An algebraic condition equivalent to strong stability of stationary solutions of nonlinear positive semidefinite programs

SIAM Journal on Optimization

Volumn 16, Issue 2, 2006, Pages 452-470

  Matsumoto, Toshihiro ^a  

^a TEIKYO UNIVERSITY OF SCIENCE   (Japan)

Author keywords

Implicit function theorem; Lipschitz continuous map; Nonlinear programming; Nonsmooth analysis; Positive semidefinite programming; Stationary index; Stationary solution; Strong stability

Indexed keywords

ALGEBRA; CONFORMAL MAPPING; FUNCTIONS; MATHEMATICAL PROGRAMMING; STABILITY; THEOREM PROVING;

IMPLICIT FUNCTION THEOREM; LIPSCHITZ CONTINUOUS MAP; NONSMOOTH ANALYSIS; POSITIVE SEMIDEFINITE PROGRAMMING; STATIONARY INDEX; STATIONARY SOLUTIONS; STRONG STABILITY;

NONLINEAR PROGRAMMING;

EID: 33646744638     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623402416797     Document Type: Article

References (18)

1. J. F. BONNANS AND A. SHAPIRO, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000.
2. F. M. CLARKE, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983.
3. H. FEDERER, Geometric Measure Theory, Springer-Verlag, New York, 1969.
4. A. GRAHAM, Kronecker Products and Matrix Calculus: With Applications, J. Wiley and Sons, New York, 1981.
5. B. IVERSEN, Cuhomology of Sheaves, Springer-Verlag, Berlin, 1986.
6. H. TH. JONGEN, D. KLATTE, AND K. TAMMER, Implicit functions and sensitivity of stationary points, Math. Programming, 49 (1990/91), pp. 123-138.
7. D. KLATTE AND B. KUMMER, Strong stability in nonlinear programming revised, J. Austral. Math. Soc. Ser. B, 40 (1999), pp. 336-352.
8. D. KLATTE AND B. KUMMER, Nonsmooth Equations in Optimization, Kluwer Academic Publishers. Dordrecht, The Netherlands, 2002.
9. M. KOJIMA, Strongly stable stationary solutions in nonlinear programs, in Analysis and Computation of Fixed Points, S. M. Robinson, ed., Academic Press, New York, 1980, pp. 93-138.
10. M. KOJIMA, Semidefinite programming and interior-point methods, Bulletin of the Japan Society for Industrial and Applied Mathematics. 6 (1996), pp. 16-25 (in Japanese).
11. 1'1 optimization, J. Optim. Theory Appl., 70 (1991), pp. 561-581.
12. 1,1-optimization, J. Math. Anal. Appl., 158 (1991), pp. 35-46.
13. A. LEVY, Solution sensitivity from general principles, SIAM J. Control Optim., 40 (2001), pp. 1-38.
14. T. MATSUMOTO, On the stability of stationary solutions of nonlinear positive Semidefinite programs, J. Oper. Res. Soc. Japan, 46 (2003), pp. 22-34.
15. J. M. ORTEGA AND W. RHEINBOLDT, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
16. J.-S. PANG. D. SUN, AND J. SUN, Semismooth homeomorphisms and strong stability of semidefinite and Lorentz complementarity problems, Math. Oper. Res., 28 (2003), pp. 39-63.
17. R. ROCKAFELLAR AND R. WETS, Variational Analysis, Springer-Verlag, Berlin, 1998.
18. E. H. ZARANTONELLO, Projections on convex sets in Hilbert space and spectral theory, in Contributions to Nonlinear Functional Analysis. E. H. Zarantonello, ed., Academic Press, New York, pp. 237-341.