A path following interior-point method for linear complementarity problems over circular cones
- 90 Downloads
Circular cones are a new class of regular cones that include the well-known second-order cones as a special case. In this paper, we study the algebraic structure of the circular cone and show that based on the standard inner product this cone is nonsymmetric while using the new-defined circular inner product this cone not only is symmetric but also the algebra associated with it, is a Euclidean Jordan algebra. Then, using the machinery of Euclidean Jordan algebras and the Nestrov–Todd search directions, we propose a primal-dual path-following interior-point algorithm for linear complementarity problems over the Cartesian product of the circular cones. The convergence analysis of the algorithm is shown and it is proved that this class of mathematical problems is polynomial-time solvable.
KeywordsLinear complementarity problem Circular cone Interior-point methods Polynomial complexity
Mathematics Subject Classification90C25 90C51
The authors would like to thank the anonymous referees for their useful comments and suggestions, which helped to improve the presentation of this paper. The authors also wish to thank Shahrekord University for financial support. The authors were also partially supported by the Center of Excellence for Mathematics, University of Shahrekord, Shahrekord, Iran.
- 11.Darvay, Z., Takács P.R.: New interior-point algorithm for symmetric optimization based on a positive-asympotic barrier function. Technical Report Operations research Report 2016-01, Eotvos Loránd University of Scincs, Budapest (2016)Google Scholar
- 20.Ma, P., Bai, Y., Chen, J.S.: A self-concordant interior point algorithm for nonsymmetric circular cone programming. Nonlinear Convex Anal. http://math.ntnu.edu.tw/~jschen/Papers/IPM-CCP.pdf
- 23.Potra, F.: An infeasible interior point method for linear complementarity problems over symmetric cones. In: Simos, T. (ed.) Proceedings of the 7th International Conference of Numerical Analysis and Applied Mathematics, Rethymno, Crete, 18–22 September 2009, pp. 1403–1406. Am. Inst. of Phys., New York (2009)Google Scholar
- 25.Skajaa, A., Ye, Y.: Homogeneous interior-point algorithm for nonsymmetric convex conic optimization. Math. Program. (2014). http://link.springer.com/journal/10107/onlineFirst/page/1
- 27.Takács, P.R., Darvay, Z.: Infeasible interior-point algorithm for symmetric optimization based on a positive-asympotic barrier function. Technical Report Operations research Report 2016-03, Eotvos Loránd University ofScincs, Budapest (2016)Google Scholar