Necessary conditions for constrained optimization problems with semicontinuous and continuous data

Transactions of the American Mathematical Society

Volumn 350, Issue 6, 1998, Pages 2409-2429

  Borwein, Jonathan M ^a   Treiman, Jay S ^a   Zhu, Qui J ^a  

^a WESTERN MICHIGAN UNIVERSITY   (United States)

Author keywords

Constrained optimization problems; Fuzzy calulus; Multipliers; Nonsmooth analysis; Normals; Suberivatives

Indexed keywords

EID: 22044439655     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9947-98-01984-9     Document Type: Article

References (29)

1. J. M. Borwein and A. loffe, Proximal analysis in smooth spaces, Set-valued Analysis 4 (1996), 1-24. MR 96m:49028
2. J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Thins. Amer. Math. Soc. 303 (1987), 517-527. MR 88k:49013
3. J. M. Borwein and H. M. Strojwas, Proximal analysis and boundaries of closed sets in Banach spaces, Part 1: Theory, Canad. J. Math. 38 (1986), 431-452. Part 2: Applications, Canid. J. Math. 39 (1987), 428-472. MR 87h:90258; MR 88f:46034
4. J. M. Borwein and Q. J. Zhu, Variational Analysis in Non-reflexive Spaces and Applications to Control Problems with L1 Perturbations, Nonlinear Analysis 28 (1997), 889-915. CMP 97:05
5. J. M. Borwein and Q. J. Zhu, Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity, SIAM J. Control and Optimization 34 (1996), 1568-1591. CMP 96:17 ' '
6. F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, Inc., New York 1983. MR 85m:49002
7. F. H. Clarke, Methods of dynamic and nonsmooth optimization, CBMS-NSF Regional conference series in applied mathematics, SIAM, Philadelphia, 1989. MR 91j:49001
8. R. Deville, G. Godefroy and V. Zizler, A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions, J. Fund. Anal. Ill (1993), 197-212. MR 94b:49010
9. R. Deville, G. Godefroy and V. Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, No. 64, John Wiley & Sons, Inc., New York 1993. MR 94d:46012
10. J. Diestel, Geometry of Banach spaces-Selected topics, Lecture Notes in Math., 485, SpringerVerlag, 1975. MR 57:1079
11. M. Fabian, J. H. Whitfield and V. Zizler, Norms with locally Lipschitz derivatives, Israel J. Math. 44 (1983), 262-276. MR 84i:46028
12. A. D. loffe, Approximate subdifferentials and applications, Part 3, Mathematica 36 (1989), 1-38. MR 90g:49012
13. A. D. loffe, Proximal analysis and approximate subdifferentials, J. London Math. Soc. 41 (1990), 175-192. MR 91i:46045
14. A. Y. Kruger, Generalized differentials of nonsmooth functions and necessary conditions for an extremum, Siberian Math. J. 26 (1985), 370-379. MR 86j:49038
15. A. Y. Kruger and B. S. Mordukhovich, Extremal points and the Euler equation in nonsmooth optimization, Dokl. Akad. Nauk BSSR 24 (1980), 684-687. MR 82b:90127
16. Y. Li and S. Shi, A generalization of Ekeland's e-variational principle and of its Borwein-Preiss smooth variant, to appear in J. Math. Anal. Appl,
17. P. D. Loewen, Optimal Control via Nonsmooth Analysis, CRM Lecture Notes Series, Amer. Math. Soc., Summer School on Control, CRM, Université de Montréal, (1992), Amer. Math. Soc., Providence, 1993. MR 94h:49003
18. B. S. Mordukhovich, Maximum principle in problems of time optimal control with nonsmooth constraints, J. Appl. Math. Mech. 40 (1976), 960-969. MR 58:7284
19. B. S. Mordukhovich, Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Soviet Math. Dokl. 22 (1980), 526-530. MR 82b:90104
20. B. S. Mordukhovich, "Approximation methods in problems of optimization and control", Nauka, Moscow, 1988. [English transi., Wiley/Interscience, to appear]. MR 89m:49001
21. B. S. Mordukhovich and Y. Shao, Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc. 348 (1996), 1235-1280. MR 96h:49036
22. R. R. Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics, No. 1364, Springer Verlag, N.Y., Berlin, Tokyo, 1988, Second Edition 1993. MR 90g:46063; MR 94f:46055
23. G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), 326-350. MR 52:14940 r
24. R. T. Rockafellar, Proximal subgradients, marginal values and augmented Lagrangians in nonconvex optimization, Math. Oper. Res. 6 (1981), 424-436. MR 83m:90088
25. R. T. Rockafellar, Extensions of subgradient calculus with applications to optimization, Nonlinear Analysis, TMA 9 (1985), 665-698. MR 87a:90148
26. J. S. Treiman, A new characterization of Clarke's tangent cone and its applications to subgradient analysis and optimization, Ph. D. Thesis, Univ. of Washington, 1983.
27. J. S. Treiman, Clarke's gradients and epsilon-subgradients in Danach spaces, Thins. Amer. Math. Soc. 294 (1986), 65-78. MR 87d:90188
28. J. S. Treiman, Lagrange multipliers for nonconvex generalized gradients with equality, inequality and set constraints, preprint.
29. Q. J. Zhu, Calculus rules for subderivatives in smooth Banach spaces, preprint.