An interior-point method for approximate positive semidefinite completions

Computational Optimization and Applications

Volumn 9, Issue 2, 1998, Pages 175-190

  Johnson, Charles R ^a   Kroschel, Brenda ^a   Wolkowicz, Henry ^b  

^a The College of William and Mary   (United States)

^b UNIVERSITY OF WATERLOO   (Canada)

Author keywords

Best nonnegative approximation; Complementarity problems; Positive definite completions; Primaldual interior point methods; Semidefinite programming

Indexed keywords

BEST NONNEGATIVE APPROXIMATION; COMPLEMENTARITY PROBLEMS; POSITIVE DEFINITE COMPLETIONS; PRIMAL-DUAL INTERIOR-POINT METHODS; SEMIDEFNITE PROGRAMMING;

ALGORITHMS; CONSTRAINT THEORY; MATHEMATICAL PROGRAMMING; PROBLEM SOLVING; VECTORS; APPROXIMATION THEORY;

MATRIX ALGEBRA;

BEST NONNEGATIVE APPROXIMATION; DUALITY THEORY; PRIMAL DUAL INTERIOR POINT METHODS; SEMIDEFINITE PROGRAMMING;

EID: 0031999150     PISSN: 09266003     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1018363021404     Document Type: Article

References (23)

1. F. Alizadeh, J.-P.A. Haeberly, and M.L. Overton, "Primal-dual interior-point methods for semidefinite programming: convergence rates, stability and numerical results," Technical Report, Courant Institute of Mathematical Sciences, 1996, NYU Computer Science Dept. Technical Report 721.
2. W.W. Barrett, C.R. Johnson, and R. Loewy, "The real positive definite completion problem: cycle completability," Memoirs of the American Mathematical Society, p. 71, 1996.
3. J.M. Borwein and H. Wolkowicz, "A simple constraint qualification in infinite dimensional programming," Mathematical Programming, vol. 35, pp. 83-96, 1986.
4. J.M. Borwein and A. Lewis, "Partially finite convex programming, Part I, duality theory," Mathematical Programming, vol. 57, pp. 15-8, 1992.
5. S. Boyd, L. El Ghaoui, E. Feron, and V. Balakrishnan, "Linear matrix inequalities in system and control theory," Studies in Applied Mathematics, SIAM: Philadelphia, PA, vol. 15, June 1994.
6. H. Dym and I. Gohberg, "Extensions of band matrices with band inverses," Linear Algebra and Its Applications, vol. 36, pp. 1-24, 1981.
7. W. Glunt, T. Hayden, C.R. Johnson, and P. Tarazaga, "Maximum determinant completions," Technical Report, Dept. of Mathematics, College of William and Mary, Williamsburg, VA, 1994.
8. B. Grone, C. Johnson, E. Marques De Sa, and H. Wolkowicz, "Positive definite completions of partial Hermitian matrices," Linear Algebra and Its Applications, vol. 58, pp. 109-124, 1984.
9. C. Helmberg, F. Rendl, R.J. Vanderbei, and H. Wolkowicz, "An interior point method for semidefinite programming," SIAM Journal on Optimization, pp. 342-361, 1996. URL: ftp://orion.uwaterloo.ca/pub/henry/ reports/sdp.ps.gz.
10. R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press: New York, 1985.
11. C.R. Johnson and B. Kroschel, "Principal submatrices, geometric multiplicities, and structured eigenvectors," SIAM Journal on Matrix Analysis and Applications, vol. 16, pp. 1004-1012, 1995.
12. C.R. Johnson and P. Tarazaga, "Approximate semidefinite matrices in a subspace," Technical Report, Dept. of Mathematics, College of William and Mary, Williamsburg, VA, 1995, to appear in SIMAX.
13. M. Kojima, M. Shida, and S. Shindoh, "Reduction of monotone linear complementarity problems over cones to linear programs over cones," Technical Report, Dept. of Information Sciences, Tokyo Institute of Technology, Tokyo, Japan, 1995.
14. M. Kojima, S. Shindoh, and S. Hara, "Interior-point methods for the monotone semidefinite linear complementarity problem in symmetric matrices," SIAM J. Optim., vol. 7, no. 1, pp. 86-125, 1997.
15. S. Kruk, M. Muramatsu, F. Rendl, R.J. Vanderbei, and H. Wolkowicz, "The Gauss-Newton direction in linear and semidefinite programming," Technical Report in progress, University of Waterloo, Waterloo, Canada, 1997.
16. D.G. Luenberger, Optimization by Vector Space Methods, John Wiley, 1969.
17. J.R. Magnus and H. Neudecker, Matrix Differential Calculus, Wiley: New York, NY, 1988.
18. p approximation," Journal of Constructive Approximation, vol. 1, pp. 93-102, 1985.
19. S. Mizuno, M. J. Todd, and Y. Ye, "On adaptive-step primal-dual interior-point algorithms for linear programming," Mathematics of Operations Research, vol. 18, no. 4, pp. 964-981, 1994.
20. J.J. Moreau, "Décomposition orthogonale d'un espace hilbertien selon deux con̂es mutuellement polaires," C.R. Acad. Sci. Paris, vol. 255, pp. 238-240, 1962.
21. Y.E. Nesterov and A.S. Nemirovsky, Interior Point Polynomial Algorithms in Convex Programming: Theory and Algorithms, SIAM Publications: SIAM, Philadelphia, USA, 1994.
22. M.L. Overton, "Large-scale optimization of eigenvalues," SIAM J. Optimization, vol. 2, pp. 88-120, 1992.
23. P.W. Smith and H. Wolkowicz, "A nonlinear equation for linear programming," Mathematical Programming, vol. 34, pp. 235-238, 1986.