Optimization and Equilibrium Problems with Equilibrium Constraints in Infinite-Dimensional Spaces

The paper is devoted to applications of modern variational f).nalysis to the study of constrained optimization and equilibrium problems in infinite-dimensional spaces. We pay a particular attention to the remarkable classes of optimization and equilibrium problems identified as MPECs (mathematical programs with equilibrium constraints) and EPECs (equilibrium problems with equilibrium constraints) treated from the viewpoint of multiobjective optimization. Their underlying feature is that the major constraints are governed by parametric generalized equations/variational conditions in the sense of Robinson. Such problems are intrinsically nonsmooth and can be handled by using an appropriate machinery of generalized differentiation exhibiting a rich/full calculus. The case of infinite-dimensional spaces is significantly more involved in comparison with finite dimensions, requiring in addition a certain sufficient amount of compactness and an efficient calculus of the corresponding sequential normal compactness (SNC) properties

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

Perturbation of error bounds

Our aim in the current article is to extend the developments in Kruger, Ngai & Théra , SIAM J. Optim. 20(6), 3280–3296 (2010) and, more precisely, to characterize , in the Banach space setting, the stability of the local and global error bound property of inequalities determined by proper lower semicontinuous under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the 'radius of error bounds'. The definitions and characterizations are illustrated by examples

About regularity properties in variational analysis and applications in optimization

Regularity properties lie at the core of variational analysis because of their importance for stability analysis of optimization and variational problems, constraint qualications, qualication conditions in coderivative and subdierential calculus and convergence analysis of numerical algorithms. The thesis is devoted to investigation of several research questions related to regularity properties in variational analysis and their applications in convergence analysis and optimization. Following the works by Kruger, we examine several useful regularity properties of collections of sets in both linear and Holder-type settings and establish their characterizations and relationships to regularity properties of set-valued mappings. Following the recent publications by Lewis, Luke, Malick (2009), Drusvyatskiy, Ioe, Lewis (2014) and some others, we study application of the uniform regularity and related properties of collections of sets to alternating projections for solving nonconvex feasibility problems and compare existing results on this topic. Motivated by Ioe (2000) and his subsequent publications, we use the classical iteration scheme going back to Banach, Schauder, Lyusternik and Graves to establish criteria for regularity properties of set-valued mappings and compare this approach with the one based on the Ekeland variational principle. Finally, following the recent works by Khanh et al. on stability analysis for optimization related problems, we investigate calmness of set-valued solution mappings of variational problems.Doctor of Philosoph

Directional metric regularity of multifunctions

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