On the directional asymptotic approach in optimization theory

• DOI: 10.48550/arxiv.2402.16530
• Publication Date: 2024-02-26
• Authors: Matúš Benko; Patrick Mehlitz
• Keywords: Asymptotic Analysis

As a starting point of our research, we show that, for fixed order $\gamma\geq 1$, each local minimizer rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary the classical sense (corresponding to stationarity $1$), satisfies conditions terms coderivative construction $\gamma$, or asymptotically stationary with respect critical direction as well $\gamma$ certain sense. By ruling out latter case constraint qualification not stronger than directional metric subregularity, end up new necessary optimality comprising mixture limiting variational tools orders $1$ and $\gamma$. These abstract findings are carved broad class geometric constraints $\gamma:=2$, visualized by examples from complementarity-constrained nonlinear semidefinite optimization. byproduct particular setting $\gamma:=1$, approach yields so-called asymptotic regularity which serve qualifications guaranteeing M-stationarity minimizers. We compare these standard Further, extend concepts pseudo- quasi-normality arbitrary set-valued mappings. It shown that properties provide sufficient validity regularity. Finally, novel coderivative-like tool used construct presence For constraints, it illustrated all appearing objects can be calculated initial data.