Necessary conditions for constrained optimization problems in smooth banach spaces and applications

SIAM Journal on Optimization

Volumn 12, Issue 4, 2002, Pages 1032-1047

  Zhu, Qiji J ^a  

^a WESTERN MICHIGAN UNIVERSITY   (United States)

Author keywords

Constrained optimization problem; Mathematical programs with equilibrium constraints; Necessary optimality condition; Singular normal vector; Smooth subdifferential; Subdifferential calculus

Indexed keywords

EID: 0036438687     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S105262340138339     Document Type: Article

References (28)

1. J.M. Borwein, J.S. Treiman, and Q.J. Zhu, Necessary conditions for constrained optimization problems with semicontinuous and continuous data, Trans. Amer. Math. Soc., 350 (1998), pp. 2409-2429.
2. J.M. Borwein and Q.J. Zhu, Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity, SIAM J. Control Optim., 34 (1996), pp. 1568-1591.
3. 1 perturbations, Nonlinear Anal., 28 (1997), pp. 889-915.
4. J.M. Borwein and Q.J. Zhu, A survey of subdifferential calculus with applications, Nonlinear Anal., 38 (1999), pp. 687-773.
5. F.H. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983; reprinted as Classics Appl. Math. 5, SIAM, Philadelphia, 1990.
6. F.H. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983; reprinted as Classics Appl. Math. 5, SIAM, Philadelphia, 1990.
7. I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), pp. 324-353.
8. A.D. Ioffe, On subdifferentiablility spaces, Ann. New York Aead. Sci., 410 (1983), pp. 107-119.
9. A.D. Ioffe, Necessary conditions for nonsmooth optimization, Math. Oper. Res., 9 (1984), pp. 159-189.
10. A.D. Ioffe, Proximal analysis and approximate subdifferentials, J. London Math. Soc., 41 (1990), pp. 175-192.
11. A.Y. Kruger and B.S. Mordukhovich, Extremal points and Euler equations in nonsmooth optimization, Dokl. Akad. Nauk. BSSR, 24 (1980), pp. 684-687 (in Russian).
12. P.D. Loewen, Optimal Control via Nonsmooth Analysis, CRM Lecture Notes Series, AMS Summer School on Control, CRM, Université de Montréal, 1992, American Mathematical Society, Providence, RI, 1993.
13. Z.Q. Lou, J.S. Pang, and D. Ralph Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, UK, 1996.
14. B.S. Mordukhovich, Approximation Methods in Problems of Optimization and Control, Nauka, Moscow, 1988 (in Russian; English translation to appear in Wiley-Interscience).
15. B.S. Mordukhovich, The extremal principle and its applications to optimization and economics, in Optimization and Related Topics, A. Rubinov and B. Glover, eds., Appl. Optim. 47, Kluwer, Dordrecht, The Netherlands, 2001, pp. 343-369.
16. B.S. Mordukhovich and Y. Shao, Extremal characterizations of Asplund spaces, Proc. Amer. Math. Soc., 124 (1996), pp. 197-205.
17. B.S. Mordukhovich and B. Wang, Necessary suboptimality and optimality conditions via variational principles, SIAM J. Control Optim, to appear.
18. H.V. Ngai and M. Théra, On Necessary Conditions for Non-Lipschitz Optimization Problems, preprint, Universite de Limoges, Limoges, France, 2000.
19. H.V. Ngai and M. Théra, Metric Regularity, Subdifferential Calculus, and Applications, Set-Valued Anal., 9 (2001), pp. 187-216.
20. J.V. Outrata, A generalized mathematical program with equilibrium constraints, SIAM J. Control Optim., 38 (2000), pp. 1623-1638.
21. R.T. Rockafellar, Proximal subgradients, marginal values and augmented Lagrangians in nonconvex optimization, Math. Oper. Res., 6 (1981), pp. 424-436.
22. J.S. Treiman, Lagrange multipliers for nonconvex generalized gradients with equality, inequality, and set constraints, SIAM J. Control Optim., 37 (1999), pp. 1313-1329.
23. J. Warga, Controllability and a multiplier rule for nondifferentiable optimization problems, SIAM J. Control Optim., 16 (1978), pp. 803-812.
24. J. Warga, Optimization and controllability without differentiability assumptions, SIAM J. Control Optim., 21 (1983), pp. 837-855.
25. J.J. Ye, Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints, SIAM J. Optim., 10 (2000), pp. 943-962.
26. 1 perturbations and applications, J. Optim. Theory Appl., 92 (1997), pp. 189-208.
27. Q.J. Zhu, Subderivatives and their applications, in Proceedings of the International Conference on Dynamical Systems and Differential Equations, Vol. II, Southwest Missouri State University, Springfield, MO, 1996, pp. 379-394.
28. Q.J. Zhu, The equivalence of several basic theorems for subdifferentials, Set-Valued Anal., 6 (1998), pp. 171-185.