Optimality conditions, approximate stationarity, and applications – a story beyond lipschitzness

¹ Federation University Australia, Centre for Informatics and Applied Optimization, School of Engineering, Information Technology and Physical Sciences, Ballarat, VIC 3353, Australia
² RMIT University, STEM College, School of Science, Melbourne, VIC 3001, Australia
³ Brandenburgische Technische Universität Cottbus-Senftenberg, Institute of Mathematics, 03046 Cottbus, Germany
⁴ University of Mannheim, School of Business Informatics and Mathematics, 68159 Mannheim, Germany

^* Corresponding author: mehlitz@b-tu.de

Received: 25 November 2021
Accepted: 30 March 2022

Abstract

Approximate necessary optimality conditions in terms of Frechet 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 Frechet 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.