2022/04/29 by Matúš Benko, Benko, Matúš, Patrick Mehlitz +1
Computer Science · Mathematics · #49J52 #49J53 #49K27 #90C22 #90C30 #90C33 #Advanced Optimization Algorithms Research #FOS: Mathematics #Optimization and Control (math.OC) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2204.13932
openalex publication_date 2022/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that, for a fixed order γ≥ 1, each local minimizer of a rather general nonsmooth optimization problem in Euclidean spaces is either M-stationary in the classical sense (corresponding to stationarity of order 1), satisfies stationarity conditions in terms of a coderivative construction of order γ, or is approximately stationary with respect to a critical direction as well as γ in a certain sense. By ruling out the latter case with a constraint qualification not stronger than directional metric subregularity, we end up with new necessary optimality conditions comprising a mixture of limiting variational tools of order 1 and γ. These abstract findings are carved out for the broad class of geometric constraints. As a byproduct, we obtain new constraint qualifications ensuring M-stationarity of local minimizers. The paper closes by illustrating these results in the context of standard nonlinear, complementarity-constrained, and nonlinear semidefinite programming.