Optimality conditions, approximate stationarity, and applications -- a story beyond Lipschitzness

TL;DR

This paper develops generalized optimality conditions for complex optimization problems with non-Lipschitz objectives, introducing new theoretical tools and applying them to control problems with sparsity terms.

Contribution

It introduces a novel framework for approximate stationarity in non-Lipschitz optimization, extending classical optimality conditions and applying them to control problems.

Findings

Established approximate optimality conditions using Fréchet subgradients.

Derived a new extremal principle for non-Lipschitz problems.

Applied conditions to control problems with sparsity-promoting terms.

Abstract

Approximate necessary optimality conditions in terms of Fr\'echet subgradients and normals for a rather general optimization problem with a potentially non-Lipschitzian objective function are established with the aid of Ekeland's variational principle, the fuzzy Fr\'echet subdifferential sum rule, and a novel notion of lower semicontinuity relative to a set-valued mapping or set. Feasible points satisfying these optimality conditions are referred to as approximately stationary. As applications, we derive a new general version of the extremal principle. Furthermore, we study approximate stationarity conditions for an optimization problem with a composite objective function and geometric constraints, a qualification condition guaranteeing that approximately stationary points of such a problem are M-stationary, and a multiplier-penalty-method which naturally computes approximately stationary points of the underlying problem. Finally, necessary optimality conditions for an optimal control problem with a non-Lipschitzian sparsity-promoting term in the objective function are established.