Abstract
Recent studies on the kernel function-based primal-dual interior-point algorithms indicate that a kernel function not only represents a measure of the distance between the iteration and the central path, but also plays a critical role in improving the computational complexity of an interior-point algorithm. In this paper, we propose a new class of parameterized kernel functions for the development of primal-dual interior-point algorithms for solving linear programming problems. The properties of the proposed kernel functions and corresponding parameters are investigated. The results lead to a complexity bounds of \({O\left(\sqrt{n}\,{\rm log}\,n\,{\rm log}\,\frac{n}{\epsilon}\right)}\) for the large-update primal-dual interior point methods. To the best of our knowledge, this is the best known bound achieved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bai Y.Q., El ghami M., Roos C.: A new efficient large-update primal-dual interior-point method based on a finite barrier. SIAM J. Optim. 13, 766–782 (2003)
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. Optim. 15, 101–128 (2004)
Bai Y.Q., Guo J.L., Roos C.: A new kernel function yielding the best known iteration bounds for primal-dual interior-point algorithms. Acta Mathematica Sinica, English Series 25, 2169–2178 (2009)
Bazaraa M.S., Sherali H.D., Shetty C.M.: Nonlinear Programming, Theory and Algorithms. Wiley, New York (2002)
Boyd S., Vandenberghe L.: Convex Optimization. Cambridge University Press, Cambridge (2004)
Cho G.M., Cho Y.Y., Lee Y.H.: A primal-dual interior-point algorithm based on a new kernel function. Anziam J. 51, 476–491 (2010)
Megiddo, N.: Pathways to the optimal set in linear programming. In: Megiddo, N. (eds.) Progress in Mathematical Programming: Interior Point and Related Methods, pp. 131–158. Springer, New York (1989) [Identical version. In: Proceedings of the 6th Mathematical Programming Symposium of Japan, pp. 1–35. Nagoya, Japan (1986)]
Neterrov, R.E., Nemirovskii, A.S.: Interior-Point Ploynomial Methods in Convex Programming. Society for Industrial and Applied Mathematics (1994)
Nemirovskii, A.S.: Lecture Notes: Interior-Point Ploynomial Methods in Convex Programming. Georgia Institute of Technology, School of Industrial and Systems Engineering, Spring Semester (2004)
Peng J., Roos C., Terlaky T.: Self-regular functions and new search directions for linear and semidefinite optimization. Math. Program. Ser. A 93, 129–171 (2002)
Peng J., Roos C., Terlaky T.: Self-Regularity. A New Paradigm for Primal-Dual Interior- Point Algorithms. Princeton University Press, Princeton (2002)
Roos C., Terlaky T., Vial J.Ph.: Theory and Algorithms for Linear Optimization. An Interior-Point Approach. Wiley, Chichester (1997)
Sonnevend, G.: An “analytic center” for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming. In: Prékopa, A., Szelezsán, J., Strazicky, B. (eds.) System Modelling and Optimization: Proceedings of the 12th IFIP-Conference held in Budapest, Hungary, September 1985, Lecture Notes in Control and Information Sciences, Vol. 84, pp. 866–876. Springer, Berlin (1986)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bai, Y., Xie, W. & Zhang, J. New parameterized kernel functions for linear optimization. J Glob Optim 54, 353–366 (2012). https://doi.org/10.1007/s10898-012-9934-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-012-9934-z