The u-Lagrangian of the maximum eigenvalue function

SIAM Journal on Optimization

Volumn 9, Issue 2, 1999, Pages 526-549

  Oustry, François ^a  

^a INRIA RHÔNE ALPES   (France)

Author keywords

Convex optimization; Eigenvalue optimization; Generalized derivative; Second order derivative

Indexed keywords

EID: 0033469859     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623496311776     Document Type: Article

References (45)

1. R. ABRAHAM, J. E. MARSDEN, AND T. RATIU, Manifolds, Tensor Analysis, and Applications, Applied Mathematical Sciences 75, 2nd ed., Springer-Verlag, Berlin, 1988.
2. F. ALIZADEH, Combinatorial Optimization with Interior Point Methods and Semi-Definite Matrices, Ph.D. thesis, University of Minnesota, Minneapolis, MN, 1991.
3. F. ALIZADEH, Optimization over the positive-definite cone: Interior-point methods and combinatorial applications, in Advances in Optimization and Parallel Computing, Panos Pardalos, ed., North-Holland, Amsterdam, 1992.
4. F. ALIZADEH, J.-P. A. HAEBERLY, AND M. L. OVERTON, Complementarity and nondegeneracy in semidefinite programming, Math. Programming Ser. B, 77 (1997), pp. 111-128.
5. 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.
6. V. I. ARNOLD, On matrices depending on parameters, Russian Math. Surveys, 26 (1971), pp. 29-43.
7. M. P. BENDSØE, A. BEN-TAL, AND J. ZOWE, Optimization methods truss geometry and topology design, Structural Optimization, 7 (1994), pp. 141-159.
8. J. F. BONNANS, R. COMINETTI, AND A. SHAPIRO, Sensivity Analysis of Optimization Problems Under Second Order Regular Constraints, Technical Report 2989, INRIA, Le Chesnay, France, 1996.
9. S. BOYD, L. EL GHAOUI, E. FERON, AND V. BALAKRISHNAN, Linear Matrix Inequalities in System and Control Theory, Studies in Applied Mathematics 15, SIAM, Philadelphia, PA, 1994.
10. S. J. COX AND M. L. OVERTON, The optimal design of columns against buckling, SIAM J. Math. Anal., 23 (1992), pp. 287-325.
11. J. CULLUM, W. E. DONATH, AND P. WOLFE, The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices, Math. Programming Study, 3 (1975), pp. 35-55.
12. A. EDELMAN, T. ARIAS, AND S. T. SMITH, The geometry of algorithms with orthogonality constraints, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 303-353.
13. M. K. H. FAN, A guadratically convergent local algorithm on minimizing the largest eigenvalue of a symmetric matrix, Linear Algebra Appl., (1993), pp. 231-253.
14. R. FLETCHER, Semi-definite matrix constraints in optimization, SIAM J. Control Optim., 23 (1985), pp. 493-522.
15. G. GOLUB AND C. VAN LOAN, Matrix Computations, Johns Hopkins University Press, Baltimore, MD, 1989.
16. C. HELMBERG AND F. RENDL, A spectral bundle method for semidefinite programming, SIAM J. Optim., submitted.
17. N. J. HICKS, Notes on Differential Geometry, Van Nostrand, Princeton, NJ, 1965.
18. J.-B. HIRIART-URRUTY AND D. YE, Sensivity analysis of all eigenvalues of a symmetric matrix, Numer. Math., 70 (1995), pp. 45-72.
19. J. B. HIRIART-URRUTY AND C. LEMARÉCHAL, Convex Analysis and Minimization Algorithms, Springer-Verlag, Berlin, 1993.
20. F. JARRE, An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices, SIAM J. Control Optim., 31 (1993), pp. 1360-1377.
21. T. KATO, Perturbation theory of matrices, J. Assoc. Comput. Mach., 5 (1958), p. 104.
22. T. KATO, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1980.
