Interior-point methods for Cartesian P_*(κ)-linear complementarity problems over symmetric cones based on the eligible kernel functions

Abstract

An interior-point method (IPM) for Cartesian P _*(κ)- linear complementarity problems over symmetric cones (SCLCP) is analysed and the complexity results are presented. The Cartesian P _*(κ)- SCLCPs have been recently introduced as the generalization of the more commonly known and more widely used monotone-SCLCPs. The IPM is based on the barrier functions that are defined by a large class of univariate functions called eligible kernel functions, which have recently been successfully used to design new IPMs for various optimization problems. Eligible barrier (kernel) functions are used in calculating the Nesterov–Todd search directions and the default step-size which lead to very good complexity results for the method. For some specific eligible kernel functions, we match the best-known iteration bound for the long-step methods while for the short-step methods the best iteration bound is matched for all cases.

Keywords:

• linear complementarity problem
• Euclidean Jordan algebras and symmetric cones
• Cartesian P _*(κ) property
• barrier and kernel functions
• interior-point method
• polynomial complexity

AMS Subject Classification :

• 90C33
• 90C51

Acknowledgements

The research of G.Q. Wang was supported by the National Natural Science Foundation of China (No. 11001169) and China Postdoctoral Science Foundation (No. 20100480604) and that of D.T. Zhu was supported by the National Natural Science Foundation of China (No. 10871130), the Ph.D. Foundation Grant (No. 20093127110005) and the Shanghai Leading Academic Discipline Project (No. T0401).

Notes

^†This paper is dedicated to Professor Florian A. Potra on the occasion of his 60th birthday.