Linear matrix inequalities with stochastically dependent perturbations and applications to chance-constrained semidefinite optimization

SIAM Journal on Optimization

Volumn 22, Issue 4, 2012, Pages 1394-1430

  Cheung, Sin Shuen ^a   So, Anthony Man Cho ^b   Wang, Kuncheng ^b  

^a Cornell University   (United States)

^b CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Chance constrained programming; Large deviation bounds; Linear matrix inequalities; Semidefinite programming; Stochastic programming

Indexed keywords

CHANCE CONSTRAINT; CHANCE-CONSTRAINED; CHANCE-CONSTRAINED PROGRAMMING; DATA PERTURBATION; IN-CONTROL; LARGE DEVIATIONS; LINEAR INEQUALITIES; PROBABILITY THEORY; SEMI-DEFINITE PROGRAMMING; SEMIDEFINITE OPTIMIZATION;

COMPUTER PROGRAMMING; CONVEX OPTIMIZATION; COVARIANCE MATRIX; SIGNAL PROCESSING; STOCHASTIC PROGRAMMING;

LINEAR MATRIX INEQUALITIES;

EID: 84871594793     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/110822906     Document Type: Article

References (65)

1. R. Ahlswede and A. Winter, Strong converse for identification via quantum channels, IEEE Trans. Inform. Theory, 48 (2002), pp. 569-579.
2. I. Bechar, A Bernstein-Type Inequality for Stochastic Processes of Quadratic Forms of Gaussian Variables, http://arxiv.org/abs/0909.3595 (2009).
3. A. Ben-Tal, L. El Ghaoui, and A. Nemirovski, Robust Optimization, Princeton Ser. Appl. Math., Princeton University Press, Princeton, NJ, 2009.
4. A. Ben-Tal and A. Nemirovski, Robust solutions of linear programming problems contaminated with uncertain data, Math. Program. Ser. A, 88 (2000), pp. 411-424.
5. A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. MPS/SIAM Ser. Optim., SIAM, Philadelphia, 2001.
6. A. Ben-Tal and A. Nemirovski, On safe tractable approximations of chance-constrained linear matrix inequalities, Math. Oper. Res., 34 (2009), pp. 1-25.
7. D. Bertsimas and D. B. Brown, Constructing uncertainty sets for robust linear optimization, Oper. Res., 57 (2009), pp. 1483-1495.
8. D. Bertsimas and M. Sim, The price of robustness, Oper. Res., 52 (2004), pp. 35-53.
9. D. Bertsimas and M. Sim, Tractable approximations to robust conic optimization problems, Math. Program. Ser. B, 107 (2006), pp. 5-36.
10. F. Blanchini, Set invariance in control, Automatica, 35 (1999), pp. 1747-1767.
11. B. Bollob'as, Modern Graph Theory, Grad. Texts in Math. 184, Springer-Verlag, New York, 1998.
12. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM Stud. Appl. Numer. Math. 15, SIAM, Philadelphia, 1994.
13. G. Calafiore and M. C. Campi, Uncertain convex programs: Randomized solutions and confidence levels, Math. Program. Ser. A, 102 (2005), pp. 25-46.
14. G. C. Calafiore and M. C. Campi, The scenario approach to robust control design, IEEE Trans. Automat. Control, 51 (2006), pp. 742-753.
15. G. C. Calafiore and L. El Ghaoui, On distributionally robust chance-constrained linear programs, J. Optim. Theory Appl., 130 (2006), pp. 1-22.
16. M. C. Campi and G. C. Calafiore, Notes on the scenario design approach, IEEE Trans. Automat. Control, 54 (2009), pp. 382-385.
17. M. C. Campi and S. Garatti, The exact feasibility of randomized solutions of uncertain convex programs, SIAM J. Optim., 19 (2008), pp. 1211-1230.
18. A. Charnes and W. W. Cooper, Chance-constrained programming, Management Sci., 6 (1959), pp. 73-79.
