and significant from the viewpoint of modern variational analysis and applications. In contrast to standard notions, it admits complete characterizations via appropriate constructions of generalized differentiation in nonconvex settings. In this paper we mainly focus on various classes of mathematical programs with equilibrium constraints (MPECs), whose principal role has been well recognized in optimization theory and its applications. Based on robust generalized differential calculus, we derive new results giving pointwise necessary and sufficient conditions for linear suboptimality in general MPECs and its important specifications involving variational and quasi variational inequalities, implicit complementarity problems, etc

Weak Sharp Minima on Riemannian Manifolds

This is the first paper dealing with the study of weak sharp minima for constrained optimization problems on Riemannian manifolds, which are important in many applications. We consider the notions of local weak sharp minima, boundedly weak sharp minima, and global weak sharp minima for such problems and obtain their complete characterizations in the case of convex problems on finite-dimensional Riemannian manifolds and their Hadamard counterparts. A number of the results obtained in this paper are also new for the case of conventional problems in linear spaces. Our methods involve appropriate tools of variational analysis and generalized differentiation on Riemannian and Hadamard manifolds developed and efficiently implemented in this paper

About [q]-regularity properties of collections of sets

We examine three primal space local Hoelder type regularity properties of finite collections of sets, namely, [q]-semiregularity, [q]-subregularity, and uniform [q]-regularity as well as their quantitative characterizations. Equivalent metric characterizations of the three mentioned regularity properties as well as a sufficient condition of [q]-subregularity in terms of Frechet normals are established. The relationships between [q]-regularity properties of collections of sets and the corresponding regularity properties of set-valued mappings are discussed.Comment: arXiv admin note: substantial text overlap with arXiv:1309.700

Conic cancellation laws and some applications

We discuss, on finite and infinite dimensional normed vector spaces, some versions of Radstr\"{o}m cancellation law (or lemma) that are suited for applications to set optimization problems. In this sense, we call our results "conic" variants of the celebrated result of Radstr\"{o}m, since they involve the presence of an ordering cone on the underlying space. Several adaptations to this context of some topological properties of sets are studied and some applications to subdifferential calculus associated to set-valued maps and to necessary optimality conditions for constrained set optimization problems are given. Finally, a stability problem is considered

Calculus of Tangent Sets and Derivatives of Set Valued Maps under Metric Subregularity Conditions

In this paper we intend to give some calculus rules for tangent sets in the sense of Bouligand and Ursescu, as well as for corresponding derivatives of set-valued maps. Both first and second order objects are envisaged and the assumptions we impose in order to get the calculus are in terms of metric subregularity of the assembly of the initial data. This approach is different from those used in alternative recent papers in literature and allows us to avoid compactness conditions. A special attention is paid for the case of perturbation set-valued maps which appear naturally in optimization problems.Comment: 17 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

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