Second-Order Subdifferential Calculus with Applications to Tilt Stability in Optimization
Boris S. Mordukhovich, R. TYRRELL ROCKAFELLAR · SIAM Journal on Optimization · 2012
This paper concerns the second-order generalized differentiation theory of variational analysis and new applications of this theory to some problems of constrained optimization in finite-dimensional spaces. The main focus is the so-called (full and partial) second-order subdifferentials of extended-real-valued functions, which are dual-type constructions generated by coderivatives of first-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 second-order chain rule for strongly and fully amenable compositions. We also calculate the second-order subdifferentials for some major classes of piecewise linear-quadratic functions. These results are applied 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.