Randomized strategies for probabilistic solutions of uncertain feasibility and optimization problems

IEEE Transactions on Automatic Control

Volumn 54, Issue 11, 2009, Pages 2545-2559

  Alamo, Teodoro ^a   Tempo, Roberto ^b   Camacho, Eduardo F ^a  

^a UNIVERSITY OF SEVILLE   (Spain)

^b POLITECNICO DI TORINO   (Italy)

Author keywords

Probabilistic robustness; Randomized algorithms; Statistical learning theory; Uncertain systems

Indexed keywords

ACCURACY PARAMETERS; BOOLEAN EXPRESSIONS; CONTROL COMMUNITY; CONVEX OPTIMIZATION PROBLEMS; DECISION VARIABLES; EXPLICIT BOUNDS; FEASIBILITY PROBLEM; NEW RESULTS; OPTIMIZATION PROBLEMS; PROBABILISTIC ROBUSTNESS; PROBABILISTIC SOLUTION; RANDOMIZED ALGORITHMS; RANDOMIZED STRATEGY; SAMPLE SIZES; SEMI-INFINITE PROGRAMMING PROBLEMS; STATISTICAL LEARNING THEORY;

CONVEX OPTIMIZATION; OPTIMIZATION; UNCERTAIN SYSTEMS;

LEARNING ALGORITHMS;

EID: 79952901713     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2009.2031207     Document Type: Article

References (40)

