- Title
- Optimality conditions, approximate stationarity, and applications 'a story beyond lipschitzness
- Creator
- Kruger, Alexander; Mehlitz, Patrick
- Date
- 2022
- Type
- Text; Journal article
- Identifier
- http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/185227
- Identifier
- vital:16631
- Identifier
-
https://doi.org/10.1051/cocv/2022024
- Identifier
- ISBN:1292-8119 (ISSN)
- 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. © The authors.
- Publisher
- EDP Sciences
- Relation
- ESAIM - Control, Optimisation and Calculus of Variations Vol. 28, no. (2022), p.; http://purl.org/au-research/grants/arc/DP160100854
- Rights
- All metadata describing materials held in, or linked to, the repository is freely available under a CC0 licence
- Rights
- https://creativecommons.org/licenses/by/4.0
- Rights
- Copyright © The authors
- Rights
- Open Access
- Subject
- 4901 Applied mathematics; 4903 Numerical and computational mathematics; Approximate stationarity; Generalized separation; Non-Lipschitzian programming; Optimality conditions; Sparse control
- Full Text
- Reviewed
- Funder
- This work is supported by the Australian Research Council, project DP160100854, and the DFG Grant Bilevel Optimal Control: Theory, Algorithms, and Applications (Grant No. WA 3636/4-2) within the Priority Program SPP 1962 (Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization). The first author benefited from the support of the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska–Curie Grant Agreement No. 823731 CONMECH, and Conicyt REDES program 180032.
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