Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T07:10:29.099Z Has data issue: false hasContentIssue false

A FULL NT-STEP INFEASIBLE INTERIOR-POINT ALGORITHM FOR SEMIDEFINITE OPTIMIZATION BASED ON A SELF-REGULAR PROXIMITY

Published online by Cambridge University Press:  28 March 2012

B. KHEIRFAM*
Affiliation:
Department of Mathematics, Azarbaijan University of Tarbiat Moallem, Tabriz, Iran (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We introduce a full NT-step infeasible interior-point algorithm for semidefinite optimization based on a self-regular function to provide the feasibility step and to measure proximity to the central path. The result of polynomial complexity coincides with the best known iteration bound for infeasible interior-point methods.

MSC classification

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2012

References

[1]Alizadeh, F., “Interior point methods in semidefinite programming with applications to combinatorial optimization”, SIAM J. Optim. 5 (1995) 1351doi:10.1137/0805002.Google Scholar
[2]de Klerk, E., Aspects of semidefinite programming. Applied optimization, 1st edn (Kluwer Academic, Dordrecht, 2002).CrossRefGoogle Scholar
[3]Horn, R. A. and Johnson, C. H. R., Topics in matrix analysis (Cambridge University Press, Cambridge, 1991).CrossRefGoogle Scholar
[4]Karisch, E. S., Rendl, F. and Clausen, J., “Solving graph bisection problems with semidefinite programming”, INFORMS J. Comput. 12 (2000) 177191doi:10.1287/ijoc.12.3.177.12637.CrossRefGoogle Scholar
[5]Kojima, M., Shida, M. and Hara, S., “Local convergence of predictor–corrector infeasible-interior-point algorithms for SDPs and SDLCPs”, Math. Program. 80 (1998) 129160; doi:10.1007/BF01581723.Google Scholar
[6]Liu, Z. Y. and Chen, Y., “A full-Newton step infeasible interior-point algorithm for linear programming based on a self-regular proximity”, J. Appl. Math. Informatics 29 (2011) 119133.Google Scholar
[7]Liu, Z. Y. and Sun, W. Y., “An infeasible interior-point algorithm with full-Newton step for linear optimization”, Numer. Algorithms 46 (2007) 173188doi:10.1007/s11075-007-9135-x.CrossRefGoogle Scholar
[8]Luo, Z. Q., Sturm, J. and Zhang, S. Z., “Superlinear convergence of a symmetric primal-dual path following algorithm for semidefinite programming”, SIAM J. Optim. 8 (1998) 5981; doi:10.1137/S1052623496299187.CrossRefGoogle Scholar
[9]Lustig, I. J., “Feasible issues in a primal-dual interior-point method for linear programming”, Math. Program. 49 (1991) 145162doi:10.1007/BF01588785.CrossRefGoogle Scholar
[10]Lütkepohl, H., Handbook of matrices (Wiley, Chichester, 1996).Google Scholar
[11]Mansouri, H. and Roos, C., “Simplified O(nL) infeasible interior-point algorithm for linear optimization using full-Newton steps”, Optim. Methods Soft. 22 (2007) 519530; doi:10.1080/10556780600816692.Google Scholar
[12]Mansouri, H. and Roos, C., “A new full-Newton step O(n) infeasible interior-point algorithm for semidefinite optimizationg”, Numer. Algorithms 52 (2009) 225255; doi:10.1007/s11075-009-9270-7.Google Scholar
[13]Nesterov, Yu. E. and Todd, M. J., “Self-scaled barriers and interior-point methods for convex programming”, Math. Oper. Res. 22 (1997) 142doi:10.1287/moor.22.1.1.Google Scholar
[14]Nesterov, Yu. E. and Todd, M. J., “Primal-dual interior-point methods for self-scaled cones”, SIAM J. Optim. 8 (1998) 324364doi:10.1137/S1052623495290209.Google Scholar
[15]Peng, J., Roos, C. and Terlaky, T., “New complexity analysis of the primal-dual method for semidefinite optimization based on the NT-direction”, J. Optim. Theory Appl. 109 (2001) 327343.CrossRefGoogle Scholar
[16]Peng, J., Roos, C. and Terlaky, T., “Self-regular functions and new search directions for linear and semidefinite optimization”, Math. Program. 93 (2002) 129171doi:10.1007/s101070200296.CrossRefGoogle Scholar
[17]Peng, J., Roos, C. and Terlaky, T., Self-regularity. A new paradigm for primal-dual interior-point algorithms (Princeton University Press, Princeton, NJ, 2002).Google Scholar
[18]Potra, F. A. and Sheng, R., “A superlinear convergent primal-dual infeasible-interior-point algorithm for semidefinite programming”, SIAM J. Optim. 8 (1998) 10071028; doi:10.1137/S1052623495294955.CrossRefGoogle Scholar
[19]Wang, G. Q., Bai, Y. Q. and Roos, C., “Primal-dual interior-point algorithms for semidefinite optimization based on a simple kernel function”, J. Math. Model. Algorithms 4 (2005) 409433; doi:10.1007/s10852–005–3561–3.CrossRefGoogle Scholar
[20]Wolkowicz, H., Saigal, R. and Vandenberghe, L., Handbook of semidefinite programming, theory, algorithm, and applications (Kluwer Academic Publishers, Dordrecht, 2000).CrossRefGoogle Scholar
[21]Zhang, Y., “On extending some primal-dual interior-point algorithms from linear programming to semidefinite programming”, SIAM J. Optim. 8 (1998) 365386; doi:10.1137/S1052623495296115.Google Scholar