Abstract
In this paper, we propose a large-update primal-dual interior point method for solving a class of linear complementarity problems based on a new kernel function. The main aspects distinguishing our proposed kernel function from the others are as follows: Firstly, it incorporates a specific trigonometric function in its growth term, and secondly, the corresponding barrier term takes finite values at the boundary of the feasible region. We show that, by resorting to relatively simple techniques, the primal-dual interior point methods designed for a specific class of linear complementarity problems enjoy the so-called best-known iteration complexity for the large-update methods.
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Ye, Y.: Interior-Point Algorithms: Theory and Analysis. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, Chichester (1997)
Cottle, R.W., Pang, J.S., Stone, R.E.: The Linear Complementarity Problem. Academic Press Inc., San Diego (1992)
Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer, New York (2003)
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)
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, vol. 538. Springer, Berlin (1991)
Peng, J., Roos, C., Terlaky, T.: Self-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms. Princeton University Press, Princeton (2002)
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., 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)
Cho, G.M., Kim, M.K.: A new large-update interior-point algorithm for \(P_*(\kappa )\)-LCPs based on kernel functions. Appl. Math. Comput. 182, 1169–1183 (2006)
Fathi-Hafshejani, S., Mansouri, H., Peyghami, M.R.: An interior-point algorithm for \(P_*(\kappa )\)-linear complementarity problem based on a new trigonometric kernel function. J. Math. Model. 5(2), 171–197 (2017)
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)
Peyghami, M.R., Amini, K.: A kernel function based interior-point methods for solving \(P_*(\kappa )\)-linear complementarity problem. Acta Math. Sin. (Engl. Ser.) 26(9), 1761–1778 (2010)
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)
Fathi-Hafshejani, S., Fakharzadeh Jahromi, A., Peyghami, M.R.: A unified complexity analysis of interior point methods for semidefinite problems based on trigonometric kernel functions. Optimization 67(1), 113–137 (2018)
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)
Peyghami, M.R., Fathi-Hafshejani, S.: Complexity analysis of an interior-point algorithm for linear optimization based on a new proximity function. Numer. Algorithms 67, 33–48 (2014)
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)
Peyghami, M.R., Fathi-Hafshejani, S., Chen, S.: A primal-dual interior-point method for semidefinite optimization based on a class of trigonometric barrier functions. Oper. Res. Lett. 44, 319–323 (2016)
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)
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)
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. Algorithms 62, 289–306 (2013)
El Ghami, 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, 500–513 (2009)
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)
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)
Roos, C., Terlaky, T., Vial, J.-P.: Theory and Algorithms for Linear Optimization: An Interior Point Approach. Springer, New York (2005)
Acknowledgements
The authors would like to thank the research councils of Shiraz University of Technology, K.N. Toosi University of Technology and York University for supporting this work. The authors would also like to thank M. Ataei for his helpful comments on the paper.
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Communicated by Yurii Nesterov.
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Fathi-Hafshejani, S., Fakharzadeh Jahromi, A., Peyghami, M.R. et al. Complexity of Interior Point Methods for a Class of Linear Complementarity Problems Using a Kernel Function with Trigonometric Growth Term. J Optim Theory Appl 178, 935–949 (2018). https://doi.org/10.1007/s10957-018-1344-z
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DOI: https://doi.org/10.1007/s10957-018-1344-z