Optimality conditions, approximate stationarity, and applications 'a story beyond lipschitzness
- Authors: Kruger, Alexander , Mehlitz, Patrick
- Date: 2022
- Type: Text , Journal article
- Relation: ESAIM - Control, Optimisation and Calculus of Variations Vol. 28, no. (2022), p.
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: 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.
Fuzzy multiplier, sum and intersection rules in non-Lipschitzian settings : decoupling approach revisited
- Authors: Fabian, Marian , Kruger, Alexander , Mehlitz, Patrick
- Date: 2024
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 532, no. 2 (2024), p.
- Relation: https://purl.org/au-research/grants/arc/DP160100854
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- Description: We revisit the decoupling approach widely used (often intuitively) in nonlinear analysis and optimization and initially formalized about a quarter of a century ago by Borwein & Zhu, Borwein & Ioffe and Lassonde. It allows one to streamline proofs of necessary optimality conditions and calculus relations, unify and simplify the respective statements, clarify and in many cases weaken the assumptions. In this paper we study weaker concepts of quasiuniform infimum, quasiuniform lower semicontinuity and quasiuniform minimum, putting them into the context of the general theory developed by the aforementioned authors. Along the way, we unify the terminology and notation and fill in some gaps in the general theory. We establish rather general primal and dual necessary conditions characterizing quasiuniform