Abstract
Recently, El Ghami (Optim Theory Decis Mak Oper Res Appl 31:331–349, 2013) proposed a primal dual interior point method for \(P_*(\kappa )\)-Linear Complementarity Problem (LCP) based on a trigonometric barrier term and obtained the worst case iteration complexity as \(O\left( (1+2\kappa )n^{\frac{3}{4}}\log \frac{n}{\epsilon }\right) \) for large-update methods. In this paper, we present a large update primal–dual interior point algorithm for \(P_{*}(\kappa )\)-LCP based on a new trigonometric kernel function. By a simple analysis, we show that our algorithm based on the new kernel function enjoys the worst case \(O\left( (1+2\kappa )\sqrt{n}\log n\log \frac{n}{\epsilon }\right) \) iteration bound for solving \(P_*(\kappa )\)-LCP. This result improves the worst case iteration bound obtained by El Ghami for \(P_*(\kappa )\)-LCP based on trigonometric kernel functions significantly.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anitescu, M., Lesaja, G., Potra, F.: An infeasible interior-point predictor-corrector algorithm for the \(P_{*}\)-geometric LCP. Appl. Math. Optim. 36, 203–228 (1997)
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)
Bai, Y.Q., El Ghami, M., Roos, C.: A new efficient large-update primal-dual interior-point methods based on a finite barrier. SIAM J. Optim. 13(3), 766–782 (2003). (electronic)
Bai, Y.Q., Lesaja, G., Roos, C.: A new class of polynomial interior-point algorithms for \(P_*(\kappa )\)-linear complementary problems. Pac. J. Optim. 4(1), 19–41 (2008)
Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press Inc., San Diego (1992)
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 Ghami, M., Roos, C.: Generic primal–dual interior point methods based on a new kernel function. RAIRO-Oper. Res. 42(2), 199–213 (2008)
Ferris, M.C., Pang, J.S.: Complementarity and variational problems state of the art. In: Proceedings of the International Conference on Complementarity Problems. SIAM, Philadelphia (1997)
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)
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.: 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., Megiddo, N., Noma, T., Yoshise, A.: A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems. In Lecture Notes in Computer Science. Volume 538, Springer, Berlin (1991)
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)
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)
Megiddo, N.: Pathways to the optimal set in linear programming. In: Megiddo, N. (ed.) Progress in Mathematical Programming: Interior Point and Related Methods, pp. 131–158. Springer, New York (1989)
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. Sin. (Engl. Ser.). 26(9), 1761–1778 (2010)
Peyghami, M.R., Hafshejani, S.F., 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, New York (2005)
Väliaho, H.: \(P_*\)-matrices are just sufficient. Linear Algebra Appl. 239, 103–108 (1999)
Wang, G.Q., Bai, Y.Q.: Polynomial interior-point algorithms for \(P_*(\kappa )\) horizontal linear complementarity problem. J. Comput. Appl. Math. 233(2), 248–263 (2009)
Wang, G.Q., Yu, C.J., Teo, K.L.: A full-Newton step feasible interior-point algorithm for \(P_*(\kappa )\)-linear complementarity problem. J. Global Optim. 59(1), 81–99 (2014)
Ye, Y.: Interior-Point Algorithms: Theory and Analysis. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, Chichester (1997)
Acknowledgments
The authors would like to thank the Research Council of K.N. Toosi University of Technology for supporting the work.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hafshejani, S.F., 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, 111–128 (2015). https://doi.org/10.1007/s12190-014-0794-1
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-014-0794-1
Keywords
- Kernel function
- \(P_*(\kappa )\)-linear complementarity problem
- Primal–dual interior point methods
- Large-update methods