Two wide neighborhood interior-point methods for symmetric cone optimization

In this paper, we present two primal–dual interior‐point algorithms for symmetric cone optimization problems. The algorithms produce a sequence of iterates in the wide neighborhood $\mathcal{N}(\mathit{\tau },\mathit{\beta })$ of the central path. The convergence is shown for a commutative class of search directions, which includes the Nesterov–Todd direction and the xs and sx directions. We derive that these two path‐following algorithms have $\begin{array}{c}\hfill \text{O}\left(\sqrt{r\text{cond}(G)}log{\mathit{ϵ}}^{‐1}\right),\text{O}\left(\sqrt{r}{\left(\text{cond}(G)\right)}^{1/4}log{\mathit{ϵ}}^{‐1}\right)\end{array}$ 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.