An implicit-function theorem for \(C^{0,1}\)-equations and parametric \(C^{1,1}\)-optimization.

*(English)*Zbl 0742.49006The article deals with locally Lipschitz functions \(\mathcal F\) from \({\mathbb{R}}^{n+m}\) into \({\mathbb{R}}^ n\). A set-valued directional derivative of \(\mathcal F\) is defined as L. Thibault’s limit set [Ann. Mat. Pura Appl., IV. Ser. 125, 157-192 (1980; Zbl 0486.46037)]. By means of it an implicit function theorem is proved and a Lipschitzian solution function is completely characterized. Thereafter a special parametric optimization problem (for which the set-valued directional derivatives of both an objective function and functions from a description of a feasible set exist and are locally Lipschitz) is considered. M. Kojima’s characterization of critical points for this problem is applied and the implicit function theorem is specified to this particular case. As a result the author obtains a complete characterization of the regular case, second order formulas for the marginal function, and some insight into the strict complementarity.

Reviewer: L.Grygarova (Praha)

##### MSC:

49J30 | Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) |

26B10 | Implicit function theorems, Jacobians, transformations with several variables |

90C31 | Sensitivity, stability, parametric optimization |

49K40 | Sensitivity, stability, well-posedness |

49J50 | FrĂ©chet and Gateaux differentiability in optimization |

##### Keywords:

locally Lipschitz functions; set-valued directional derivative; implicit function theorem; parametric optimization
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\textit{B. Kummer}, J. Math. Anal. Appl. 158, No. 1, 35--46 (1991; Zbl 0742.49006)

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