Spectral bundle methods for non-convex maximum eigenvalue functions: First-order methods

Mathematical Programming

Volumn 104, Issue 2-3, 2005, Pages 701-727

  Noll, Dominikus ^a   Apkarian, Pierre ^b  

^a INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

^b ONERA   (France)

Author keywords

subgradient; Bilinear matrix inequality (BMI); Eigenvalue optimization; First order spectral bundle method; Linear matrix inequality (LMI)

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; FUNCTIONS; MATHEMATICAL PROGRAMMING; MATRIX ALGEBRA; OPTIMIZATION; PROBLEM SOLVING;

∈-SUBGRADIENT; BILINEAR MATRIX INEQUALITY (BMI); EIGENVALUE OPTIMIZATION; FIRST-ORDER SPECTRAL BUNDLE METHOD; LINEAR MATRIX INEQUALITY (LMI);

EIGENVALUES AND EIGENFUNCTIONS;

EID: 27244452234     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-005-0634-z     Document Type: Article

References (73)

1. Apkarian, P., Noll, D., Tuan, H.D.: A prototype primal-dual LMI interior point algorithm for non-convex robust control problems. Rapport interne 01-08 (2002), UPS-MIP, CNRS UMR 5640. http://mip.upstlse.fr/publi/publi.html
2. ∞, control design via an augmented Lagrangian method. Int. J. Robust and Nonlin. Control 13 (12), 1137-1148 (2003)
3. ∞ synthesis. European J. Control, 10, 527-538 (2004)
4. Apkarian, P., Noll, D.: Controller design vianonsmooth multi-directional search. SIAM J. on Control and Optimization, to appear (2005)
5. ∞-synthesis. Submitted
6. Apkarian, P., Tuan, H.D.: Concave programming in control theory. J. Global Optim. 15, 343-370 (1999)
7. Apkarian, P., Tuan, H.D.: Robust control via concave minimization - local and global algorithms. IEEE Trans. Autom. Control 45, 299-305 (2000)
8. Beran, E.B., Vandenberghe, L., Boyd, S.: A global BMI algorithm based on the generalized Benders decomposition. Proc. European Control Conf., Belgium, 1997
9. Boyd, S., ElGhaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory, vol. 15 of Studies in Appl. Math. SIAM, Philadelphia, 1994
10. Boyd, S., Vandenberghe, L.: Semidefinite programming relaxations of non-convex problems in control and combinatorial optimization
11. Clarke, F.H.: Optimization and Nonsmooth Analysis. John Wiley, New York, 1983
12. Cullum, J., Donath, W.E., Wolfe, P.: The minimization of certain nondifferential sums of eigenvalues of symmetric matrices. Math. Progr. Stud. 3, 35-55 (1975)
13. ElGhaoui, L., Balakrishnan, V.: Synthesis of fixed-structure controllers via numerical optimization. Proc. IEEE Conf. Dec. Contr., December 1994
14. Fares, B., Noll, D., Apkarian, P.: An augmented Lagrangian method for a class of LMI-constrained problems in robust control theory. Intern. J. Control 74, 348-360 (2001)
15. Fares, B., Noll, D., Apkarian, P.: Robust control via successive semidefinite programming. SIAM J. Control Optim. 40, 1791-1820 (2002)
16. Fletcher, R.: Semidefinite constraints in optimization. SIAM J. Control Optim. 23, 493-513 (1985)
17. Fletcher, R., Leyffer, S.: A bundle filter method for nonsmooth nonlinear optimization. University of Dundee, Report NA/195, 1999
18. Fuduli, A., Gaudioso, M., Giallombardo, G.: Minimizing nonconvex nonsmooth functions via cutting planes and proximity control. SIAM J. Optim. 14 (3), 743-756 (2004)
19. Fuduli, A., Gaudioso, M., Giallombardo, G.: A DC piecewise affine model and a bundling technique in nonconvex nonsmooth minimization. Optimization Methods and Software 19 (1), 89-102 (2004)
20. Fukuda, M., Kojima, M.: Branch-and-cut algorithms for the bilinear matrix inequality eigenvalue problem. Research Report on Mathematical and Computational Sciences. Series B: Operations Research, 1999
21. Grigoriadis, K.M., Skelton, R.E.: Low-order control design for LMI problems using alternating projection methods. Automatica 32, 1117-1125 (1996)
