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.
Keywords: Approximate stationarity, generalized separation, non-Lipschitzian programming, optimality conditions, Sparse control
@article{COCV_2022__28_1_A42_0,
author = {Kruger, Alexander Y. and Mehlitz, Patrick},
title = {Optimality conditions, approximate stationarity, and applications {\textendash} a story beyond lipschitzness},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022024},
mrnumber = {4445585},
zbl = {1494.49005},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2022024/}
}
TY - JOUR AU - Kruger, Alexander Y. AU - Mehlitz, Patrick TI - Optimality conditions, approximate stationarity, and applications – a story beyond lipschitzness JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2022024/ DO - 10.1051/cocv/2022024 LA - en ID - COCV_2022__28_1_A42_0 ER -
%0 Journal Article %A Kruger, Alexander Y. %A Mehlitz, Patrick %T Optimality conditions, approximate stationarity, and applications – a story beyond lipschitzness %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2022024/ %R 10.1051/cocv/2022024 %G en %F COCV_2022__28_1_A42_0
Kruger, Alexander Y.; Mehlitz, Patrick. Optimality conditions, approximate stationarity, and applications – a story beyond lipschitzness. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 42. doi: 10.1051/cocv/2022024
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