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

TL;DR

This paper develops a second-order subdifferential calculus in variational analysis and applies it to analyze tilt stability of local minimizers in constrained optimization problems.

Contribution

It introduces an extended second-order subdifferential calculus and applies it to tilt stability analysis in nonlinear and extended nonlinear programming.

Findings

Developed a new second-order subdifferential calculus.

Derived conditions for tilt stability in constrained optimization.

Applied calculus to piecewise linear-quadratic functions.

Abstract

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.