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An implicit-function theorem for $$C^{0,1}$$-equations and parametric $$C^{1,1}$$-optimization. (English) Zbl 0742.49006
The 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.

##### 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 Fréchet and Gateaux differentiability in optimization
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##### References:
  Aubin, J.P, Differential calculus of set-valued maps. an update, ()  Aubin, J.P; Ekeland, I, Applied nonlinear analysis, (1984), Wiley New York  Clarke, F.H, On the inverse function theorem, Pacific J. math., 64, No. 1, 97-102, (1976) · Zbl 0331.26013  Clarke, F.H, Optimization and nonsmooth analysis, (1983), Wiley New York · Zbl 0727.90045  Cornet, B; Laroque, G, Lipschitz properties of solutions in mathematical programming, () · Zbl 0595.90081  Fiacco, A.V, Introduction to sensitivity and stability analysis in nonlinear programming, (1983), Academic Press New York · Zbl 0543.90075  Frankowska, H, High order inverse function theorems, () · Zbl 0701.49040  Guddat, J; Jongen, H.T; Nowack, D, Parametric optimization: pathfollowing with jumps, (), 43-53  Jongen, H.Th; Weber, G.-W, On parametric nonlinear programming, () · Zbl 0745.90067  Jongen, H.Th; Klatte, D; Tammer, K, Implicit functions and sensitivity of stationary points, () · Zbl 0715.65034  Jongen, H.Th; Moebert, T; Tammer, K, On iterated minimization in nonconvex optimization, Math. oper. res., 11, 679-691, (1986) · Zbl 0626.90080  Jourani, A; Thibault, L, Approximate subdifferential and metric regularity: the finite dimensional case, (1989), Dept. of Mathematics, Univ. Pau, preprint · Zbl 0714.49023  Klatte, D; Tammer, K, On second-order sufficient optimality conditions for C1, 1-optimization problems, Optimization, 19, No. 2, 169-180, (1988) · Zbl 0647.49014  Klatte, D; Kummer, B; Walzebok, R, Conditions for optimality and strong stability in nonlinear programs without assuming twice differentiability of data, ()  Kojima, M, Strongly stable stationary solutions in nonlinear programs, () · Zbl 0478.90062  \scB. Kummer, Lipschitzian inverse functions, directional derivatives and application in C1,1-optimization, J. Optim. Theory Appl., to appear. · Zbl 0795.49012  Kummer, B, Newton’s method for non-differentiable functions, (), 114-125  Pang, J.S, Newton’s method for B-differentiable equations, () · Zbl 0716.90090  Pang, J.S, Solution differentiability and continuation of Newton’s method for variational inequality problems over polyhedral sets, () · Zbl 0681.49011  Pshenichny, B.N, Implicit function theorems for multivalued mappings, ()  Robinson, S.M, Strongly regular generalized equations, Math. oper. res., 5, 43-62, (1980) · Zbl 0437.90094  Robinson, S.M, Bundle-based decomposition: conditions for convergence, () · Zbl 0675.90068  Robinson, S.M, An implicit-function theorem for B-differentiable functions, ()  Thibault, L, Subdifferentials of compactly Lipschitzian vector-valued functions, Ann. mat. pura appl. (4), 125, 157-192, (1980) · Zbl 0486.46037  Thibault, L, On generalized differentials and subdifferentials of Lipschitz vector-valued functions, Nonlinear anal. theory mathods appl., 6, No. 10, 1037-1053, (1982) · Zbl 0492.46036
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