On smoothing of parametric minimax-functions and generalized max-functions via regularization

Journal of Convex Analysis

Volumn 5, Issue 2, 1998, Pages 199-220

  Levitin, E ^a   Tichatschke, R ^b  

^a INSTITUTE FOR SYSTEMS ANALYSIS   (Russian Federation)

^b UNIVERSITY OF TRIER   (Germany)

Author keywords

Convex parametric optimization; Duality theory; Non diffefentiable optimization; Parametric minimax problems; Regularization; Semi infinite optimization

Indexed keywords

EID: 22444451843     PISSN: 09446532     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (32)

1. K. Beer: Solution of Large Scale Linear Optimization Problems, Deutscher Verlag der Wissenschaften, Berlin, 1977.
2. K. Beer: Decomposition approach in optimization, in: K. H. Elster (ed.): Modern Mathematical Methods of Optimization, Akademie Verlag, Berlin (1993) 116-131.
3. B. S. Darkhovsky, E. S. Levitin: On differential properties of the value function of a finite dimensional parametric min-max-problem, Kibernetika, Kiev, Part. I: 4 (1976) 58-69,
4. B. S. Darkhovsky, E. S. Levitin: On differential properties of the value function of a finite dimensional parametric min-max-problem, Kibernetika, Kiev, Part II: 5 (1976) 115-123. (in Russian)
5. V. F. Demyanov: Minimax: Directional Differentiation, University of Leningrad, Leningrad, 1975. (in Russian)
6. V. F. Demyanov, A. M. Rubinov: Quasidifferential Calculus, Optimization Software, Inc. Publ. Division, New York, 1986.
7. V. F. Demyanov, I. S. Zabrodin: Directional differentiability of a continuous maximum function of quasidifferentiable functions, Math. Prog. Study 29 (1986) 108-117.
8. V. F. Demyanov, D. Pallaschke (eds.): Nondifferential Optimization Methods and Applications, Springer-Verlag, Berlin et al., 1987.
9. I. Ekeland, R. Temam: Convex Analysis and Variational Problems, North-Holland, Elsevier, 1976.
10. Y. B. Germeyer: Games with Non-antagonistic Interests, Nauka, Moscow, 1976. (in Russian)
11. E. G. Gol'stein: Konvexe Optimierung, WTG, Bd. 122, Akademie Verlag, Berlin, 1973.
12. E. G. Gol'stein, N. V. Tretyakov: Modified Lagrange Functions, J. Wiley & Sons, New York, 1995.
13. R. Hettich, G. Still: Second order optimality conditions for generalized semi-infinite programming problems, Optimization 34 (1995) 195-211.
14. H. Th. Jongen, J. Rueckmann, O. Stein: Generalized semi-infinite optimization: A first order optimality condition and examples, No 95-24, Dept. of Mathematics, University of Trier, Trier, 1995. (Preprint)
15. A. Kaplan, R. Tichatschke: Stable Methods for Ill-posed Variational Problems, Akademie Verlag, Berlin, 1994.
16. A. Kaplan, R. Tichatschke: On a class of terminal variational problems, Proceedings of the PARAOPT-Conference, Enschede, June 1995, in: Parametric Optimization and Related Topics IV, (eds. J. Guddat, H. Th. Jongen, F. Nozicka, G. Still and F. Twilt), Verlag Peter Lang, Frankfurt/M. (1996) 185-199.
17. A. Kaplan; R. Tichatschke: On the numerical treatment of a class of semi-infinite terminal problems, Optimization 41 (1997) 1-36.
18. S. Karlin: Mathematical Methods and Theory in Games, Programming and Economics, Pergamon Press, London - Paris, 1959.
19. L. S. Lasdon: Optimization Theory for Large Systems, Macmillan Comp., London - New York, 1975.
20. E. S. Levitin, B. T. Polyak: Convergence of minimizing sequences in conditional extremum problems, Soviet Math. Doklady 7 (1966) 764-767.
21. E. S. Levitin: Second-order conditions in nonsmooth mathematical programming problems with applications in minimax problems with binding variables, Soviet Math. Doklady 20 (1979) 59-63.
22. E. S. Levitin: Decomposition choice of the decent direction in a block programming problem with binding variables and constraints, in: Models and Methods of Optimization, (ed. E.R. Smolyakov) issue 7, pp. 43-54, All-Union Institute for System Research (VNIISI); Moscow 1990. (in Russian)
23. E. S. Levitin: Perturbation Theory in Mathematical Programming and its Applications, J. Wiley & Sons, Chichester et al., 1994.
24. E. S. Levitin: Differential properties of parametric minimum functions and extremal maps, Proceedings of the PARAOPT-Conference, Enschede, June 1995, in: Parametric Optimization and Related Topics IV, (eds. J. Guddat, H. Th. Jongen, F. Nozicka, G. Still and F. Twilt), Verlag Peter Lang, Frankfurt/M. (1996) 213-226.
25. E. S. Levitin, R. Tichatschke: A branch-and-bound approach for solving a class of generalized semi-infinite programming problems, Journal of Global Optimization 13 (1998) 3.
26. B. Luderer: The quasidifferential of an optimal value function in non-smooth programming, Optimization 20 (1989) 597-613.
27. B. Luderer: Directional derivative estimates for the optimal value function of the quasidifferentiable programming problem, Math. Programming 51 (1991) 333-348.
28. B. T. Polyak: Introduction to Optimization, Optimization Software, Inc. Publ. Division, New York, 1987.
29. A. Shapiro: Second-order derivatives of extremal value functions and optimality conditions for semi-infinite programs, Mathematics of Operation Research 10 (1985) 207-219.
30. A. Shapiro: Sensitivity analysis of parametrized programs via generalized equations, SIAM J. Control and Optimization 32 (1994) 553-571.
31. V. D. Skarin: Regularization of minimax problems appearing in convex programming, Journal Comp. Maths. Math. Phys. 17 (1977). (in Russian)
32. A. N. Tikhonov, V. J. Arsenin: Methods for Solving Ill-posed Problems, J. Wiley & Sons, New York, 1977.