## A convergence analysis of the scaling-invariant primal-dual path-following algorithms for second-order cone programming.(English)Zbl 0957.90129

Summary: In this paper we study primal-dual path-following algorithms for Second-Order Cone Programming problems (SOCP). We extend the standard long-step/semilong-step/short-step primal-dual path-following alogorithms for LP and SDP to SOCP, and prove that the long-step algorithm using the NT direction and the HRVW/KSH/M direction have $$O(n \log \varepsilon^{-1})$$ iteration-complexity and $$O(n^{3/2} \log \varepsilon^{-1})$$ iteration-complexity, respectively, to reduce the duality gap by a factor of $$1/\varepsilon$$, where $$n$$ is the number of the second-order cones. We also show that the short and semilong-step algorithms using the NT direction and the HRVW/KSH/M direction have $$O(\sqrt{n} \log \varepsilon^{-1})$$ and $$O(n \log \varepsilon^{-1})$$ iteration-complexities, respectively.

### MSC:

 90C51 Interior-point methods 90C30 Nonlinear programming
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### References:

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