A randomized cutting plane scheme with geometric convergence: Probabilistic analysis and SDP applications

Proceedings of the IEEE Conference on Decision and Control

Volumn , Issue , 2008, Pages 3044-3049

  Dabbene, F ^a   Shcherbakov, P ^b   Polyak, B T ^b  

^a POLITECNICO DI TORINO   (Italy)

^b IPU   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

CONVEX OPTIMIZATION; MARKOV CHAINS;

CONVEX OPTIMIZATION PROBLEMS; INTERIOR POINT ALGORITHM; MARKOV CHAIN MONTE CARLO ALGORITHMS; OPTIMIZATION PROBLEMS; PROBABILISTIC ANALYSIS; PROBABILISTIC ESTIMATES; RANDOMIZED APPROACH; RATE OF CONVERGENCE;

MONTE CARLO METHODS;

EID: 62949214229     PISSN: 07431546     EISSN: 25762370     Source Type: Conference Proceeding    
DOI: 10.1109/CDC.2008.4739188     Document Type: Conference Paper

References (20)

1. D. Bertsimas and S. Vempala. Solving convex programs by random walks. Journal of the ACM, 51(4):540-556, 2004.
2. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, 1994.
3. G. Calafiore. Random walks for probabilistic robustness. In Conf. Decison and Control, pages 5316-5321, 2004.
4. G. Calafiore and F. Dabbene. A probabilistic analytic center cutting plane method for feasibility of uncertain LMIs. Automatica, 43(12):2022-2033, 2006.
5. F. Dabbene. A randomized cutting plane scheme for convex optimization. In Multiconf. on Systems and Control. San Antonio, 2008.
6. B. Grünbaum. Partitions of mass-distributions and convex bodies by hyperplanes. Pacific J. Math., 10:1257-1261, 1960.
7. A.Y. Levin. On an algorithm for the minimization of convex functions over convex functions. Soviet Math. Doklady, 6:286-290, 1965.
8. L. Lovász. Hit-and-Run mixes fast. Mathematical Programming, 86:443-461, 1999.
9. L. Lovász and S. Vempala. Hit-and-Run from a corner. SIAM Journal on Computing, 35(4):985-1005, 2006.
10. Y. Nesterov and A.S. Nemirovski. Interior Point Polynomial Algorithms in Convex Programming. SIAM Journal on Applied Mathematics, Philadelphia, 1994.
11. D.J. Newman. Location of the maximum on unimodal surfaces. Journal of the ACM, 12(3):395-398, 1965.
12. B.T. Polyak and P.S. Shcherbakov. The D-decomposition technique for linear matrix inequalities. Automat. Remote Control, 67(11):1847-1861, 2006.
13. B.T. Polyak and P.S. Shcherbakov. A randomized method for solving semidefinite programs. In Proc. of the 9th IFAC Workshop ALCOSP'07, 2007. Available from IPACS Electronic Library at http://lib.physcon.ru/.
14. L. Rademacher. Approximating the centroid is hard. In Proc. of the 24th Annual Symposium on Computationa Geometry (SoCG-07), 2007.
15. J. Radon. Über eine Erweiterung des Begriffs der konvexen Functionen, mit einer Anwendung auf die Theorei der konvexen Körper. S. B. Akad. Wiss. Wien, 125:241-258, 1916.
16. M. Rudelson. Random vectors in the isotropic position. J. Funct. Anal., 164:60-72, 1999.
17. R.L. Smith. Efficient Monte-Carlo procedures for generating points uniformly distributed over bounded regions. Operations Research, 32:1296-1308, 1984.
18. R. Tempo, G. Calafiore, and F. Dabbene. Randomized Algorithms for Analysis and Control of Uncertain Systems. Communications and Control Engineering Series. Springer-Verlag, London, 2004.
19. V.F. Turchin. On the computation of multidimensional integrals by the Monte-Carlo method. Theory of Probability and Appl., 16(4):720-724, 1971.
20. S. Vempala. Geometric random walks: a survey. Combinatorial and Computational Geometry, 52:573-611, 2005.