23. K. C. KIWIEL, A linearization algorithm for optimizing control systems subject to singular value inequalities, IEEE Trans. Automat. Control, AC-31 (1986), pp. 595-602.
24. P. LANCASTER, On eigenvalues of matrices dependent on a parameter, Numer. Math., 6 (1964), pp. 377-387.
25. C. LEMARÉCHAL, F. OUSTRY, AND C. SAGASTIZÁBAL, The U-Lagrangian of a convex function, Trans. Amer. Math. Soc. (1996), to appear.
26. A. S. LEWIS AND M. L. OVERTON, Eigenvalue optimization, Acta Numerica, 5 (1996), pp. 149-190.
27. R. MIFFLIN AND C. SAGASTIZÁBAL, VU-derivatives for convex max-functions, Math. Oper. Res. (1997), submitted.
28. A. NEMIROVSKY, On Normal Self-concordant Barriers and Long-step Interior Point Methods, Technical report, Optimization Laboratory, Faculty of Industrial Engineering and Management, Technion, Israel Institute of Technology, Technion City, Haifa, Israel, 1997.
29. A. NEMIROVSKI AND P. GAHINET, The projective method for solving linear matrix inequalities, Math. Programming Ser. B, 77 (1997), pp. 163-190.
30. A. NEMIROVSKY AND D. YUDIN, Problem Complexity and Method Efficiency in Optimization, John Wiley, New York, 1983.
31. YU. NESTEROV, Interior-point methods: An old and new approach to nonlinear programming, Math. Programming, 79 (1997), pp. 285-297.
32. YU. NESTEROV AND A. NEMIROVSKY, A General Approach to Polynomial-time Algorithms Design for Convex Programming, Technical report, Centr. Econ. & Math. Inst., USSR Academy of Sciences, Moscow, USSR, 1988.
33. YU. NESTEROV AND A. NEMIROVSKII, Interior-Point Polynomial Algorithms in Convex Programming, SIAM Stud. Appl. Math. 13, SIAM, Philadelphia, PA, 1994.
34. YU. NESTEROV AND M. J. TODD, Self-scaled cones and interior-points methods in nonlinear programming, Math. Oper. Res., 22 (1997), pp. 1-42.
35. F. OUSTRY, A second-order bundle method to minimize the maximum eigenvalue function, Math. Programming (1997), submitted.
36. M. L. OVERTON, On minimizing the maximum eigenvalue of a symmetric matrix, SIAM J. Matrix Anal. Appl., 9 (1988), pp. 256-268.
37. M. L. OVERTON, Large-scale optimization of eigenvalues, SIAM J. Optim., 2 (1992), pp. 88-120.
38. M. L. OVERTON AND R. S. WOMERSLEY, Second derivatives for optimizing eigenvalues of symmetric matrices, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 667-718.
39. M. L. OVERTON AND X. YE, Toward second-order methods for structured nonsmooth optimization, in Advances in Optimization and Numerical Analysis, S. Gomez and J.-P. Hennart, eds., Kluwer Academic Publishers, Norwell, MA, 1994, pp. 97-109.
40. E. POLAK AND Y. WARDI, Nondifferentiable optimization algorithm for designing control systems having singular value inequalities, Automatica, 18 (1982), pp. 267-283.
41. F. A. POTRA AND R. SHENG, A superlinearly convergent primal-dual infeasible-interior-point algorithm for semidefinite programming, SIAM J. Optim. 8 (1998), pp. 1007-1028.
42. H. SCHRAMM AND J. ZOWE, A version of the bundle idea for minimizing a nonsmooth function: Conceptual idea, convergence analysis, numerical results, SIAM J. Optim., 2 (1992), pp. 121-152.
43. A. SHAPIRO, First and second order analysis of nonlinear semidefimte programs, Math. Programming Ser. B, 77 (1997), pp. 301-320.
44. A. SHAPIRO AND M. K. H. FAN, On eigenvalue optimization, SIAM J. Optim., 5 (1995), pp. 552-568.
45. L. VANDENBERGHE AND S. BOYD, Primal-dual potential reduction method for problems involving matrix inequalities, Math. Programming, Series B, 69 (1995), pp. 205-236.