Abstract
Kernel functions play an important role in the design and complexity analysis of interior point algorithms for solving convex optimization problems. They determine both search directions and the proximity measure between the iterate and the central path. In this paper, we introduce a primal-dual interior point algorithm for solving \(P_*(\kappa ) \)-horizontal linear complementarity problems based on a new kernel function that has a trigonometric function in its barrier term. By using some simple analysis tools, we present some properties of the new kernel function. Our analysis shows that the algorithm meets the best known complexity bound i.e., \(O\left( (1+2\kappa )\sqrt{n}\log n\log \frac{n}{\varepsilon }\right) \) for large-update methods. Finally, we present some numerical results illustrating the performance of the algorithm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Asadi, S., Mansouri, H.: Polynomial interior-point algorithm for \(P_*(\kappa )\)-horizontal linear complementarity problems. Numer. Algorithms 63(2), 385–398 (2013)
Asadi, S., Mansouri, H., Zangiabadi, M.: A class of path-following interior-point methods for \(P_*(\kappa )\)-horizontal linear complementarity problems. J. Oper. Res. Soc. China 3(1), 17–30 (2015)
Asadi, S., Zangiabadi, M., Mansouri, H.: An infeasible interior-point algorithm with full-Newton steps for \(P_*(\kappa )\)-horizontal linear complementarity problems based on a kernel function. J. Appl. Math. Comput. 50(1), 15–37 (2016)
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(1), 101–128 (2004)
Bouafia, M., Benterki, D., Yassine, A.: An efficient primal-dual interior point method for linear programming problems based on a new kernel function with a trigonometric barrier term. J. Optim. Theory Appl. 170(2), 528–545 (2016)
Cai, X., Wang, G., Zhang, Z.: Complexity analysis and numerical implementation of primal-dual interior-point methods for convex quadratic optimization based on a finite barrier. Numer. Algorithims 62, 289–306 (2013)
El Ghami, M., Guennoun, Z.A., Boula, S., Steihaug, T.: Interior-point methods for linear optimization based on a kernel function with a trigonometric barrier term. J. Comput. Appl. Math. 236, 3613–3623 (2012)
El. Ghamia, M., Ivanov, I., Melissen, J.B.M., Roos, C., Steihaug, T.: A polynomial-time algorithm for linear optimization based on a new class of kernel functions. J. Comput. Appl. Math. 224(2), 500–513 (2009)
El Ghami, M.: Primal-dual interior-point methods for \(P_*(\kappa )\)-linear complementarity problem based on a kernel function with a trigonometric barrier term. Optim. Theory Decis. Mak. Oper. Res. Appl. 31, 331–349 (2013)
Fathi-Hafshejani, S., Fatemi, M., Peyghami, M.R.: An interior-point method for \(P_*(\kappa )\)-linear complementarity problem based on a trigonometric kernel function. J. Appl. Math. Comput. 48, 111–148 (2015)
Karmarkar, N.K.: A new polynomial-time algorithm for linear programming. In: Proceedings of the 16th Annual ACM Symposium on Theory of Computing, pp. 302–311 (1984)
Kheirfam, B.: A new search direction for full-newton stepinterior-point method in \(P_*(\kappa )\)-HLCP. J. Numer. Funct. Anal. Optim. 40(10), 1169–1181 (2019)
Kheirfam, B.: Primal-dual interior-point algorithm for semidefinite optimization based on a new kernel function with trigonometric barrier term. Numer. Algorithms 61, 659–680 (2012)
Kojima, M., Mizuno, S., Yoshise, A.: A primal-dual interior point algorithm for linear programming. In: Megiddo, N. (ed.) Progress in Mathematical Programming: Interior Point and Related Methods, pp. 29–47. Springer, New York (1989)
Lee, Y.H., Cho, Y.Y., Cho, G.M.: Kernel function based interior-point methodsfor horizontal linear complementarity problems. J. Inequal. Appl. 2013, 215–229 (2013)
Lesaja, G., Roos, C.: Unified analysis of kernel-based interior-point methods for \(P_*(\kappa )\)-linear complementarity problems. SIAM J. Optim. 20, 3014–3039 (2010)
Peng, J., Roos, C., Terlaky, T.: Self-Regularity a New Paradigm for Primal-Dual Interior-Point Algorithms. Princeton University Press, Princeton (2002)
Peyghami, M.R., Amini, K.: A kernel function based interior-point methods for solving \(P_{\ast }(\kappa )\)-linear complementarity problem. Acta Math. Sinica (Engl. Ser.) 26(9), 1761–1778 (2010)
Peyghami, M.R., Fathi-Hafshejani, S.: Complexity analysis of an interior point algorithm for linear optimization based on a new poriximity function. Numer. Algorithims 67, 33–48 (2016)
Peyghami, M.R., Fathi-Hafshejani, S., Shirvani, L.: Complexity of interior-point methods for linear optimization based on a new trigonometric kernel function. J. Comput. Appl. Math. 255, 74–85 (2014)
Roos, C., Terlaky, T., Vial, J.-P.: Theory and Algorithms for Linear Optimization: An Interior Point Approach. Springer, NewYork (2005)
Sonnevend, G.: An analytic center for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming. In: Prakopa, A., Szelezsan, J., Strazicky, B. (eds.) Lecture Notes in Control and Information Sciences, vol. 84, pp. 866–876. Springer, Berlin (1986)
Wang, G.Q., Bai, Y.Q.: Polynomial interior-point algorithm for \(P_*(\kappa )\) horizontal linear complementarity problem. J. Comput. Appl. Math. 233, 248–263 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Fathi Hafshejani, S., Fakharzadeh Jahromi, A. An interior point method for \(P_{*}(\kappa )\)-horizontal linear complementarity problem based on a new proximity function. J. Appl. Math. Comput. 62, 281–300 (2020). https://doi.org/10.1007/s12190-019-01284-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-019-01284-9
Keywords
- Trigonometric kernel function
- Horizontal linear complementarity problems
- Primal-dual interior point methods
- Large-update methods