A second-order bundle method to minimize the maximum eigenvalue function

Mathematical Programming, Series B

Volumn 89, Issue 1, 2000, Pages 1-33

  Oustry, François ^a  

^a INRIA   (France)

Author keywords

Convex optimization; Eigenvalue optimization; Second order bundle methods; Semidefinite programming

Indexed keywords

EID: 0002928322     PISSN: 00255610     EISSN: None     Source Type: Journal    
DOI: 10.1007/PL00011388     Document Type: Article

References (44)

1. Alizadeh, F., Haeberly, J.-P.A., Overton, M.L. (1997): Complementarity and nondegeneracy in semidefinite programming. Math. Program. 77, 111-128
2. Alizadeh, F., Haeberly, J.-P.A., Overton, M.L. (1998): Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results. SIAM J. Optim. 8, 746-768
3. Arnold, V.I. (1971): On matrices depending on parameters. Russ. Math. Surveys 26, 29-43
4. Bellman, R., Fan, K. (1963): On systems of linear inequalities in Hermitian matrix variables. In: Klee, V.L., ed., Convexity, volume 7 of Proceedings of Symposia in Pure Mathematics, pp. 1-11. American Mathematical Society
5. Boyd, S., El Ghaoui, L., Feron, E., Balakrishnan, V. (1994): Linear Matrix Inequalities in System and Control Theory, volume 15 of Studies in Applied Mathematics. SIAM, Philadelphia, PA, June 1994
6. Cullum, J., Donath, W.E., Wolfe, P. (1975): The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices. Math. Program. Study 3, 35-55
7. Delorme, C., Poljak, S. (1993): Laplacian eigenvalues and the maximum cut problem. Math. Program. 62, 557-574
8. Demjanov, V.F., Malozenov, V.N. (1974): Introduction to Minimax. Wiley & Sons
9. Edelman, A., Arias, T., Smith, S.T. (1998): The geometry of algorithms with orthogonality constraints. SIAM J. Matrix. Anal. Appl. 20, 303-353
10. Fletcher, R. (1985): Semi-definite matrix constraints in optimization. SIAM J. Control Optim. 23, 493-523
11. Forsgren, A. (2000): Optimality conditions for nonconvex semidefinite programming. Math. Program. 88, 105-128
12. Gilbert, E.G. (1966): An iterative procedure for computing the minimum of a quadratic form on a convex set. SIAM J. Control 4, 61-80
13. Helmberg, C., Oustry, F. (2000): Bundle methods to minimize the maximum eigenvalue function. In: Lieven, Vandenberghe, Saigal, R., Wolkovicz, H., eds., Handbook on Semidefinite Programming. Theory, Algorithms and Applications, volume 27 of International Series in Operations Research and Management Science. Kluwer Academic Publisher, March 2000
14. Helmberg, C., Rendl, F. (2000): A spectral bundle method for semidefinite programming. SIAM J. Optim. 10, 673-696
15. Hiriart-Urruty, J.-B., Ye, D. (1995): Sensivity analysis of all eigenvalues of a symmetric matrix. Numer. Math. 70, 45-72
16. Hiriart-Urruty, J.B., Lemaréchal, C. (1993): Convex Analysis and Minimization Algorithms. Springer-Verlag. Two volumes
17. Horn, R.A., Johnson, C.R. (1991): Topics in Matrix Analysis. Cambridge University Press
18. Kato, T. (1980): Perturbation Theory for Linear Operators. Springer-Verlag, New York
19. Kiwiel, K.C. (1986): A linearization algorithm for optimizing control systems subject to singular value inequalities. IEEE Trans. Autom. Control AC-31, 595-602
20. Kiwiel, K.C. (1990): Proximity control in bundle methods for convex nondifferentiable minimization. Math. Program. 46, 105-122
21. Kojima, M., Shida, M., Shindoh, S. (1998): Local convergence of predictor-corrector infeasible-interior-point algorithms for sdps and sdlcps. Math. Program. 80, 129-161
22. Lancaster, P. (1964): On eigenvalues of matrices dependent on a parameter. Numer. Math. 6, 377-387
23. Lemaréchal, C., Oustry, F. (1999): Nonsmooth algorithms to solve semidefinite programs. In: El Ghaoui, L., Niculescu, S.-I., eds., Advances in LMI methods in Control, Advances in Design and Control series. SIAM
24. Lemaréchal, C., Oustry, F., Sagastizábal, C. (2000): The U-Lagrangian of a convex function. Transact. Amer. Math. Soc. 352
25. Lewis, A.S., Overton, M.L. (1996): Eigenvalue optimization. Acta Numer. 5, 149-190
26. Nemirovsky, A., Gahinet, P. (1997): The projective method for solving linear matrix inequalities. Math. Program. 77, 163-190
27. Nesterov, Yu. (1997): Interior-point methods: An old and new approach to nonlinear programming. Math. Program. 79, 285-297, October 1997
28. Nesterov, Yu. (1997): Quality of semidefinite relaxation for nonconvex quadratic optimization. CORE Discussion, Paper # 9719
29. Nesterov, Yu. (1998): Private communication. Center of Operations Research and Econometrics, Université Catholique de Louvain, Belgium, May 1998
30. Nesterov, Yu., Nemirovsky, A. (1988): A general approach to polynomial-time algorithms design for convex programming. Technical report, Centr. Econ. & Math. Inst., USSR Academy of Sciences, Moscow, USSR, 1988
31. Nesterov, Yu., Nemirovsky, A. (1994): Interior-point polynomial methods in convex programming: Theory and applications, volume 13 of Studies in Applied Mathematics. SIAM, Philadelphia, PA
32. Oustry, F. (1998): Vertical developments of a convex function. J. Convex Anal. 5, 153-170
33. Oustry, F. (1999): The U-Lagrangian of the maximum eigenvalue function. SIAM J. Optim. 9, 526-549
34. Overton, M.L. (1992): Large-scale optimization of eigenvalues. SIAM J. Optim. 2, 88-120
35. Overton, M.L., Womersley, R.S. (1995): Second derivatives for optimizing eigenvalues of symmetric matrices. SIAM J. Matrix Anal. Appl. 16, 667-718, July 1995
36. Overton, M.L., Ye., X. (1994): Toward second-order methods for structured nonsmooth optimization. In: Gomez, S., Hennart, J.-P., eds., Advances in Optimization and Numerical Analysis, pp. 97-109. Kluwer Academic Publishers
37. Polak, E., Wardi, Y. (1982): Nondifferentiable optimization algorithm for designing control systems having singular value inequalities. Automatica 18, 267-283
38. Potra, F.A., Sheng, R. (1998): A superlinearly convergent primal-dual infeasible-interior-point algorithm for semidefinite programming. SIAM J. Optim. 8, 1007-1028
39. Schramm, H., Zowe, J. (1992): A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results. SIAM J. Optim. 2, 121-152
40. Schwartz, L. (1967): Cours d'analyse, Vol. 1. Hermann, Paris
41. Shapiro, A., Fan, M.K.H. (1995): On eigenvalue optimization. SIAM J. Optim. 5, 552-568
42. Trefethen, L.N. (1997): Pseudospectra of linear operators. SIAM Rev. 39, 383-406
43. Trefethen, L.N., Bau, III D. (1997): Numerical Linear Algebra. SIAM, April 1997
44. Ye, D.Y. (1993): Sensitivity analysis of the greatest eigenvalue of a symmetric matrix via the ε-subdifferential of the associated convex quadratic form. J. Optim. Theory Appl. 76, February 1993