On Solving Large-Scale Finite Minimax Problems Using Exponential Smoothing

Journal of Optimization Theory and Applications

Volumn 148, Issue 2, 2011, Pages 390-421

  Pee, E Y ^a   Royset, J O ^a  

^a NAVAL POSTGRADUATE SCHOOL   (United States)

Author keywords

Exponential smoothing technique; Finite minimax; Rate of convergence; Run time complexity

Indexed keywords

APPROXIMATION THEORY;

EXPONENTIAL SMOOTHING; EXPONENTIAL SMOOTHING TECHNIQUE; FINITE MINIMAX; LARGE-SCALES; MINIMAX; MINIMAX PROBLEM; RATE OF CONVERGENCE; RUN TIME COMPLEXITY; SMOOTHING ALGORITHMS; SMOOTHING TECHNIQUES;

ITERATIVE METHODS;

EID: 79952247904     PISSN: 00223239     EISSN: 15732878     Source Type: Journal    
DOI: 10.1007/s10957-010-9759-1     Document Type: Article

References (36)

1. Polak, E.: On the mathematical foundations of nondifferentiable optimization in engineering design. SIAM Rev. 29, 21-89 (1987).
2. Polak, E., Salcudean, S., Mayne, D. Q.: Adaptive control of ARMA plants using worst case design by semi-infinite optimization. IEEE Trans. Autom. Control 32, 388-397 (1987).
3. Cai, X., Teo, K., Yang, X., Zhou, X.: Portfolio optimization under a minimax rule. Manag. Sci. 46(7), 957-972 (2000).
4. Demyanov, V. F., Malozemov, V. N.: Introduction to Minimax. Wiley, New York (1974).
5. Panier, E. R., Tits, A. L.: A globally convergent algorithm with adaptively refined discretization for semi-infinite optimization problems arising in engineering design. IEEE Trans. Autom. Control 34(8), 903-908 (1989).
6. Zhou, J. L., Tits, A. L.: An SQP algorithm for finely discretized continuous minimax problems and other minimax problems with many objective functions. SIAM J. Optim. 6(2), 461-487 (1996).
7. Polak, E., Royset, J. O., Womersley, R. S.: Algorithms with adaptive smoothing for finite minimax problems. J. Optim. Theory Appl. 119(3), 459-484 (2003).
8. Obasanjo, E., Tzallas-Regas, G., Rustem, B.: An interior-point algorithm for nonlinear minimax problems. J. Optim. Theory Appl. 144, 291-318 (2010).
9. Zhu, Z., Cai, X., Jian, J.: An improved SQP algorithm for solving minimax problems. Appl. Math. Lett. 22(4), 464-469 (2009).
10. Sturm, J. F., Zhang, S.: A dual and interior-point approach to solve convex min-max problems. In: Du, D. Z., Pardalos, P. M. (eds.) Minimax and Applications, pp. 69-78. Kluwer Academic, Dordrecht (1995).
11. Luksan, L., Matonoha, C., Vlcek, J.: Primal interior-point method for large sparse minimax optimization. Technical Report 941, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, Czech Republic (2005).
12. Ye, F., Liu, H., Zhou, S., Liu, S.: A smoothing trust-region Newton-CG method for minimax problem. Appl. Math. Comput. 199(2), 581-589 (2008).
13. Polak, E., Womersley, R. S., Yin, H. X.: An algorithm based on active sets and smoothing for discretized semi-infinite minimax problems. J. Optim. Theory Appl. 138, 311-328 (2008).
14. Li, X.: An entropy-based aggregate method for minimax optimization. Eng. Optim. 18, 277-285 (1992).
15. Xu, S.: Smoothing method for minimax problems. Comput. Optim. Appl. 20, 267-279 (2001).
16. Kort, B. W., Bertsekas, D. P.: A new penalty function algorithm for constrained minimization. In: Proceedings 1972 IEEE Conf. Decision and Control, vol. 82, pp. 343-362 (1972).
17. Nemirovski, A. S., Yudin, D. B.: Problem Complexity and Method Efficiency in Optimization. Wiley, New York (1983).
18. Drezner, Z.: On the complexity of the exchange algorithm for minimax optimization problems. Math. Program. 38(2), 219-222 (1987).
19. Wiest, E. J., Polak, E.: On the rate of convergence of two minimax algorithms. J. Optim. Theory Appl. 71(1), 1-30 (1991).
20. Nesterov, Yu.: Complexity estimates of some cutting plane methods based on the analytic barrier. Math. Program. 69(1), 149-176 (1995).
21. Ariyawansa, K. A., Jiang, P. L.: On complexity of the translational-cut algorithm for convex minimax problems. J. Optim. Theory Appl. 107(2), 223-243 (2000).
22. Nesterov, Yu., Vial, J. Ph.: Augmented self-concordant barriers and nonlinear optimization problems with finite complexity. Math. Program. 99(1), 149-174 (2004).
23. Nesterov, Yu.: Introductory Lectures on Convex Optimization: A Basic Course (Applied Optimization). Kluwer Academic, Dordrecht (2004).
24. Polak, E.: Optimization. Algorithms and Consistent Approximations. Springer, New York (1997).
25. Urruty, H., Baptiste, J.: Convex Analysis and Minimization Algorithms 1. Fundamentals. Springer, Berlin (1996).
26. Gill, P. E., Murray, W., Wright, M. H.: Numerical Linear Algebra and Optimization. Addison-Wesley, Redwood (1991).
27. Monteiro, R. D. C., Adler, I.: Interior path following primal-dual algorithms. Part II: Convex quadratic programming. Math. Program. 44(1), 43-66 (1989).
28. Rockafellar, R. T., Wets, R. J. B.: Variational Analysis. Springer, Heidelberg (1998).
29. Polak, E.: Smoothing techniques for the solution of finite and semi-infinite min-max-min problems. In: Pillo, G. D., Murli, A. (eds.) High Performance Algorithms and Software for Nonlinear Optimization. Kluwer Academic, Dordrecht (2003).
30. Polak, E.: On the convergence of the Pshenichnyi-Pironneau-Polak minimax algorithm with an active set strategy. J. Optim. Theory Appl. 138(2), 305-309 (2008).
31. Mathworks Inc.: MATLAB 7 Getting Started Guide. Natick, MA (2009).
32. Tomlab Optimization Inc.: User's Guide for TOMLAB/CPLEX v12. 1. Pullman, WA (2009).
33. Gill, P. E., Hammarling, S. J., Murray, W., Saunders, M. A., Wright, M. H.: User's Guide for LSSOL (Version 1. 0): A Fortran Package for Constrained Linear Least-Squares and Convex Quadratic Programming. Stanford, CA (1986).
34. Lawrence, C., Zhou, J. L., Tits, A. L.: User's Guide for CFSQP Version 2. 5: A C Code for Solving (Large Scale) Constrained Nonlinear (Minimax) Optimization Problems, Generating Iterates Satisfying All Inequality Constraints. Technical Report (1997).
35. Brualdi, R. A.: Introductory Combinatorics. Prentice-Hall, Upper Saddle River (2004).
36. Zhu, Z., Zhang, K.: A superlinearly convergent sequential quadratic programming algorithm for minimax problems. Chin. J. Numer. Math. Appl. 27(4), 15-32 (2005).