Volume 28, 2022
|Number of page(s)||35|
|Published online||29 June 2022|
Optimality conditions, approximate stationarity, and applications – a story beyond lipschitzness
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: firstname.lastname@example.org
Accepted: 30 March 2022
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.
Mathematics Subject Classification: 49J52 / 49J53 / 49K27 / 90C30 / 90C48
Key words: Approximate stationarity / generalized separation / non-Lipschitzian programming / optimality conditions / sparse control
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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