Second-Order Subdifferential Calculus with Applications to Tilt Stability in Optimization

Abstract

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 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 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.

Tilt Stability, Uniform Quadratic Growth, and Strong Metric Regularity of the Subdifferential

“…For more details on the theory of generalized Hessians see [8,9]. This is significant since there is now a fairly effective calculus of this second-order nonsmooth object [10], thereby providing the means of identifying stable strong local minimizers in many instances of practical importance.…”

Stability of Saddle Points Via Explicit Coderivatives of Pointwise Subdifferentials

“…It can also be used to directly characterize the convergence of certain basic optimization methods [4,25,26]. Metric regularity is also closely related to the concept of tilt-stability, mainly studied in finite dimensions, see, e.g., [14,15,31,40,43], but recently also in infinite dimensions [36,39]. An extended concept incorporating tilt stability is that of full stability [30,37].…”

“…We remark that due to the linear dependence of the optimality conditions 0 ∈ J y (u, v) on y, the stability with respect to y can be seen as a form of tilt-stability [14,15,30,31,36,37,40,43] for saddle-point systems.…”

Conditioning of linear-quadratic two-stage stochastic optimization problems

“…Following [10], for (x, z) ∈ R n × R m and any u ∈ ∂ x f (x, z) the (extended) partial second-order subdifferential of f is a multifunction…”