19. A. Charnes, W. W. Cooper, and G. H. Symonds, Cost horizons and certainty equivalents: An approach to stochastic programming of heating oil, Management Sci., 4 (1958), pp. 253- 263.
20. W. Chen and M. Sim, Goal-driven optimization, Oper. Res., 57 (2009), pp. 342-357.
21. W. Chen, M. Sim, J. Sun, and C.-P. Teo, From CVaR to uncertainty set: Implications in joint chance-constrained optimization, Oper. Res., 58 (2010), pp. 470-485.
22. X. Chen, M. Sim, and P. Sun, A robust optimization perspective on stochastic programming, Oper. Res., 55 (2007), pp. 1058-1071.
23. E. Delage and Y. Ye, Distributionally robust optimization under moment uncertainty with application to data-driven problems, Oper. Res., 58 (2010), pp. 595-612.
24. L. El 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.
25. E. Erdoǧan and G. Iyengar, Ambiguous chance constrained problems and robust optimization, Math. Program. Ser. B, 107 (2006), pp. 37-61.
26. H. Föllmer and A. Schied, Stochastic Finance: An Introduction in Discrete Time, 2nd ed., de Gruyter Stud. Math. 27, Walter de Gruyter, Berlin, 2004.
27. D. J. H. Garling, Inequalities: A Journey into Linear Analysis, Cambridge University Press, New York, 2007.
28. S. Golden, Lower bounds for the Helmholtz function, Phys. Rev., 137 (1965), pp. B1127- B1128.
29. M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming, http://cvxr.com/cvx (2011).
30. F. Hansen and G. K. Pedersen, Jensen's operator inequality, Bull. London Math. Soc., 35 (2003), pp. 553-564.
31. D. L. Hanson, Some results relating moment generating functions and convergence rates in the law of large numbers, Ann. Math. Statist., 38 (1967), pp. 742-750.
32. R. Henrion and C. Strugarek, Convexity of chance constraints with independent random variables, Comput. Optim. Appl., 41 (2008), pp. 263-276.
33. W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc., 58 (1963), pp. 13-30.
34. S. Janson, Large deviations for sums of partly dependent random variables, Random Structures Algorithms, 24 (2004), pp. 234-248.
35. S. Kataoka, A stochastic programming model, Econometrica, 31 (1963), pp. 181-196.
36. C. M. Lagoa, On the convexity of probabilistically constrained linear programs, in Proceedings of the 38th IEEE Conference on Decision and Control, 1999, pp. 516-521.
37. C. M. Lagoa, X. Li, and M. Sznaier, Probabilistically constrained linear programs and risk- adjusted controller design, SIAM J. Optim., 15 (2005), pp. 938-951.
38. Q. Li, W.-K. Ma, and A. M.-C. So, Safe convex approximation to outage-based MISO secrecy rate optimization under imperfect CSI and with artificial noise, in Proceedings of the 45th Annual Asilomar Conference on Signals, Systems, and Computers, 2011, pp. 207-211.
39. J. Luedtke and S. Ahmed, A sample approximation approach for optimization with probabilistic constraints, SIAM J. Optim., 19 (2008), pp. 674-699.
40. C. Lund and M. Yannakakis, On the hardness of approximating minimization problems, J. ACM, 41 (1994), pp. 960-981.
41. B. L. Miller and H. M. Wagner, Chance constrained programming with joint constraints, Oper. Res., 13 (1965), pp. 930-945.
42. K. Natarajan, D. Pachamanova, and M. Sim, Constructing risk measures from uncertainty sets, Oper. Res., 57 (2009), pp. 1129-1141.
43. A. Nemirovski, On tractable approximations of randomly perturbed convex constraints, in Proceedings of the 42nd IEEE Conference on Decision and Control, 2003, pp. 2419-2422.
