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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Necessary conditions
for constrained optimization problems
with semicontinuous and continuous data


Authors: Jonathan M. Borwein, Jay S. Treiman and Qiji J. Zhu
Journal: Trans. Amer. Math. Soc. 350 (1998), 2409-2429
MSC (1991): Primary 49J52; Secondary 49J40, 49J50, 58C20
DOI: https://doi.org/10.1090/S0002-9947-98-01984-9
MathSciNet review: 1433112
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Abstract: We consider nonsmooth constrained optimization problems with semicontinuous and continuous data in Banach space and derive necessary conditions without constraint qualification in terms of smooth subderivatives and normal cones. These results, in different versions, are set in reflexive and smooth Banach spaces.


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  • 1. J. M. Borwein and A. Ioffe, 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, Trans. 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, Canad. 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 $L^1$ 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. Funct. Anal. 111 (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, Springer-Verlag, 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. Ioffe, Approximate subdifferentials and applications, Part 3, Mathematica 36 (1989), 1-38. MR 90g:49012
  • 13. A. D. Ioffe, 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 $\varepsilon$-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 transl., 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
  • 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 Banach spaces, Trans. 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.

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Additional Information

Jonathan M. Borwein
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada
Email: jborwein@cecm.sfu.ca

Jay S. Treiman
Affiliation: Department of Mathematics and Statistics, Western Michigan University, Kalamazoo, Michigan 49008
Email: treiman@math-stat.wmich.edu

Qiji J. Zhu
Affiliation: Department of Mathematics and Statistics, Western Michigan University, Kalamazoo, Michigan 49008
Email: zhu@math-stat.wmich.edu

DOI: https://doi.org/10.1090/S0002-9947-98-01984-9
Keywords: Constrained optimization problems, multipliers, nonsmooth analysis, subderivatives, normals, fuzzy calculus
Received by editor(s): September 14, 1995
Received by editor(s) in revised form: August 9, 1996
Additional Notes: Research of the first author was supported by NSERC and by the Shrum Endowment at Simon Fraser University
Article copyright: © Copyright 1998 American Mathematical Society

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