\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
In this paper,we present two primalï¿½??dual interior-point algorithms for symmetric
cone optimization problems. The algorithms produce a sequence of iterates in
the wide neighborhood N(ï¿½?, Î²) of the central path. The convergence is shown for a
commutative class of search directions, which includes the Nesterovï¿½??Todd direction
and the xs and sx directions.We derive that these two path-following algorithms have
\[\mbox{O}\br{\sqrt{r\cond(G)}\log \varepsilon^{-1}},
\mbox{O}\br{\sqrt{r}\br{\cond(G)}^{1/4}\log \varepsilon^{-1}}\]
iteration complexity bounds, respectively. The obtained complexity bounds are the best
result in regard to the iteration complexity bound in the context of the path-following
methods for symmetric cone optimization. Numerical results show that the algorithms
are efficient for this kind of problems.
\end{document}