MPCC: strong stability of \(m\)-stationary points
type: Item
instance of: scholarly article
title: MPCC: strong stability of \(m\)-stationary points
zbMATH Open document ID: 1479.90200
author: Daniel Hernández Escobar
author: Harald Günzel
author: Jan-Joachim Rückmann
published in: Set-Valued and Variational Analysis
publication date: 2021-08-19T00:00:00Z
review text: Let \(f,r_{m},s_{m}:\mathbb{R}^{n}\rightarrow \mathbb{R}\) be twice continuously differentiable. For the MPCC consisting in minimizing \(f(x)\) subject to \(\min \{r_{m}(x),s_{m}(x)\}=0,\) \(m=1,\dots,l\), the authors generalize the notion of strong stationarity introduced for standard nonlinear programs by \textit{M. Kojima} [in: Analysis and computation of fixed points, Proc. Symp., Univ. Wis. 1979, 93--138 (1980; Zbl 0478.90062)]. Assuming that the MPCC-LICQ holds at a feasible point \(\overline{x}\) of the MPCC (which means that the gradients at \(\overline{x}\) of the functions \(r_{m}\) and \(s_{m}\) that vanish at \(\overline{x}\) are linearly independent), the main result of the paper establishes an algebraic characterization of the strong stationarity of \(\overline{x}\), involving the Hessian matrix at \(\overline{x}\) of the MPCC-Lagrange function \(\mathbf{L}^{cc}\) defined by \[ \mathbf{L}^{cc}(x,\rho ,\sigma )=f(x)-\sum_{m=1}^{l}[\rho _{m}r_{m}(x)+\sigma _{m}s_{m}(x)]. \]
Mathematics Subject Classification ID: 90C33
Mathematics Subject Classification ID: 90C46
Mathematics Subject Classification ID: 90C31
zbMATH DE Number: 7383624
zbMATH Keywords: strong stability
zbMATH Keywords: mathematical programs with complementarity constraints
zbMATH Keywords: M-stationarity: MPCC-LICQ
reviewed by: Juan-Enrique Martínez-Legaz
MaRDI profile type: Publication
full work available at URL: s11228-021-00592-2
OpenAlex ID: W3166579098
cites work: On stability of M-stationary points in mpccs
cites work: A smoothing method for mathematical programs with equilibrium constraints
cites work: On M-stationary points for mathematical programs with equilibrium constraints
cites work: MPCC: on necessary conditions for the strong stability of C-stationary points
cites work: MPCC: strongly stable C-stationary points when the number of active constraints is n + 1
cites work: Differential Topology
cites work: Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints
cites work: A note on sensitivity of value functions of mathematical programs with complementarity constraints
cites work: Q5691990
cites work: An Active-Set Newton Method for Mathematical Programs with Complementarity Constraints
cites work: On inertia and Schur complement in optimization
cites work: Nonlinear optimization in finite dimensions. Morse theory, Chebyshev approximation, transversality, flows, parametric aspects
cites work: Q4828904
cites work: On Stability of the Feasible Set of a Mathematical Problem with Complementarity Problems
cites work: MPCC: Critical Point Theory
cites work: Characterization of strong stability for C-stationary points in MPCC
cites work: Mathematical Programs with Equilibrium Constraints: Enhanced Fritz John-conditions, New Constraint Qualifications, and Improved Exact Penalty Results
cites work: A new regularization method for mathematical programs with complementarity constraints with strong convergence properties
cites work: Strong stability in nonlinear programming revisited
cites work: Q3936512
cites work: Interior Methods for Mathematical Programs with Complementarity Constraints
cites work: Mathematical Programs with Equilibrium Constraints
cites work: Optimality Conditions for a Class of Mathematical Programs with Equilibrium Constraints
cites work: Strongly Regular Generalized Equations
cites work: On existence and uniqueness of stationary points in semi-infinite optimization
cites work: Mathematical programs with complementarity constraints: stationarity, optimality, and sensi\-tivity.
cites work: Topological aspects of nonsmooth optimization.
cites work: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints
DOI: 10.1007/S11228-021-00592-2
description: scientific article; zbMATH DE number 7383624