On necessary optimality conditions in a class of optimization problems

Jiří V. Outrata

Aplikace matematiky (1989)

  • Volume: 34, Issue: 6, page 466-474
  • ISSN: 0862-7940

Abstract

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In the paper necessary optimality conditions are derived for the minimization of a locally Lipschitz objective with respect to the consttraints x S , 0 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.

How to cite

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Outrata, Jiří V.. "On necessary optimality conditions in a class of optimization problems." Aplikace matematiky 34.6 (1989): 466-474. <http://eudml.org/doc/15602>.

@article{Outrata1989,
abstract = {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.},
author = {Outrata, Jiří V.},
journal = {Aplikace matematiky},
keywords = {Clarke regular graph; necessary conditions; tangent cone; locally Lipschitz objective function; set-valued map; Clarke normal cone; generalized gradient; contingent cone; Clarke regular graph; necessary conditions; tangent cone; locally Lipschitz objective function; set-valued map; Clarke normal cone; generalized gradient; contingent cone},
language = {eng},
number = {6},
pages = {466-474},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On necessary optimality conditions in a class of optimization problems},
url = {http://eudml.org/doc/15602},
volume = {34},
year = {1989},
}

TY - JOUR
AU - Outrata, Jiří V.
TI - On necessary optimality conditions in a class of optimization problems
JO - Aplikace matematiky
PY - 1989
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 34
IS - 6
SP - 466
EP - 474
AB - 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.
LA - eng
KW - Clarke regular graph; necessary conditions; tangent cone; locally Lipschitz objective function; set-valued map; Clarke normal cone; generalized gradient; contingent cone; Clarke regular graph; necessary conditions; tangent cone; locally Lipschitz objective function; set-valued map; Clarke normal cone; generalized gradient; contingent cone
UR - http://eudml.org/doc/15602
ER -

References

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  1. J.-P. Aubin I. Ekeland, Applied Nonlinear Analysis, Wiley, New York 1984. (1984) Zbl0641.47066MR0749753
  2. J. M. Borwein, Multivalued convexity: a unified approach to equality and inequality constraints, Math. Programming 13 (1977), 163-180. (1977) Zbl0375.90062
  3. F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York 1983. (1983) Zbl0582.49001MR0709590
  4. P. H. Dien P. H. Sach, Further properties of the regularity of inclusion systems, Preprint 87-21, Inst. of Mathematics, Hanoi 1987. (1987) Zbl0702.49012
  5. J.-В. Hiriart-Urruty, 10.1137/0316019, SIAM J. Control Optim. 16(1978), 301-316. (1978) Zbl0385.90099MR0493610DOI10.1137/0316019
  6. A. D. Ioffe, 10.1137/0317019, SIAM J. Control Optim. 17 (1979), 245-250. (1979) Zbl0417.49027MR0525025DOI10.1137/0317019
  7. B. N. Pschenichnyi, Convex set-valued mappings and their adjoints, Kibernetika 3 (1972), 94-102 (in Russian). (1972) 
  8. B. N. Pschenichnyi, Convex Analysis and Extremal Problems, Nauka, Moscow 1982 (in Russian). (1982) 
  9. S. M. Robinson, Generalized equations and their solutions. Part II: Applications to nonlinear programming, Univ. Wisconsin-Madison, Technical Summary Rep. # 2048, 1980. (1980) 
  10. R. T. Rockafellar, Directional differentiability of the optimal value function in a nonlinear programming problem, Math. Prog. Study 21 (1984), 213-226. (1984) Zbl0546.90088MR0751251
  11. P. H. Sach, 10.1080/02331938808843311, Optimization 19 (1988), 13 - 27. (1988) Zbl0648.49016MR0926215DOI10.1080/02331938808843311

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