Robust solutions to uncertain semidefinite programs

SIAM Journal on Optimization

Volumn 9, Issue 1, 1998, Pages 33-52

  El Ghaoui, Laurent ^a   Oustry, Francois ^a   Lebret, Hervé ^a  

^a ENSTA PARISTECH   (France)

Author keywords

Convex optimization; Regularization; Robustness; Semidefinite programming; Uncertainty

Indexed keywords

EID: 0032252302     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623496305717     Document Type: Article

References (37)

1. F. ALIZADEH, Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM J. Optim., 5 (1995), pp. 13-51.
2. F. ALIZADEH, J.-P. A. HAEBERLY, AND M. L. OVERTON, Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results, SIAM J. Optim., 8 (1998), pp. 746-768.
3. H. ATTOUCH, Viscosity Solutions of Optimization Problems, Tech. Report 07, Dépt. des Sciences Mathématiques, Université Montpellier 2, France, 1994.
4. A. BEN-TAL, L. E. GHAOUI, AND A. NEMIROVSKI, Robust semidefinite programming, in Semidefinite Programming and Applications, to appear.
5. A. BEN-TAL AND A. NEMIROVSKI, Robust Convex Programming, Tech. Report 1/95, Optimization Laboratory, Faculty of Industrial Engineering and Management, Technion, Israel Institute of Technology, Technion City, Haifa 32000, Israel, 1995; Math. Oper. Res., to appear.
6. A. BEN-TAL AND A. NEMIROVSKI, Robust Solutions to Uncertain Linear Programs, Tech. Report 6/95, Optimization Laboratory, Faculty of Industrial Engineering and Management, Technion, Israel Institute of Technology, Technion City, Haifa 32000, Israel, 1995; Oper. Res. Lett., to appear.
7. A. BEN-TAL AND A. NEMIROVSKI, Robust truss topology design via semidefinite programming, SIAM J. Optim., 7 (1997), pp. 991-1016.
8. A. BEN-TAL AND A. NEMIROVSKI, Robust convex programming, IMA J. Numer. Anal., 1998, to appear.
9. B. BEREANU, Some Numerical Methods in Stochastic Programming Under Risk and Uncertainty, Academic Press, New York, 1980, Ch. 11, pp. 169-205.
10. J. F. BONNANS, R. COMINETTI, AND A. SHAPIRO, Sensitivity Analysis of Optimization Problems under Second Order Regular Constraints, Tech. Report 2989, INRIA, 1996; Math. Oper. Res., to appear.
11. S. BOYD AND L. EL GHAOUI, Method of centers for minimizing generalized eigenvalues, Linear Algebra and Appl., special issue on Numerical Linear Algebra Methods in Control, Signals and Systems, 188 (1993), pp. 63-111.
12. S. BOYD, L. EL GHAOUI, E. FERON, AND V. BALAKRISHNAN, Linear Matrix Inequalities in System and Control Theory, Studies in Applied Mathematics series, SIAM, Philadelphia, PA, 1994.
13. S. P. BOYD AND L. VANDENBERGHE, Introduction to convex optimization with engineering applications, lecture notes for ee392x, Stanford University, Stanford, CA, 1996. Available via anonymous ftp at isl.stanford.edu/pub/boyd.
14. S. CHANDRASEKARAN, G. H. GOLUB, M. GU, AND A. H. SAYED, An efficient algorithm for a bounded errors-in-variables model, SIAM J. Matrix Anal. Appl., to appear.
15. M. DEMPSTER, Stochastic Programming, Academic Press, New York, 1980.
16. 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.
17. M. K. H. FAN, A. L. TITS, AND J. C. DOYLE, Robustness in the presence of mixed parametric uncertainty and unmodeled dynamics, IEEE Trans. Automat. Control, 36 (1991), pp. 25-38.
18. J. W. HELTON AND H. J. WOERDEMAN, Symmetric Hankel operators: Minimal norm extensions and eigenstructures, Linear Algebra Appl., 185 (1993), pp. 1-19.
19. M. KOJIMA, M. SHIDA, AND S. SHINDOH, Global and Local Convergence of Predictor-Corrector-Interior-Point Algorithm for Semidefinite Programming, Tech. Report B-308, Dept. of Information Sciences, Tokyo Institute of Technology, 2-12-1 Oh-Okayama, Meguru-ku, Tokyo 152, Japan, 1995.
20. H. LEBRET, Synthèse de diagrammes de réseaux d'antennes par optimisation convexe, Ph.D. thesis, Université de Rennes I, Nov. 1994.
21. C. LEMARÉCHAL, F. OUSTRY, AND C. SAGASTIZÁBAL, The U-Lagrangian of a convex function, Trans. Amer. Math. Soc., to appear.
22. A. S. LEWIS AND M. L. OVERTON, Eigenvalue optimization, Acta Numerica, 5 (1996), pp. 149-190.
23. M. S. LOBO, L. VANDENBERGHE, S. BOYD, AND H. LEBRET, Second-order cone programming: Interior-point methods and engineering applications, Linear Algebra Appl., submitted.
24. Y. NESTEROV AND A. NEMIROVSKII, Interior Point Polynomial Methods in Convex Programming: Theory and Applications, Studies in Applied Mathematics series, SIAM, Philadelphia, PA, 1994.
25. F. OUSTRY, The U-Lagrangian of the maximum eigenvalue function, SIAM J. Optim., to appear.
26. F. A. POTRA AND R. SHENG, On homogeneous interior-point algorithms for semidefinite programming, Optim. Methods Softw., 9 (1998), pp. 161-184.
27. S. ROBINSON, Stability theorems for systems of inequalities, part II: Differentiable nonlinear systems, SIAM J. Numer. Anal., 13 (1976), pp. 497-513.
28. R. T. ROCKAFELLAR, Convex Analysis, 2nd ed., Princeton Univ. Press, Princeton, NJ, 1970.
29. R. RUBINSTEÌN AND A. SHAPIRO, Discrete Event Systems, John Wiley, New York, 1993.
30. A. SHAPIRO, Perturbation theory of nonlinear programs when the set of optimal solutions is not a singleton, Appl. Math. Optim., 18 (1988), pp. 215-229.
31. A. SHAPIRO, Perturbation analysis of optimization problems in Banach spaces, Numer. Funct. Anal. Optim., 13 (1992), pp. 97-116.
32. A. SHAPIRO, First and second order analysis of nonlinear semidefinite programs. Semidefinite programming, Math. Programming Ser. B, 77 (1997), pp. 301-320.
33. A. SHAPIRO AND M. K. H. FAN, On eigenvalue optimization, SIAM J. Optim., 5 (1995), pp. 552-568.
34. A. TIKHONOV AND V. ARSENIN, Solutions of Ill-Posed Problems, John Wiley, New York, 1977.
35. L. VANDENBERGHE AND S. BOYD, Semidefinite programming, SIAM Rev., 38 (1996), pp. 49-95.
36. V. A. YAKUBOVICH, The solution of certain matrix inequalities in automatic control theory, Soviet Math. Dokl., 3 (1962), pp. 620-623.
37. K. ZHOU, J. DOYLE, AND K. GLOVER, Robust and Optimal Control, Prentice-Hall, Upper Saddle River, NJ, 1995.