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Clarke regular graph; necessary conditions; tangent cone; locally Lipschitz objective function; set-valued map; Clarke normal cone; generalized gradient; contingent cone
In the paper necessary optimality conditions are derived for the minimization of a locally Lipschitz objective with respect to the consttraints $x \in S, 0 \in F(x)$, where $S$ is a closed set and $F$ is a set-valued map. No convexity requirements are imposed on $F$. The conditions are applied to a generalized mathematical programming problem and to an abstract finite-dimensional optimal control problem.
[1] J.-P. Aubin I. Ekeland: Applied Nonlinear Analysis. Wiley, New York 1984. MR 0749753
[2] J. M. Borwein: Multivalued convexity: a unified approach to equality and inequality constraints. Math. Programming 13 (1977), 163-180.
[3] F. H. Clarke: Optimization and Nonsmooth Analysis. Wiley, New York 1983. MR 0709590 | Zbl 0582.49001
[4] P. H. Dien P. H. Sach: Further properties of the regularity of inclusion systems. Preprint 87-21, Inst. of Mathematics, Hanoi 1987.
[5] J.-В. Hiriart-Urruty: Gradients generalisés de fonctions marginales. SIAM J. Control Optim. 16(1978), 301-316. DOI 10.1137/0316019 | MR 0493610 | Zbl 0385.90099
[6] A. D. Ioffe: Necessary and sufficient conditions for a local minimum. Part 1: A reduction theorem and first order conditions. SIAM J. Control Optim. 17 (1979), 245-250. DOI 10.1137/0317019 | MR 0525025
[7] B. N. Pschenichnyi: Convex set-valued mappings and their adjoints. Kibernetika 3 (1972), 94-102 (in Russian).
[8] B. N. Pschenichnyi: Convex Analysis and Extremal Problems. Nauka, Moscow 1982 (in Russian).
[9] S. M. Robinson: Generalized equations and their solutions. Part II: Applications to nonlinear programming. Univ. Wisconsin-Madison, Technical Summary Rep. # 2048, 1980.
[10] R. T. Rockafellar: Directional differentiability of the optimal value function in a nonlinear programming problem. Math. Prog. Study 21 (1984), 213-226. DOI 10.1007/BFb0121219 | MR 0751251 | Zbl 0546.90088
[11] P. H. Sach: Regularity, calmness and support principle. Optimization 19 (1988), 13 - 27. DOI 10.1080/02331938808843311 | MR 0926215 | Zbl 0648.49016
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