44. A. Nemirovski, Sums of random symmetric matrices and quadratic optimization under orthogonality constraints, Math. Program. Ser. B, 109 (2007), pp. 283-317.
45. A. Nemirovski and A. Shapiro, Convex approximations of chance constrained programs, SIAM J. Optim., 17 (2006), pp. 969-996.
46. R. I. Oliveira, Sums of random Hermitian matrices and an inequality by Rudelson, Electron. Comm. Probab., 15 (2010), pp. 203-212.
47. D. Petz, A survey of certain trace inequalities, in Functional Analysis and Operator Theory, J. Zem'anek, ed., Banach Center Publ. 30, Polish Academy of Sciences, Warsaw, Poland, 1994, pp. 287-298.
48. J. Pint'er, Deterministic approximations of probability inequalities, Math.Methods Oper.Res., 33 (1989), pp. 219-239.
49. A. Pr'ekopa, Logarithmic concave measures with application to stochastic programming, Acta Sci. Math., 32 (1971), pp. 301-316.
50. A. Pr'ekopa, Contributions to the theory of stochastic programming, Math. Program., 4 (1973), pp. 202-221.
51. A. Pr'ekopa, Programming under probabilistic constraints with a random technology matrix, Math. Oper. Statist. Ser. Optim., 5 (1974), pp. 109-116.
52. A. Pr'ekopa, Stochastic Programming, Math. Appl., Kluwer Academics, Dordrecht, Netherlands, 1995.
53. A. Pr'ekopa, K. Yoda, and M. M. Subasi, Uniform quasi-convexity in probabilistic constrained stochastic programming, Oper. Res. Lett., 39 (2011), pp. 188-192.
54. C. W. Scherer and C. W. J. Hol, Matrix sum-of-squares relaxations for robust semi-definite programs, Math. Program. Ser. B, 107 (2006), pp. 189-211.
55. A. Shapiro, D. Dentcheva, and A. Ruszczy'nski, Lectures on Stochastic Programming: Modeling and Theory, MPS/SIAM Ser. Optim., SIAM, Philadelphia, 2009.
56. A. M.-C. So, Moment inequalities for sums of random matrices and their applications in optimization, Math. Program. Ser. A, 130 (2011), pp. 125-151.
57. R. Tempo, G. Calafiore, and F. Dabbene, Randomized Algorithms for Analysis and Control of Uncertain Systems, Comm. Control Engrg. Ser., Springer-Verlag, London, 2005.
58. C. J. Thompson, Inequality with applications in statistical mechanics, J. Math. Phys., 6 (1965), pp. 1812-1813.
59. J. A. Tropp, User-friendly tail bounds for sums of random matrices, Found. Comput. Math., 12 (2012), pp. 389-434.
60. C. van de Panne and W. Popp, Minimum-cost cattle feed under probabilistic protein constraints, Management Sci., 9 (1963), pp. 405-430.
61. K.-Y. Wang, T.-H. Chang, W.-K. Ma, A. M.-C. So, and C.-Y. Chi, Probabilistic SINR constrained robust transmit beamforming: A Bernstein-type inequality based conservative approach, in Proceedings of the 2011 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2011), 2011, pp. 3080-3083.
62. K.-Y. Wang, A. M.-C. So, T.-H. Chang, W.-K. Ma, and C.-Y. Chi, Outage Constrained Robust Transmit Optimization for Multiuser MISO Downlinks: Tractable Approximations by Conic Optimization, http://www.se.cuhk.edu.hk/?manchoso/ papers/rbf rar.pdf (2011).
63. R. W. Yeung, Information Theory and Network Coding, Springer Science+Business Media, New York, 2008.
64. S. Zymler, D. Kuhn, and B. Rustem, Worst-Case Value-at-Risk of Non-linear Portfolios, Management Sci., to appear.
65. S. Zymler, D. Kuhn, and B. Rustem, Distributionally robust joint chance constraints with second-order moment information, Math. Program., (2011).