Sparse quasi-Newton updates with positive definite matrix completion

Mathematical Programming

Volumn 115, Issue 1, 2008, Pages 1-30

  Yamashita, Nobuo ^a  

^a KYOTO UNIVERSITY   (Japan)

Author keywords

Large scale problems; Positive definite matrix completion; Quasi Newton method; Sparsity

Indexed keywords

APPROXIMATION THEORY; COMPUTATIONAL COMPLEXITY; CONVERGENCE OF NUMERICAL METHODS; MATRIX ALGEBRA;

LARGE SCALE PROBLEMS; POSITIVE DEFINITE MATRIX COMPLETION; QUASI-NEWTON METHOD;

PROBLEM SOLVING;

EID: 43949114859     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-007-0137-1     Document Type: Article

References (22)

1. Blair J.R.S. and Peyton B. (1993). An introduction to chordal graphs and clique trees. In: George, A., Gilbert, J.R. and Liu, J.W.H. (eds) Graph Theory and Sparse Matrix Computation, pp 1-29. Springer, New York
2. Coleman T.F., Garbow B. and Moré J.J. (1985). Software for estimating sparse Hessian matrices. ACM Trans. Math. Softw. 11: 363-378
3. Conn, A.R., Gould, N.I.M., Toint, Ph.L.: LANCELOT: a Fortran package for large-scale nonlinear optimization (Release A). In: Springer Series in Computational Mathematics, vol. 17. Springer, New York (1992)
4. Moré J.J. and Dennis J.E. (1974). A characterization of superlinear convergence and its application to quasi-Newton methods. Math. Comput. 28(126): 549-560
5. Schnabel R.B. and Dennis J.E. (1983). Numerical methods for unconstrained optimization and nonlinear equations. Prentice-Hall Inc., New Jersey
6. Fletcher R. (1991). A new result for quasi-Newton formulae. SIAM J. Optim. 1: 18-21
7. Fletcher R. (1995). An optimal positive definite update for sparse Hessian matrices. SIAM J. Optim. 5: 192-218
8. Fukuda M., Kojima M., Murota K. and Nakata K. (2000). Exploiting sparsity in semidefinite programming via matrix completion I: general framework. SIAM J. Optim. 11: 647-674
9. George A. and Liu J.W.H. (1981). Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall, Englewood Cliff
10. Gould N.I.M., Orban D. and Toint Ph.L. (2003). CUTEr, a constrained and unconstrained testing environment: revisited. ACM Trans. Math. Softw. 29: 373-394
11. Griewank A. and Toint Ph.L. (1982). On the unconstrained optimization of partially separable functions. In: Powell, M.J.D. (eds) Nonlinear Optimization 1981, pp 301-312. Academic, London
12. Griewank A. and Toint Ph.L. (1982). Partitioned variable metric updates for large structured optimization problems. Numer. Math. 39: 119-137
13. Griewank A. and Toint Ph.L. (1984). On the existence of convex decomposition of partially separable functions. Math. Program. 28: 25-29
14. Grone R., Johnson C.R., Sá E.M. and Wolkowicz H. (1984). Positive definite completions of partial Hermitian matrices. Linear Algebra Appl. 58: 109-124
15. Liu D.C. and Nocedal J. (1989). On the limited memory BFGS method for large scale optimization. Math. Program. 45: 503-528
16. Nocedal J. (1980). Updating quasi-Newton matrices with limited storage. Math. Comput. 35: 773-782
17. Nocedal J. and Wright S.J. (1999). Numerical Optimization. Springer, New York
18. Powell M.J.D. and Toint Ph.L. (1979). On the estimation of sparse Hessian matrices. SIAM J. Numer. Anal. 16: 1060-1074
19. Powell M.J.D. and Toint Ph.L. (1981). The Shanno-Toint procedure for updating sparse symmetric matrices. IMA J. Numer. Anal. 1: 403-413
20. Schnabel R.B. and Toint Ph.L. (1983). Forcing sparsity by projecting with respect to a non-diagonally weighted Frobenius norm. Math. Program. 25(1): 125-129
21. Sorensen D.C. (1982). Collinear scaling and sequential estimation in sparse optimization algorithm. Math. Program. Stud. 18: 135-159
22. Toint P.L. (1977). On sparse and symmetric matrix updating subject to a linear equation. Math. Comput. 31: 954-961