Critical multipliers in variational systems via second-order generalized differentiation.

*(English)*Zbl 1407.90314The following equation system is studied:
\[
\Psi(x,v) = 0, \;\;v \in \partial\theta(\Phi),
\]
where \(\Phi: \mathbb R^n \longrightarrow \mathbb R^m\), \(\Psi : \mathbb R^n \times \mathbb R^m \longrightarrow \mathbb R^l\), \(\theta: \mathbb R^m \longrightarrow \overline{\mathbb R}\) is an extended real-valued function, \(\partial\) denotes an appropriate subdifferential. The authors use mainly a convex function \(\theta\) so that \(\partial\) is the classical subdifferential.

The authors introduce the notion of critical and non-critical mulltipliers for the variational system. A complete general characterization of the critical and non-critical multipliers is obtained. Especially comprehensive results are presented for the case that \(\theta\) belongs to the class of convex piecewise linear functions.

The authors introduce the notion of critical and non-critical mulltipliers for the variational system. A complete general characterization of the critical and non-critical multipliers is obtained. Especially comprehensive results are presented for the case that \(\theta\) belongs to the class of convex piecewise linear functions.

Reviewer: Karel Zimmermann (Praha)

##### MSC:

90C31 | Sensitivity, stability, parametric optimization |

49J52 | Nonsmooth analysis |

49J53 | Set-valued and variational analysis |

##### Keywords:

variational systems; composite optimization; critical noncritical multipliers; noncritical multipliers; generalized differentiation; piecewise linear functions; robust isolated calmness; Lipschitzian stability##### References:

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