Second-order subdifferential calculus with applications to tilt stability in optimization

The paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finitedimensional spaces. The main attention is paid to the so-called (full and partial) second-order subdifferentials of extended-real-valued functions, which are dual-type constructions generated by coderivatives of frst-order subdifferential mappings. We develop an extended second-order subdifferential calculus and analyze the basic second-order qualification condition ensuring the fulfillment of the principal secondorder chain rule for strongly and fully amenable compositions. The calculus results obtained in this way and computing the second-order subdifferentials for piecewise linear-quadratic functions and their major specifications are applied then to the study of tilt stability of local minimizers for important classes of problems in constrained optimization that include, in particular, problems of nonlinear programming and certain classes of extended nonlinear programs described in composite terms

Relative Well-Posedness of Constrained Systems with Applications to Variational Inequalities

The paper concerns foundations of sensitivity and stability analysis, being primarily addressed constrained systems. We consider general models, which are described by multifunctions between Banach spaces and concentrate on characterizing their well-posedness properties that revolve around Lipschitz stability and metric regularity relative to sets. The enhanced relative well-posedness concepts allow us, in contrast to their standard counterparts, encompassing various classes of constrained systems. Invoking tools of variational analysis and generalized differentiation, we introduce new robust notions of relative coderivatives. The novel machinery of variational analysis leads us to establishing complete characterizations of the relative well-posedness properties with further applications to stability of affine variational inequalities. Most of the obtained results valid in general infinite-dimensional settings are also new in finite dimensions.Comment: 25 page

Lipchitzian Stability of Parametric Variational Inequalities Over Generalized Polyhedra in Banach Spaces

This paper concerns the study of solution maps to parameterized variational inequalities over generalized polyhedra in reflexive Banach spaces. It has been recognized that generalized polyhedral sets are significantly different from the usual convex polyhedra in infinite dimensions and play an important role in various applications to optimization, particularly to generalized linear programming. Our main goal is to fully characterize robust Lipschitzian stability of the aforementioned solutions maps entirely via their initial data. This is done on the base of the coderivative criterion in variational analysis via efficient calculations of the coderivative and related objects for the systems under consideration. The case of generalized polyhedra is essentially more involved in comparison with usual convex polyhedral sets and requires developing elaborated techniques and new proofs of variational analysis