The rate of convergence of the augmented Lagrangian method for nonlinear semidefinite programming

Mathematical Programming

Volumn 114, Issue 2, 2008, Pages 349-391

  Sun, Defeng ^a   Sun, Jie ^a   Zhang, Liwei ^b  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^b DALIAN UNIVERSITY OF TECHNOLOGY   (China)

Author keywords

Nonlinear semidefinite programming; Rate of convergence; The augmented Lagrangian method; Variational analysis

Indexed keywords

CONSTRAINT THEORY; NONLINEAR PROGRAMMING; OPTIMIZATION; THEOREM PROVING;

AUGMENTED LAGRANGIAN METHOD; NONLINEAR SEMIDEFINITE PROGRAMMING; RATE OF CONVERGENCE;

CONVERGENCE OF NUMERICAL METHODS;

EID: 41149143510     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-007-0105-9     Document Type: Article

References (50)

1. Arnold V.I. (1971). On matrices depending on parameters. Russ. Math. Surv. 26: 29-43
2. Arrow K.J. and Solow R.M. (1958). Gradient methods for constrained maxima with weakened assumptions. In: Arrow, K.J., Hurwicz, L. and Uzawa, H. (eds) Studies in Linear and Nonlinear Programming., pp 165-176. Stanford University Press, Stanford
3. Bertsekas D.P. (1976). On penalty and multiplier methods for constrained minimization. SIAM J. Cont. Optim. 14: 216-235
4. Bertsekas D.P. (1982). Constrained Optimization and Lagrange Multiplier Methods. Academic, New York
5. Bonnans J.F., Cominetti R. and Shapiro A. (1998). Sensitivity analysis of optimization problems under second order regularity constraints. Math. Oper. Res. 23: 803-832
6. Bonnans J.F., Cominetti R. and Shapiro A. (1999). Second order optimality conditions based on parabolic second order tangent sets. SIAM J. Optim. 9: 466-493
7. Bonnans J.F., Ramírez C.H.(2005). Perturbation analysis of second order cone programming problems. Math. Programm. Ser. B 104: 205-227
8. Bonnans J.F. and Shapiro A. (2000). Perturbation Analysis of Optimization Problems. Springer, New York
9. Chen X., Qi H.-D. and Tseng P. (2003). Analysis of nonsmooth symmetric-matrix functions with applications to semidefinite complementarity problems. SIAM J. Optim. 13: 960-985
10. Clarke F.H. (1983). Optimization and Nonsmooth Analysis. Wiley, New York
11. Conn A.R., Gould N.I.M. and Toint Ph. L. (1991). A globally convergent augmented Lagrangian algorithm for optimization with general constraints and simple bounds. SIAM J. Numer. Anal. 28: 545-572
12. Contesse-Becker L. (1993). Extended convergence results for the method of multipliers for non-strictly binding inequality constraints. J. Optim. Theory Appl. 79: 273-310
13. Debreu G. (1952). Definite and semidefinite quadratic forms. Econometrica 20: 295-300
14. Diehl M., Jarre F. and Vogelbusch C. (2006). Loss of superlinear convergence for an SQP-type method with conic constraints. SIAM J. Optim. 16: 1201-1210
15. Eaves B.C. (1971). On the basic theorem of complementarity. Math. Programm. 1: 68-75
16. Faraut J. and Korányi A. (1994). Analysis on Symmetric Cones. Clarendon Press, London
17. Golshtein E.G. and Tretyakov N.V. (1989). Modified Lagrangians and Monotone Maps in Optimization. Wiley, New York
18. Golub G. and Van Loan C.F. (1996). Matrix Computations. The Johns Hopkins University Press, Baltimore
19. Hestenes M.R. (1969). Multiplier and gradient methods. J. Optim. Theory Appl. 4: 303-320
20. Higham N.J. (1998). Computing a nearest symmetric positive semidefinite matrix. Linear Algebra Appl. 103: 103-118
21. Ito K. and Kunisch K. (1990). The augmented Lagrangian method for equality and inequality constraints in Hilbert spaces. Math. Program. 46: 341-360
22. Kummer B. (1988). Newton's method for non-differentiable functions. In: Guddat, J. (eds) Advances in Mathematical Optimization, pp 114-125. Akademie-Verlag, Berlin
