A unified kernel function approach to polynomial interior - point algorithms for the Cartesian P ∗ ( κ ) - SCLCP ∗

Recently, Bai et al. [Bai Y.Q., Ghami M. El, Roos C., 2004. A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization. SIAM Journal on Optimization, 15(1), 101-128.] provided a unified approach and comprehensive treatment of interior-point methods for linear optimization based on the class of eligible kernel functions. In this paper we generalize the analysis presented in the above paper to the Cartesian P∗(κ)-linear complementarity problem over symmetric cones via the machinery of the Euclidean Jordan algebras. The symmetry of the resulting search directions is forced by using the Nesterov-Todd scaling scheme. The iteration bounds for the algorithms are performed in a systematic scheme, which highly depend on the choice of the eligible kernel functions. Moreover, we derive the iteration bounds that match the currently best known iteration bounds for largeand small-update methods, namely O((1 + 2κ) √ r log r log r ε ) and O((1 + 2κ) √ r log r ε ), respectively, where r denotes the rank of the associated Euclidean Jordan algebra and ε the desired accuracy.