The strong second-order sufficient condition and constraint nondegeneracy in nonlinear semidefinite programming and their implications

Mathematics of Operations Research

Volumn 31, Issue 4, 2006, Pages 761-776

  Sun, Defeng ^a  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Constraint nondegeneracy; Nonlinear semidefinite programming; Strong regularity; Strong second order sufficient condition

Indexed keywords

CONSTRAINT THEORY; OPERATIONS RESEARCH; PROBLEM SOLVING;

CONSTRAINT NONDEGENERACY; CONSTRAINT QUALIFICATION; NONLINEAR SEMIDEFINITE PROGRAMMING; STRONG SECOND-ORDER SUFFICIENT CONDITION;

NONLINEAR PROGRAMMING;

EID: 33746089809     PISSN: 0364765X     EISSN: 15265471     Source Type: Journal    
DOI: 10.1287/moor.1060.0195     Document Type: Article

References (42)

1. Bonnans, J. F., H. Ramírez C. 2005a. Perturbation analysis of second order cone programming problems. Math. Programming, Ser. B 104 205-227.
2. Bonnans, J. F., H. Ramírez C. 2005b. Strong regularity of semidefinite programming problems. Techincal Report CMM-DIM B-05/06137, Department of Mathematical Engineering, Universida de Chile.
3. Bonnans, J. F., A. Shapiro. 1998. Optimization problems with perturbations: A guided tour. SIAM Rev. 40 202-227.
4. Bonnans, J. F., A. Shapiro. 2000. Perturbation Analysis of Optimization Problems. Springer, New York.
5. Bonnans, J. F., A. Sulem. 1995. Pseudopower expansion of solutions of generalized equations and constrained optimization problems. Math. Programming 70 123-148.
6. Bonnans, J. F., R. Cominetti, A. Shapiro. 1998. Sensitivity analysis of optimization problems under second order regularity constraints. Math. Oper. Res. 23 803-832.
7. Bonnans, J. F., R. Cominetti, A. Shapiro. 1999. Second order optimality conditions based on parabolic second order tangent sets. SIAM J. Optim. 9 466-493.
8. Clarke, F. H. 1976. On the inverse function theorem. Pacific J. Math. 64 97-102.
9. Clarke, F. H. 1983. Optimization and Nonsmooth Analysis. John Wiley and Sons, New York.
10. Dontchev, A. L., R. T. Rockafellar. 1996. Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6 1087-1105.
11. Eaves, B. C. 1971. On the basic theorem of complementarity. Math. Programming 1 68-75.
12. Facchinei, F., J. S. Pang. 2003. Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. I. Springer, New York.
13. Gauvin, J. 1977. A necessary and sufficient regularity condition to have bounded multipliers in nonconvex programming. Math. Programming 12 136-138.
14. Gowda, M. S. 2004. Inverse and implicit function theorems for H-differentiable and semismooth functions. Optim. Methods Software 19 443-461.
15. Higham, N. J. 1988. Computing a nearest symmetric positive semidefinite matrix. Linear Algebra Its Appl. 103 103-118.
16. Jongen, H. Th., J. Rückmann, K. Tammer. 1990. Implicit functions and sensitivity of stationary points. Math. Programming 49 123-138.
17. Jongen, H. Th., T. Mobert, J. Rückmann, K. Tammer. 1987. On inertia and Schur complement in optimization. Linear Algebra Its Appl. 95 97-109.
18. Klatte, D., B. Kummer. 2002. Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications. Kluwer Academic Publishers, Boston, MA.
19. Kojima, M. 1980. Strongly stable stationary solutions in nonlinear programs. S. M. Robinson, ed. Analysis and Computation of Fixed Points. Academic Press, New York, 93-138.
20. 1,1-optimization. J. Optim. Theory Appl. 70 559-580.
21. Lloyd, N. G. 1978. Degree Theory. Cambridge University Press, Cambridge, UK.
22. Mangasarian, O. L., S. Fromovitz. 1967. The Fritz-John necessary optimality conditions in the presence of equality and inequality constraints. J. Math. Anal. Appl. 7 37-47.
23. Meng, F., D. Sun, G. Zhao. 2005. Semismoothness of solutions to generalized equations and the Moreau-Yosida regularization. Math. Programming, Ser. B 104 561-581.
24. Mifflin, R. 1977. Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15 957-972.
25. Ortega, J. M., W. C. Rheinboldt. 1970. Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.
26. Pang, J. S., D. Sun, J. Sun. 2003. Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems. Math. Oper. Res. 28 39-63.
27. Qi, L., J. Sun. 1993. A nonsmooth version of Newton's method. Math. Programming 58 353-367.
28. Robinson, S. M. 1975. Stability theory for systems of inequalities, Part I: Linear systems. SIAM J. Numer. Anal. 12 754-769.
29. Robinson, S. M. 1976. First order conditions for general nonlinear optimization. SIAM J. Appl. Math. 30 597-607.
30. Robinson, S. M. 1980. Strongly regular generalized equations. Math. Oper. Res. 5 43-62.
31. Robinson, S. M. 1984. Local structure of feasible sets in nonlinear programming, Part II: Nondegeneracy. Math. Programming Stud. 22 217-230.
32. Robinson, S. M. 1987. Local structure of feasible sets in nonlinear programming, Part III: Stability and sensitivity. Math. Programming Stud. 30 45-66.
33. Robinson, S. M. 1991. An implicit-function theorem for a class of nonsmooth functions. Math. Oper. Res. 16 292-309.
34. Robinson, S. M. 2003. Constraint nondegeneracy in variational analysis. Math. Oper. Res. 28 201-232.
35. Rockafellar, R. T. 1970. Convex Analysis. Princeton University Press, Princeton, NJ.
36. Rockafellar, R. T., R. J-B. Wets. 1998. Variational Analysis. Springer, New York.
37. Shapiro, A. 1982. Rank-reducibility of a symmetric matrix and sampling theory of minimum trace factor analysis. Psychometrika 47 187-199.
38. Shapiro, A. 1997. First and second order analysis of nonlinear semidefinite programs. Math. Programming, Ser. B 77 301-320.
39. Shapiro, A. 2003. Sensitivity analysis of generalized equations. J. Math. Sci. 115 2554-2565.
40. Sun, D., J. Sun. 2002. Semismooth matrix valued functions. Math. Oper. Res. 27 150-169.
41. Tseng, P. 1998. Merit functions for semi-definite complementarity problems. Math. Programming 83 159-185.
42. Zarantonello, E. H. 1971. Projections on convex sets in Hilbert space and spectral theory I and II. E. H. Zarantonello, ed. Contributions to Nonlinear Functional Analysis. Academic Press, New York, 237-424.