22. Goh, K.-C., Safonov, M.G., Papavassilopoulos, O.P.: Global optimization for the biaffine matrix inequality problem. J. Global Optim. 7, 365-380 (1995)
23. Helmberg, C.: A cutting plane algorithm for large scale semidefinite relaxations. ZIB-Report ZR-01-26. Padberg Festschrift. The sharpest cut. SIAM, 2001
24. Helmberg, C., Kiwiel, K.C.: A spectral bundle method with bounds. Math. Programming 93 (2), 173-194 (2002)
25. Helmberg, C., Oustry, F.: Bundle methods to minimize the maximum eigenvalue function. In: L. Vanden-berghe, R. Saigal, H. Wolkowitz, (eds.), Handbook of Semidefinite Programming. Theory, Algorithms and Applications, vol. 27, Int. Series in Oper. Res. and Management Sci. Kluwer Acad. Publ., 2000
26. Helmberg, C., Rendl, F.: A spectral bundle method for semidefinite programming. SIAM J. Optim. 10 (3), 673-696 (2000)
27. Helmberg, C., Rendl, F.: Solving quadratic (0, 1)-problems by semidefinite programs and cutting planes. Math. Programming 82, 291-315 (1998)
28. Helton, J.W., Merino, O.: Coordinate optimization for bi-convex matrix inequalities. Proc. IEEE Conf. Decision and Control, San Diego, 1997, pp. 3609-3613
29. Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms, part II. Springer Verlag, Berlin, 1993
30. Hoi, C.W.J., Scherer, C.W., van der Meché, E.G., Bosgra, O.H.: A nonlinear SDP approach to fixed-order controller synthesis and comparison with two other methods applied to an active suspension system, submitted, 2002
31. Iwasaki, T.: The dual iteration for fixed order control. Proc. Am. Control Conf., 1997, pp. 62-66
32. Kiwiel, K.C.: Methods of descent for nondifferentiable optimization. Lect. Notes in Math. vol. 1133, Springer Verlag, Berlin, 1985
33. Kiwiel, K.C.: A linearization algorithm for optimizing control systems subject to singular value inequalities. IEEE Trans. Autom. Control AC-31, 1986, pp. 595-602
34. Kiwiel, K.C.: A constraint linearization method for nondifferentiable convex minimization. Numerische Mathematik 51, 395-414 (1987)
35. Kiwiel, K.C.: Proximity control in bundle methods for convex nondifferentiable optimization. Math. Programming 46, 105-122 (1990)
36. Kiwiel, K.C.: Restricted-step and Levenberg-Marquardt techniques in proximal bundle methods for nonconvex nondifferentiable optimization. SIAM J. Optim. 6, 227-249 (1996)
37. Krishnan, K., Mitchell, J.E.: Cutting plane methods for semidefinite programming. Submitted, 2002
38. Lemaréchal, C.: Extensions diverses des méthodes de gradient et applications. Thése d'Etat, Paris, 1980
39. Lemaréchal, C.: Bundle methods in nonsmooth optimization. In: Nonsmooth Optimization, Proc. IIASA Workshop 1977, C. Lemaréchal, R. Mifflin (eds.), Pergamon Press, 1978
40. Lemaréchal, C.: Nondifferentiable Optimization, chapter VII In: Handbooks in Operations Research and Management Science, vol. 1, Optimization, G.L. Nemhauser, A.H.G. Rinnooy Kan, M.J. Todd (eds.), North Holland, 1989
41. Lemaréchal, C., Nemirovskii, A., Nesterov, Y.: New variants of bundle methods. Math. Programming 69, 111-147(1995)
42. Lemaréchal, C., Oustry, F.: Nonsmooth algorithms to solve semidefinite programs. In: L. El Ghaoui, L. Niculescu (eds.), Advances in LMI methods in Control, SIAM Advances in design and Control Series, 2000
43. Miller, S.A., Smith, R.S.: Solving large structured semidefinite programs using an inexact spectral bundle method. Proc. IEEE Conf. Dec. Control, 2000, pp. 5027-5032
44. Miller, S.A., Smith, R.S.: A bundle method for efficiently solving large structured linear matrix inequalities. Proc. Am. Control Conf., Chicago, 2000
45. Nemirovskii, A.: Several NP-Hard problems arising in robust stability analysis. Mathematics of Control, Signals, and Systems 6 (1), 99-105 (1994)
46. Noll, D., Apkarian, P.: Spectral bundle methods for nonconvex maximum eigenvalue functions: second-order methods. Math. Programming, Series B 104, 729-747 (2005)