23. 1,1-optimization. J. Optim. Theory Appl. 70: 559-580
24. e ib 1.1: COnstraint Matrix-optimization Problem Library-a collection of test examples for nonlinear semidefinite programs, control system design and related problems. Technical Report, Department of Mathematics, University of Trier, Germany (2005)
25. Löwner K. (1934). Über monotone matrixfunktionen. Mathematische Zeitschrift 38: 177-216
26. Kocvara M. and Stingl M. (2004). Solving nonconvex SDP problems of structural optimization with stability control. Optim. Methods Softw. 19: 595-609
27. Meng F., Sun D. and Zhao G. (2005). Semismoothness of solutions to generalized equations and the Moreau-Yosida regularization. Math. Program. Ser. B 104: 561-581
28. Mifflin R. (1977). Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15: 957-972
29. Pang J.-S., Sun D. and Sun J. (2003). Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems. Math. Oper. Res. 28: 39-63
30. Pennanen T. (2002). Local convergence of the proximal point algorithm and multiplier methods without monotonicity. Math. Oper. Res. 27: 170-191
31. Powell M.J.D. (1972). A method for nonlinear constraints in minimization problems. In: Fletcher, R. (eds) Optimization, pp 283-298. Academic, New York
32. Qi L. and Sun J. (1993). A nonsmooth version of Newton's method. Math. Program. 58: 353-367
33. Robinson S.M. (1984). Local structure of feasible sets in nonlinear programming, part II: nondegeneracy. Math. Program. Study 22: 217-230
34. Rockafellar R.T. (1973). A dual approach to solving nonlinear programming problems by unconstrained optimization. Math. Program. 5: 354-373
35. Rockafellar R.T. (1973). The multiplier method of Hestenes and Powell applied to convex programming. J. Optim. Theory Appl. 12: 555-562
36. Rockafellar R.T. (1976). Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14: 877-898
37. Rockafellar R.T. (1976). Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1: 97-116
38. Rockafellar R.T. (1993). Lagrange multipliers and optimality. SIAM Rev. 35: 183-238
39. Rockafellar R.T. and Wets R.J.-B (1998). Variational Analysis. Springer, New York
40. Shapiro A. (1990). On concepts of directional differentiability. J. Optim. Theory Appl. 66: 477-487
41. Shapiro A. (2003). Sensitivity analysis of generalized equations. J. Math. Sci. 115: 2554-2565
42. Shapiro A. and Sun J. (2001). Some properties of the augmented Lagrangian in cone constrained optimization. Math. Oper. Res. 29: 479-491
43. Sun D. (2001). A further result on an implicit function theorem for locally Lipschitz functions. Oper. Res. Lett. 28: 193-198
44. Sun D. (2006). The strong second order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications. Math. Oper. Res. 31: 761-776
45. Sun D. and Sun J. (2002). Semismooth matrix valued functions. Math. Oper. Res. 27: 150-169
46. Sun, D., Sun, J.: Löwner's operator and spectral functions in Euclidean Jordan algebras. Department of Mathematics, National University of Singapore (2004)
47. Tretyakov N.V. (1973). A method of penalty estimates for convex programming problems. Èkonomika i Matematicheskie Metody 9: 525-540
48. Tseng P. (1998). Merit functions for semi-definite complementarity problems. Math. Program. 83: 159-185
49. Tsing N.-K., Fan M.K.H. and Verriest E.I. (1994). On analyticity of functions involving eigenvalues. Linear Algebra Appl. 207: 159-180
50. Zarantonello E.H. (1971). Projections on convex sets in Hilbert space and spectral theory I and II. In: Zarantonello, E.H. (eds) Contributions to Nonlinear Functional Analysis, pp 237-424. Academic, New York