The exact feasibility of randomized solutions of uncertain convex programs

SIAM Journal on Optimization

Volumn 19, Issue 3, 2008, Pages 1211-1230

  Campi, M C ^a   Garatti, S ^b  

^a UNIVERSITY OF BRESCIA   (Italy)

^b POLITECNICO DI MILANO   (Italy)

Author keywords

Chance constrained; Convex optimization; Randomized methods; Robust optimization; Semi infinite programming; Uncertain optimization

Indexed keywords

CHANCE-CONSTRAINED; RANDOMIZED METHODS; ROBUST OPTIMIZATION; SEMI-INFINITE PROGRAMMING; UNCERTAIN OPTIMIZATION;

CONVEX OPTIMIZATION;

CONSTRAINED OPTIMIZATION;

EID: 61349159944     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/07069821X     Document Type: Article

References (27)

1. A. Ben-Tal and A. Nemirovski, Robust truss topology design via semidefinite programming, SIAM J. Optim., 7 (1997), pp. 991-1016.
2. A. Ben-Tal and A. Nemirovski, Robust convex optimization, Math. Oper. Res., 23 (1998), pp. 769-805.
3. A. Ben-Tal and A. Nemirovski, Robust solutions of uncertain linear programs, Oper. Res. Lett., 25 (1999), pp. 1-13.
4. A. Ben-Tal, A. Nemirovski, and C. Roos, Robust solutions of uncertain quadratic and conicquadratic problems, SIAM J. Optim., 13 (2002), pp. 535-560.
5. G. Calafiore and M. C. Campi, Uncertain convex programs: Randomized solutions and confidence levels, Math. Program., 102 (2005), pp. 25-46.
6. G. Calafiore and M. C. Campi, The scenario approach to robust control design, IEEE Trans. Automat. Control, 51 (2006), pp. 742-753.
7. G. Calafiore and F. Dabbene, A probabilistic framework for problems with real structured uncertainty in systems and control, Automatica, 38 (2002), pp. 1265-1276.
8. G. Calafiore, F. Dabbene, and R. Tempo, Randomized algorithms for probabilistic robustness with real and complex structured uncertainty, IEEE Trans. Automat. Control, 45 (2000), pp. 2218-2235.
9. G. Calafiore and B. Polyak, Stochastic algorithms for exact and approximate feasibility of robust LMIs, IEEE Trans. Automat. Control, 46 (2001), pp. 1755-1759.
10. A. Charnes and W. W. Cooper, Chance constrained programming, Management Sci., 6 (1959), pp. 73-79.
11. D. P. de Farias and B. Van Roy, On constraint sampling in the linear programming approach to approximate dynamic programming, Math. Oper. Res., 29 (2004), pp. 462-478.
12. D. Dentcheva, Optimization models with probabilistic constraints, in Probabilistic and Randomized Methods for Design under Uncertainty, G. Calafiore and F. Dabbene, eds., Springer, London, 2005, pp. 47-96.
13. E. Erdogan and G. Iyengar, Ambiguous chance constrained problems and robust optimization, Math. Program. Ser. B, 107 (2006), pp. 37-61.
14. L. El Ghaoui and G. Calafiore, Robust filtering for discrete-time systems with bounded noise and parametric uncertainty, IEEE Trans. Automat. Control, 46 (2001), pp. 1084-1089.
15. L. El Ghaoui and H. Lebret, Robust solutions to least-squares problems with uncertain data, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 1035-1064.
16. L. El Ghaoui, F. Oustry, and H. Lebret, Robust solutions to uncertain semidefinite programs, SIAM J. Optim., 9 (1998), pp. 33-52.
17. L. El Ghaoui and S. I. Niculescu, Robust decision problems in engineering: A linear matrix inequality approach, in Advances in Linear Matrix Inequality Methods in Control, L. El Ghaoui and S. I. Niculescu, eds., SIAM, Philadelphia, 1999.
18. V. L. Levin, Application of E. Helly's theorem to convex programming, problems of best approximation and related questions, SB. Math., 8 (1969), pp. 235-247.
19. R. LUCCHETTI, Convexity and Well-posed Problems, CMS Books Math., Springer, New York, 2006.
20. A. Nemirovski and A. Shapiro, Scenario approximations of chance constraints, in Probabilistic and Randomized Methods for Design under Uncertainty, G. Calafiore and F. Dabbene, eds., Springer, London, 2005, pp. 3-48.
21. A. Prékopa, Stochastic Programming, Kluwer, Boston, 1995.
22. A. Prékopa, Probabilistic programming, in Stochastic Programming, A. Ruszczyǹski and A. Shapiro, eds., Handbooks Oper. Res. Managment Sci. 10, Elsevier, London, UK, 2003, pp. 267-352.
23. W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
24. A. Ruszczyǹski and A. Shapiro, Stochasting programming models, in Stochastic Programming, A. Ruszczyǹski and A. Shapiro, eds., Handbooks Oper. Res. Managment Sci. 10, Elsevier, London, UK, 2003, pp. 1-64.
25. A. Ruszczyǹski and A. Shapiro, Lectures on Stochastic Programming, preprint, 2008.
26. A. N. Shiryaev, Probability, Springer, New York, 1996.
27. R. L. Smith, Efficients Monte-Carlo procedures for generating points uniformly distributed over bounded regions, Oper. Res., 32 (1984), pp. 1296-1308.