Abstract
In this paper, we establish the polynomial complexity of a primal-dual path-following interior point algorithm for solving semidefinite optimization (SDO) problems. The proposed algorithm is based on a new kernel function which differs from the existing kernel functions in which it has a double barrier term. With this function we define a new search direction and also a new proximity function for analyzing its complexity. We show that if q 1 > q 2 > 1, the algorithm has \(O((q_1 + 1)n^{\frac{{q_1 + 1}} {{2(q_1 - q_2 )}}} \log \tfrac{n} {e}) \) and \(O((q_1 + 1)^{\frac{{3q_1 - 2q_2 + 1}} {{2(q_1 - q_2 )}}} \sqrt n \log \tfrac{n} {e}) \) complexity results for large- and small-update methods, respectively.
Similar content being viewed by others
References
Achache, M.: A new primal-dual path-following method for convex quadratic programming. Comput. Appl. Math., 25(1), 97–110 (2006)
Alizadeh, F.: Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM Journal. Optim., 5(1), 13–51 (2006)
Bai, Y. Q., El Ghami, M., Roos, C.: A comparative study of kernel functions for primal-dual interior point algorithms in linear optimization. SIAM J. Optimiz., 15(1), 101–128 (2004)
Bai, Y. Q., Wang, G. Q., Roos, C.: Primal-dual interior point interior point algorithms for second-order cone optimization based on kernel functions. Nonlinear Anal., 70, 3548–3602 (2009)
Bai, Y. Q., Roos, C., El Ghami, M.: A primal-dual interior point method for linear optimization based on a new proximity function. Optimiz. Methods Software, 17(6), 985–1008 (2002)
Choi, B. K., Lee, G. M.: On complexity analysis of the primal-dual interior point method for semidefinite optimization problem based on a new proximity function. Nonlinear Anal., 71(12), 2540–2550 (2009)
El Ghami, M., Steihaug, T., Roos, C.: Primal-dual IPMs for semidefinite optimization based on finite barrier functions, Report N0341, Department of Informatics, University of Bergen, Bergen, Norway, 2006
El Ghami, M.: New primal-dual interior-point methods based on kernel functions. Ph.D. thesis, Delft University, Netherland, 2005
Liu, L., Li, S.: A new kind of simple kernel function yielding good iteration bounds for for primal-dual interior point methods. J. Comput. Appl. Math., 235(9), 2944–2955 (2011)
Cho, G. M.: A new primal-dual interior point method for linear optimization. J. KSIAM, 13(1), 41–53 (2009)
Peng, J., Roos, C., Terlaky, T.: A new class of polynomial primal-dual methods for linear and semidefinite optimization. European J. Operations Research, 143(2), 234–256 (2002)
Peng, J., Roos, C., Terlaky, T.: Self-regular functions and new search directions for linear and semidefinite optimization. Math. Programming, 93, 129–117 (2002)
Peng, J., Roos, C., Terlaky, T.: Self-regularity. A New Paradigm for Primal-Dual Interior Point Algorithm, Princeton University Press, Princeton, 2002
Roos, C., Terlaky, T., Vial, J. Ph.: Theory and Algorithms for Linear Optimization. An Interior Point Approach. John Wiley and Sons, Chister, U.K., 1997
Todd, M. J.: A study of search directions in primal-dual interior point methods for semidefinite programming. Optim. Methods Softw., 11, 1–46 (1999)
Vanderberghe, L., Boyd, S.: Semidefinite programming. SIAM Review, 38(1), 49–95 (1996)
Horn, R. A., Johnson, C. R.: Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Achache, M. Complexity analysis of an interior point algorithm for the semidefinite optimization based on a kernel function with a double barrier term. Acta. Math. Sin.-English Ser. 31, 543–556 (2015). https://doi.org/10.1007/s10114-015-1314-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-015-1314-4
Keywords
- Semidefinite optimization
- kernel functions
- primal-dual interior point methods
- large and small-update algorithms
- complexity of algorithms