An inexact primal-dual path following algorithm for convex quadratic SDP

Mathematical Programming

Volumn 112, Issue 1, 2008, Pages 221-254

  Toh, Kim Chuan ^a,b  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^b SINGAPORE MIT ALLIANCE   (Singapore)

Author keywords

Inexact search direction; Interior point method; Krylov iterative method; Semidefinite least squares; Semidefinite programming

Indexed keywords

ALGORITHMS; EIGENVALUES AND EIGENFUNCTIONS; ITERATIVE METHODS; LEAST SQUARES APPROXIMATIONS; MATHEMATICAL OPERATORS; PROBLEM SOLVING;

INEXACT SEARCH DIRECTIONS; INTERIOR POINT METHODS; KRYLOV ITERATIVE METHODS; QUADRATIC SEMIDEFINITE PROGRAMMING (QSDP); SEMIDEFINITE LEAST SQUARES;

QUADRATIC PROGRAMMING;

EID: 34547439695     PISSN: 00255610     EISSN: 14364646     Source Type: Journal    
DOI: 10.1007/s10107-006-0088-y     Document Type: Conference Paper

References (32)

1. Alfakih A.Y., Khandani A. and Wolkowicz H. (1999). Solving Euclidean distance matrix completion problems via semidefinite programming. Comput. Optim. Appl. 12: 13-30
2. Alizadeh F., Haeberly J.-P.A. and Overton M.L. (1997). Complementarity and nondegeneracy in semidefinite programming. Math. Program. 77: 111-128
3. Alizadeh F., Haeberly J.-P.A. and Overton M.L. (1998). Primal-dual interior-point methods for semidefinite programming: convergence results, stability and numerical results. SIAM J. Optim. 8: 746-768
4. Anjos, M.F., Higham, N.J., Takouda, P.L., Wolkowicz, H.: A semidefinite programming approach for the nearest correlation matrix problem. Research Report, Department of Combinatorics and Optimization, University of Waterloo (2003)
5. Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 1-137 (2005)
6. Benzi M. and Simoncini V. (2006). On the eigenvalues of a class of saddle point matrices. Numer. Math. 103: 173-196
7. Bergamaschi L., Gondzio J. and Zilli G. (2004). Preconditioning indefinite systems in interior point methods for optimization. Comput. Optim. Appl. 28: 149-171
8. Berman H.M., Westbrook J., Feng Z., Gilliland G., Bhat T.N., Weissig H., Shindyalov I.N. and Bourne P.E. (2000). The protein data bank. Nucl. Acids Res. 28: 235-242
9. Bhatia R. and Kittaneh F. (1990). Norm inequalities for partitioned operators and an application. Math. Ann. 287: 719-726
10. Burer S., Monteiro R.D.C. and Zhang Y. (2002). Solving a class of semidefinite programs via nonlinear programming. Math. Program. 93: 97-122
11. Freund, R.W., Nachtigal, N.M.: A new Krylov-subspace method for symmetric indefinite linear systems. In: Ames W.F. (ed.) Proceedings of the 14th IMACS World Congress on Computational and Applied Mathematics, Atlanta, USA, pp. 1253-1256 (1994)
12. Fujisawa K., Kojima M. and Nakata K. (1997). Exploiting sparsity in primal-dual interior-point method for semidefinite programming. Math. Program. 79: 235-253
13. Halicka M., Roos C. and Klerk E. (2005). Limiting behaviour of the central path in semidefinite optimization. Optim. Methods Softw. 20: 99-113
14. Higham N.J. (2002). Computing the nearest correlation matrix-a problem from finance. IMA J. Numer. Anal. 22: 329-343
15. Horn, R., Johnson, C. Matrix Analysis. Cambridge University Press, Cambridge (1998)
16. Keller C., Gould N.I.M. and Wathen A.J. (2000). Constraint preconditioning for indefinite linear systems. Matrix Anal. Appl. 21: 1300-1317
17. Kojima M., Shindoh S. and Hara S. (1997). Interior-point methods for the monotone linear complementarity problem in symmetric matrices. SIAM J. Optim. 7: 86-125
18. Krislock N., Lang J., Varah J., Pai D.K. and Seidel H.-P. (2004). Local compliance estimation via positive semidefinite constrained least sqaures. IEEE Trans. Robot. 20: 1007-1011
19. Langville A.N. and Stewart W.J. (2004). A Kronecker product approximate preconditioner for SANs. Numer. Linear Algebra Appl. 11: 723-752
20. Luo Z.-Q., Sturm J.F. and Zhang S. (1998). Superlinear convergence of a symmetric primal-dual path following algorithm for semidefinite programming. SIAM J. Optim. 8: 59-81
21. Malick J. (2004). A dual approach to semidefinite least-squares problems. SIAM J. Matrix Anal. Appl. 26: 272-284
22. Monteiro R.D.C. and Zanjácomo P.R. (1999). Implementation of primal-dual methods for semidefinite programming based on Monteiro and Tsuchiya Newton directions and their variants. Optim. Methods Softw. 11(12): 91-140
23. Nie J.W. and Yuan Y.X. (2001). A predictor-corrector algorithm for QSDP combining Dikin-type and Newton centering steps. Ann. Oper. Res. 103: 115-133
24. Phoon K.K., Toh K.C., Chan S.H. and Lee F.H. (2002). An efficient diagonal preconditioner for finite element solution of Biot's consolidation equations. Int. J. Numer. Methods in Eng. 55: 377-400
25. Qi H. and Sun D. (2006). A quadratically convergent Newton method for computing the nearest correlation matrix. SIAM J. Matrix Anal. Appl. 28: 360-385
26. Saad Y. (1996). Iterative Methods for Sparse Linear Systems. PWS Publishing Company, Boston
27. Todd M.J., Toh K.C. and Tütüncü R.H. (1998). On the Nesterov-Todd direction in semidefinite programming. SIAM J. Optim. 8: 769-796
28. Toh K.C., Phoon K.K. and Chan S.H. (2004). Block preconditioners for symmetric indefinite linear systems. Int. J. Numer. Methods Eng. 60: 1361-1381
29. Toh, K.C., Tütüncü, R.H., Todd, M.J.: Inexact primal-dual path-following algorithms for a special class of convex quadratic SDP and related problems. Pacific J. Optim. (in press)
30. Van Loan C.F. (2000). The ubiquitous Kronecker product. J. Comput. Appl. Math. 123: 85-100
31. Wathen, A.J., Fischer, B., Silvester, D.J.: The convergence of iterative solution methods for symmetric and indefinite linear systems. In: Griffiths, D.F., Watson, G.A., (eds.) Numerical Analysis 1997. Addison Wesley Longman, Harlow, pp. 230-243 (1997)
32. Zhou G.L. and Toh K.C. (2004). Polynomiality of an inexact infeasible interior point algorithm for semidefinite programming. Math. Program. 99: 261-282