A minimax theorem with applications to machine learning, signal processing, and finance

SIAM Journal on Optimization

Volumn 19, Issue 3, 2008, Pages 1344-1367

  Kim, Sbung Jean ^a   Boyd, Stephen ^a  

^a Stanford University   (United States)

Author keywords

Convex optimization; Minimax theorem; Robust optimization

Indexed keywords

CONVEX SET; DESIGN VARIABLES; EFFICIENT METHOD; FISHER LINEAR DISCRIMINANT ANALYSIS; MACHINE-LEARNING; MATCHED FILTERING; MINIMAX THEOREM; PORTFOLIO SELECTION; POSITIVE DEFINITE; ROBUST BEAMFORMING; ROBUST OPTIMIZATION; SADDLE POINT; STATISTICAL MODELS;

DISCRIMINANT ANALYSIS; FACE RECOGNITION; OPTIMIZATION; ROBOT LEARNING; SET THEORY; SIGNAL PROCESSING; THEOREM PROVING;

CONVEX OPTIMIZATION;

EID: 69649090538     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/060677586     Document Type: Article

References (31)

1. S. BENSON and Y. Ye, DSDP5: Software for Semidefinite Programming, http://wwwunix.mcs.anl.gov/DSDP/(2005).
2. D. Bertsekas, A. Nedić, and A. Ozdaglar, Convex Analysis and Optimization, Athena Scientific, Cambridge, MA, 2003.
3. D. Bertsimas and I. Popescu, Optimal inequalities in probability theory: A convex optimization approach, SIAM J. Optim., 15 (2005), pp. 780-804.
4. M. Best and P. Grauer, On the sensitivity of mean-variance-efficient portfolios to changes in asset means: Some analytical and computational results, Rev. Financ. Stud., 4 (1991), pp. 315-342.
5. F. Black and R. Litterman, Global portfolio optimization, Financ. Analysts J., 48 (1992), pp. 28-43.
6. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, 1994.
7. S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004.
8. J. Cochrane, Asset Pricing, 2nd ed., Princeton University Press, Princeton, NJ, 2001.
9. O. Costa and J. Paiva, Robust portfolio selection using linear matrix inequalities, J. Econom. Dynam. Control, 26 (2002), pp. 889-909.
10. L. E. Ghaoui, M. Oks, and F. Oustry, Worst-case value-at-risk and robust portfolio optimization: A conic programming approach, Oper. Res., 51 (2003), pp. 543-556.
11. D. Goldfarb and G. IYENGAR, Robust portfolio selection problems, Math. Oper. Res., 28 (2003), pp. 1-38.
12. B. Halldórsson and R. Tütüncü, An interior-point method for a class of saddle point problems, J. Optim. Theory Appl., 116 (2003), pp. 559-590.
13. T. Hastie, R. Tibshirani, and J. Friedman, The Elements of Statistical Learning, Springer Ser. Statist., Springer-Verlag, New York, 2001.
14. P. Jorion, Bayes-Stein estimation for portfolio analysis, J. Financ. Quant. Anal., 21 (1979), pp. 279-292.
15. S. Kassam and V. Poor, Robust techniques for signal processing: A survey, Proc. IEEE, 73 (1985), pp. 433-481.
16. S.-J. Kim and S. Boyd, Two-fund separation under model mis-specification, manuscript. Available from http://www.stanford.edu/~boyd/rob-two-fund-sep.html.
17. S.-J. Kim, A. Magnani, and S. Boyd, Robust Fisher discriminant analysis, in Adv. Neural Inform. Process. Syst., MIT Press, Cambridge, MA, 2006.
18. M. Lobo and S. Boyd, The worst-case risk of a portfolio, manuscript. Available from http://faculty.fuqua.duke.edu/2000.
19. D. Luenberger, Investment Science, Oxford University Press, New York, 1998.
20. H. Markowitz, Portfolio selection, J. Finance, 7 (1952), pp. 77-91.
21. R. MiCHAUD, The Markowitz optimization enigma: Is 'optimized' optimal?, Financ. Analysts J., 45 (1989), pp. 31-42.
22. A. Mutapcic, S.-J. Kim, and S. Boyd, Array signal processing with robust rejection constraints via second-order cone programming, in Proceedings of the 40th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA, 2006.
23. A. Roy, Safety first and the holding of assets, Econometrica, 20 (1952), pp. 413-449.
24. W. Sharpe, The Sharpe ratio, J. Portfolio Manag., 21 (1994), pp. 49-58.
25. M. SiON, On general minimax theorems, Pacific J. Math., 8 (1958), pp. 171-176.
26. J. Sturm, Using SBDUMI 1.02, A Matlab Toolbox for Optimization Over Symmetric Cones, http://www.fewcal.kub.nl/sturm/software/sedumi.html/(2001).
27. K. Toh, R. H. Tütüncü, and M. Todd, SDPTZ version 3.02. A Matlab software for semidefinite-quadratic-linear programming, http://www.math.nus.edu.sg/~mattohkc/sdpt3.html/(2002).
28. H. Van Trees, Detection, Estimation, and Modulation Theory, Part I, Wiley-Interscience, New York, 2001.
29. L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Rev., 38 (1996), pp. 49-95.
30. L. Vandenberghe, S. Boyd, and K. Comanor, Generalized Chebyshev bounds via semidefinite programming, SIAM Rev., 49 (2007), pp. 52-64.
31. S. Verdú and H. Poor, On minimax robustness: A general approach and applications, IEEE Trans. Inform. Theory, 30 (1984), pp. 328-340.