Second-order optimality conditions for general nonconvex optimization problems and variational analysis of disjunctive systems

TL;DR

This paper develops second-order sufficient optimality conditions for broad nonconvex constrained problems and provides a variational analysis framework for disjunctive systems, enabling effective computation of second-order objects.

Contribution

It introduces new second-order optimality conditions applicable to general nonconvex problems and offers a variational analysis approach for disjunctive systems.

Findings

Conditions are sufficient for local minimizers under basic assumptions.

Provides a method to compute second-order objects in disjunctive systems.

Enhances understanding of second-order analysis in nonconvex optimization.

Abstract

In this paper, we propose second-order sufficient optimality conditions for a very general nonconvex constrained optimization problem, which covers many prominent mathematical programs.Unlike the existing results in the literature, our conditions prove to be sufficient, for an essential local minimizer of second order, under merely basic smoothness and closedness assumptions on the data defining the problem.In the second part, we propose a comprehensive first- and second-order variational analysis of disjunctive systems and demonstrate how the second-order objects appearing in the optimality conditions can be effectively computed in this case.