1. T. Alamo, R. Tempo, and E. F. Camacho, "Improved sample size bounds for probabilistic robust control design: A pack-based strategy", in Proc. 46th IEEE Conf. Decision Control, New Orleans, LA, Dec. 2007, pp. 6178-6183. (Pubitemid 351278126)
2. T. Alamo, R. Tempo, and E. F. Camacho, "Revisiting statistical learning theory for uncertain feasibility and optimization problems", in Proc. 46th IEEE Conf. Decision Control, New Orleans, LA, Dec. 2007, pp. 2135-2140. (Pubitemid 351277470)
3. T. Alamo, R. Tempo, and E. F. Camacho, "Statistical learning theory: A pack-based strategy for uncertain feasibility and optimization problems", in Recent Advances in Learning and Control, V. D. Blondel, S. P. Boyd, and H. Kimura, Eds. London, U. K.: Springer Verlag, 2008.
4. T. Alamo, R. Tempo, D. R. Ramirez, and E. F. Camacho, "A new vertex result for robustness problems with interval matrix uncertainty", Syst. Control Lett., vol. 57, pp. 474-481, 2008. (Pubitemid 351509642)
5. T. Alamo, R. Tempo, D. Rodríguez, and E. F. Camacho, "A sequentially optimal randomized algorithm for robust LMI feasibility problems", in Proc. Eur. Control Conf., Kos, Greece, Jul. 2007, pp. 3620-3627.
6. B. D. O. Anderson and J. B. Moore, Optimal Control: Linear Quadratic Methods. Englewood Cliffs, NJ: Prentice-Hall, 1990.
7. T. Basar and G. J. Olsder, Dynamic Noncooperative Game Theory. Philadelphia, PA: SIAM, 1999.
8. E.-W. Bai, R. Tempo, and M. Fu, "Worst-case properties of the uniform distribution and randomized algorithms for robustness analysis", Math. Control, Signals, Syst., vol. 11, pp. 183-196, 1998. (Pubitemid 128493150)
9. B. R. Barmish and C. M. Lagoa, "The uniform distribution: A rigorous justification for its use in robustness analysis", Math. Control, Signals, Syst., vol. 10, pp. 203-222, 1997. (Pubitemid 127551900)
10. V. Blondel and J. N. Tsitsiklis, "A survey of computational complexity results in system and control", Automatica, vol. 36, pp. 1249-1274, 2000.
11. G. Calafiore and M. C. Campi, "Uncertain convex programs: Randomized solutions and confidence levels", Math. Program., vol. 102, pp. 25-46, 2005. (Pubitemid 40134690)
12. G. Calafiore and M. C. Campi, "The scenario approach torobust control design", IEEE Trans. Autom. Control, vol. 51, no. 5, pp. 742-753, May 2006. (Pubitemid 43792636)
13. G. Calafiore and F. Dabbene, "A probabilistic analytic center cutting plane method for feasibility of uncertain LMIs", Automatica, vol. 43, pp. 2022-2033, 2007. (Pubitemid 350137263)
14. G. Calafiore and B. T. Polyak, "Stochastic algorithms for exact and approximate feasibility of robust LMIs", IEEE Trans. Autom. Control, vol. 46, no. 11, pp. 1755-1759, Nov. 2001. (Pubitemid 33120659)
15. M. C. Campi and S. Garatti, "The exact feasibility of randomized solutions of robust convex programs", SIAM J. Optim., vol. 19, pp. 1211-1230, 2008.
16. H. Chernoff, "A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations", Annals Math. Stat., vol. 23, pp. 493-507, 1952.
17. Y. Fujisaki, F. Dabbene, and R. Tempo, "Probabilistic design of LPV control systems", Automatica, vol. 39, pp. 1323-1337, 2003.
18. Y. Fujisaki and Y. Kozawa, "Probabilistic robust controller design: Probable near minimax value and randomized algorithms", in Probabilistic and Randomized Methods for Design under Uncertainty, G. Calafiore and F. Dabbene, Eds. London, U. K.: Springer Verlag, 2006.
19. W. Hoeffding, "Probability inequalities for sums of bounded random variables", J. Amer. Stat. Assoc., vol. 58, pp. 13-30, 1963.
20. H. Ishii, T. Basar, and R. Tempo, "Randomized algorithms for synthesis of switching rules for multimodal systems", IEEE Trans. Autom. Control, vol. 50, no. 6, pp. 754-767, Jun. 2005. (Pubitemid 40902701)
21. S. Kanev, B. De Schutter, and M. Verhaegen, "An ellipsoid algorithm for probabilistic robust controller design", Syst. Control Lett., vol. 49, pp. 365-375, 2003.
22. M. Karpinski and A. Macintyre, "Polynomial bounds for VC dimension of sigmoidal and general pfaffian neural networks", J. Comput. Syst. Sci., vol. 54, pp. 169-176, 1997. (Pubitemid 127437202)
23. V. Koltchinskii, C. T. Abdallah, M. Ariola, P. Dorato, and D. Panchenko, "Improved sample complexity estimates for statistical learning control of uncertain systems", IEEE Trans. Autom. Control, vol. 45, no. 12, pp. 2383-2388, Dec. 2000. (Pubitemid 32191005)
24. S. Mendelson, "A few notes on statistical learning theory", in Advanced Lectures in Machine Learning, S. Mendelson and A. J. Smola, Eds. London, U. K.: Springer Verlag, 2003.
25. Y. Oishi, "Polynomial-time algorithms for probabilitic solutions of parameter-dependent linear matrix inequalities", Automatica, vol. 43, pp. 538-545, 2007. (Pubitemid 46209059)
26. J. M. Parrondo and C. van den Broeck, "Vapnik-Chervonenkis bounds for generalization", J. Phys. A, vol. 26, pp. 2211-2223, 1993.
27. B. T. Polyak and R. Tempo, "Probabilistic robust design with linear quadratic regulators", Syst. Control Lett., vol. 43, pp. 343-353, 2001. (Pubitemid 32663763)
28. N. Sauer, "On the density of family of sets", J. Combinatorial Theory, vol. 13, no. A, pp. 145-147, 1972.
29. C. Scherer, "LMI relaxations in robust control", Eur. J. Control, vol. 12, no. 1, pp. 3-29, 2006.
30. E. D. Sontag, "VC dimension of neural networks", in Neural Networks and Machine Learning, C. M. Bishop, Ed. New York: Springer Verlag, 1998.
31. R. Tempo, E.-W. Bai, and F. Dabbene, "Probabilistic robustness analysis: Explicit bounds for the minimum number of samples", Syst. Control Lett., vol. 30, pp. 237-242, 1997. (Pubitemid 127454009)
32. R. Tempo, G. Calafiore, and F. Dabbene, "Randomized algorithms for analysis and control of uncertain systems", in Communications and Control Engineering Series. London, U. K.: Springer Verlag, 2005.
33. J. S. Tyler and F. B. Tuteur, "The use of a quadratic performance index to design multivariable invariant plants", IEEE Trans. Autom. Control, vol. 11, no. 1, pp. 84-92, Jan. 1966.
34. V. N. Vapnik, Statistical Learning Theory. New York: Wiley, 1998.
35. V. N. Vapnik and A. Ya. Chervonenkis, "On the uniform convergence of relative frequencies of events to their probabilities", Theory Probab. Appl., vol. 16, pp. 264-280, 1971.
36. V. N. Vapnik and A. Ya. Chervonenkis, "Necessary and sufficient conditions for the uniform convergence of means to their expectations", Theory Probab. Appl., vol. 26, no. 3, pp. 532-553, 1981.
37. M. Vidyasagar, A Theory of Learning and Generalization: With Applications to Neural Networks and Control Systems. London, U. K.: Springer Verlag, 1997.
38. M. Vidyasagar, "Statistical learning theory and randomized algorithms for control", Control Syst. Mag., vol. 18, no. 6, pp. 69-85, 1998. (Pubitemid 128566741)
39. M. Vidyasagar, "Randomized algorithms for robust controller synthesis using statistical learning theory", Automatica, vol. 37, pp. 1515-1528, 2001. (Pubitemid 32737008)
40. M. Vidyasagar and V. D. Blondel, "Probabilistic solutions to some NP-hard matrix problems", Automatica, vol. 37, no. 9, pp. 1397-1405, 2001. (Pubitemid 32691035)