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An arc-search infeasible-interior-point method for symmetric optimization in a wide neighborhood of the central path. (English) Zbl 06737238
Summary: In this paper, we propose a new arc-search infeasible-interior-point method for symmetric optimization using a wide neighborhood of the central path. The proposed algorithm searches for optimizers along the ellipses that approximate 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. The complexity bound is $$\mathcal {O}(r^{5/4}\log \varepsilon ^{-1})$$ for the Nesterov-Todd direction, and $$\mathcal {O}(r^{7/4}\log \varepsilon ^{-1})$$ for the $$xs$$ and $$sx$$ directions, where $$r$$ is the rank of the associated Euclidean Jordan algebra and $$\varepsilon$$ is the required precision. The obtained complexity bounds coincide with the currently best known theoretical complexity bounds for the short step path-following algorithm. Some limited encouraging computational results are reported.

##### MSC:
 90C Mathematical programming
Netlib; SDPLIB
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