MPCC: strong stability of \(m\)-stationary points
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 \[ \mathbf{L}^{cc}(x,\rho ,\sigma )=f(x)-\sum_{m=1}^{l}[\rho _{m}r_{m}(x)+\sigma _{m}s_{m}(x)]. \]
mathematical programs with complementarity constraints
scientific article; zbMATH DE number 7383624
scientific article