47. Noll, D., Torki, M., Apkarian, P.: Partially augmented Lagrangian method for matrix inequalities. SIAM J. on Optimization 15 (1), 161-184 (2004)
48. Overton, M.L.: On minimizing the maximum eigenvalue of a symmetric matrix. SIAM J. Matrix Anal. Appl. 9 (2), 256-268 (1988)
49. Overton, M.L.: Large-scale optimization of eigenvalues. SIAM J. Optim. 2 (1), 88-120 (1992)
50. Overton, M.L., Womersley, R.S.: Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices. Math. Programming 62, 321-357 (1993)
51. Overton, M.L., Womersley, R.S.: Second derivatives for optimizing eigenvalues of symmetric matrices. SIAM J. Matrix Anal. Appl. 16 (3), 697-718 (1995)
52. Haeberly, J.-P.A., Overton, M.L.: A hybrid algorithm for optimizing eigenvalues of symmetric definite pencils. SIAM J. Matrix Anal. Appl. 15, 1141-1156 (1994)
53. Oustry, F.: Vertical development of a convex function. J. Convex Analysis 5, 153-170 (1998)
54. Oustry, F.: The U-Lagrangian of the maximum eigenvalue function. SIAM J. Optim. 9 (2), 526-549 (1999)
55. Oustry, F.: A second-order bundle method to minimize the maximum eigenvalue function. Math. Programming Series A 89 (1), 1-33 (2000)
56. Polak, E.: On the mathematical foundations of nondifferentiable optimization in engineering design. SIAM Review 29, 21-89 (1987)
57. Polak, E.: Optimization: Algorithms and consistent approximations. Springer Verlag, Applied Mathematical Sciences, 1997
58. Polak, E., Wardi, Y.: Nondifferentiable optimization algorithm for designing control systems having singular value inequalities. Automatica 18 (3), 267-283 (1982)
59. Rotunno, M., de Callafon, R.A.: A bundle method for solving the fixed order control problem. Proc. IEEE Conf. Dec. Control, Las Vegas, 2002, pp. 3156-3161
60. Scherer, C.: A full block S-procedure with applications. Proc. IEEE Conf. on decision and Control, San Diego, USA, 1997, pp. 2602-2607
61. Scherer, C.: Robust mixed control and linear parameter-varying control with full block scalings. SIAM Advances in Linear Matrix Inequality Methods in Control Series, L. El Ghaoui, S.I. Niculescu (eds.), 2000
62. Schramm, H., Zowe, J.: A version of the bundle method for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results. SIAM J. Optim. 2, 121-152 (1992)
63. Shapiro, A.: Extremal problems on the set of nonnegative matrices. Lin. Algebra Appl. 67, 7-18 (1985)
64. Shapiro, A.: First and second order analysis of nonlinear semidefinite programs. Math. Programming Ser. B 77, 301-320 (1997)
65. Shapiro, A., Fan, M.K.H.: On eigenvalue optimization. SIAM J. Optim. 5(30), 552-569 (1995)
66. Steward, G.W., Ji-guang Sun: Matrix Perturbation Theory. Computer Science and Scientific Computing, Academic Press, 1990
67. Thevenet, J.-B., Apkarian, P., Noll, D.: Description and user's guide of specSDP, a Matlab-interfaced Fortran code to solve linear and nonlinear SDP constrained optimization programs. Technical Report, UPS-MIP, 2005
68. Thevenet, J.-B., Apkarian, P., Noll, D.: Reduced-order output feedback control design with specSDP, a code for linear and nonlinear SDP problems. ICCA05, Budapest, 2005
69. Thevenet, J.-B., Noll, D., Apkarian, P.: Nonlinear spectral SDP method for BMI-constrained problems: Applications to control design. Informatics in Systems and Control, Automation and Robotics, Kluwer Academic Publishers, 2004
70. Torki, M.: Second order directional derivative of all eigenvalues of a symmetric matrix. Nonlin. Analysis, Theory, Methods and Appl. 46, 1133-1500 (2001)
71. Tuan, H.D., Apkarian, P., Nakashima, Y.: A new Lagrangian dual global optimization algorithm for solving bilinear matrix inequalities. Proc. Amer. Control Conf., 1999
72. 1 Framework. Int. J. Control 73 (2), 166-176 (2000)
73. Wolfe, Ph.: A method of conjugate subgradients for minimizing nondifferentiable convex functions. Math. Programming Studies 3, 145-173 (1975)