On the use of augmented Lagrangians in the solution of generalized semi-infinite min-max problems

Computational Optimization and Applications

Volumn 31, Issue 2, 2005, Pages 173-192

  Polak, Elijah ^a   Royset, Johannes O ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b NAVAL POSTGRADUATE SCHOOL   (United States)

Author keywords

Augmented Lagrangian functions; Generalized minimax; Nonsmooth optimization algorithms; Semi infinite optimization

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; INTEGRATION; NUMERICAL METHODS; PROBABILITY; PROBLEM SOLVING; SET THEORY;

AUGMENTED LAGRANGIAN FUNCTIONS; GENERALIZED MINIMAX; NONSMOOTH OPTIMIZATION ALGORITHMS; SEMI-INFINITE OPTIMIZATION;

OPTIMIZATION;

EID: 21144454232     PISSN: 09266003     EISSN: 15732894     Source Type: Journal    
DOI: 10.1007/s10589-005-2179-8     Document Type: Article

References (33)

1. M.S. Bazaraa, H.D. Sherall, and C.M. Shetty, Nonlinear Programming. Theory and Algorithms, 2nd edition, Wiley: New York, NY, 1993.
2. D.P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods, Academic Press: New York, NY, 1982.
3. J.F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Verlag: New York, NY, 2000.
4. J. V. Burke, "Calmness and exact penalization," SIAM J. Control and Optimization, vol. 29, no. 2, pp. 493-497, 1991.
5. F. Clarke, Optimization and Nonsmooth Analysis, Wiley: New York, NY, 1983.
6. O. Ditlevsen and H.O. Madsen, Structural Reliability Methods, Wiley: New York, NY, 1996.
7. O. Fujiwara, S.-P. Han, and O.L. Mangasarian, "Local duality of nonlinear programs," SIAM J. Control and Optimization, vol. 22, pp. 162-169, 1984.
8. C. Gonzaga and E. Polak, "On constraint dropping schemes and optimality functions for a class of outer approximations algorithms," SIAM J. Control and Optimization, vol. 17, no. 4, pp. 477-493, 1979.
9. T.J. Graettinger and B.H. Krogh, "The acceleration radius: A global performance measure for robotic manipulators," IEEE J. of Robotics and Automation, vol. 4, pp. 60-69, 1988.
10. W.W. Hager and D.L. Presler, "Dual techniques for minimax," SIAM J. Control and Optimization, vol. 25, pp. 660-685, 1987.
11. R. Hettich and G. Still, "Semi-infinite programming models in robotics," Parametric Optimization and Related Topics II, J. Goddat et al. (Eds.), Akademie Verlag, Berlin, 1991, pp. 112-118.
12. H.Th. Jongen, J.-J. Ruckmann, and O. Stein, "Generalized semi-infinite optimization: A first order optimality condition and examples," Mathematical Programming, vol. 83, pp. 145-158, 1998.
13. A. Kaplan and R. Tichatschk, "On the numerical treatment of a class of semi-infinite terminal problems," Optimization, vol. 41, pp. 1-36, 1997.
14. J. Kyparisis, "On the uniquenss of Kuhn-Tucker multipliers in nonlinear programming," Mathematical Programming, vol. 32, pp. 242-246, 1985.
15. E. Levitin, "Reduction of generalized semi-infinite programming problems to semi-infinite or piece-wise smooth programming problems," Preprint no. 8-2001, University of Trier, Germany, 2001.
16. E. Levitin and R. Tichatschke, "A branch-and-bound approach for solving a class of generalized semi-infinite programming problems," J. Global Optimization, vol. 13, pp. 299-315, 1998.
17. X. Li, "An entropy-based aggregate method for minimax optimization," Engineering Optimization, vol. 18, pp. 277-285, 1997.
18. Matlab reference manual, version 5.3, (R11), MathWorks, Inc., Natick, Mass., 1999.
19. G. Di Pillo, "Exact penalty methods," in Algorithms for Continuous Optimization: The State of the Art, E. Spedicato (Ed.), Kluwer Academic Pub., Dordrecht, 1994.
20. E. Polak, Optimization. Algorithms and Consistent Approximations, Springer Verlag: New York, NY, 1997.
21. B.N. Pshenichnyi and Yu. M. Danilin, Numerical Methods in Extremal Problems (Chislennye Melody v Ekstremal' nykh Zadachakh), Nauka, Moscow, 1975.
22. R.T. Rockafellar and R.J.-B. Wets, Variational Analysis, Springer-Verlag: New York, NY, 1997.
23. R.T. Rockafellar, "Augmented Lagrange multiplier functions and duality in nonconvex programming," SIAM J. Control, vol. 12. no. 2, pp. 268-285, 1974.
24. J. O. Royset, A. Der Kiureghian, and E. Polak, "Reliability-based optimal structural design by the decoupling approach," Reliability Engineering and System Safety, Elsevier Science, vol. 73, no. 3, 2001, pp. 213-221.
25. J.O. Royset, E. Polak, and A. Der Kiureghian, "Adaptive approximations and exact penalization for the solution of generalized semi-infinite min-max problems," SIAM J. Optimization, vol. 14, no. 1, pp. 1-34, 2003.
26. J.-J. Ruckmann and A. Shapiro, "On first-order optimality conditions in generalized semi-infinite programming," J. Optimization Theory and Applications, vol. 101, no. 3, pp. 677-91, 1999.
27. O. Stein, Bi-Level Strategies in Semi-Infinite Programming, Kluwer Academic: Boston, 2003.
28. O. Stein, "First order optimality conditions for degenerate index sets in generalized semi-infinite programming," Mathematics of Operations Research, vol. 26, pp. 565-582, 2001.
29. O. Stein and G. Still, "On optimality conditions for generalized semi-infinite programming problems," J. Optimization Theory and Applications, vol. 104, pp. 443-458, 2000.
30. G. Still, "Generalized semi-infinite programming: Theory and methods," European J. of Operations Research, vol. 119, pp. 301-313, 1999.
31. G. Still, "Generalized semi-infinite programming: Numerical aspects," Optimization, vol. 49, no. 3, pp. 223-242, 2001.
32. J.L. Zhou and A.L. Tits, "An SQP algorithm for finely discretized continuous minimax problems and other minimax problems with many objective functions," SIAM Journal on Optimization, vol. 6, pp. 461-487, 1996.
33. G.-W. Weber, "Generalized semi-infinite optimization: On some foundations," Vychislitel'nye Tekhnologii, vol. 4, pp. 41-61, 1999.