A primal-dual interior-point algorithm with arc-search for semidefinite programming

Abstract

We propose a wide neighborhood primal-dual interior-point algorithm with arc-search for semidefinite programming. In every iteration, the algorithm constructs an ellipse and searches an $\varepsilon$-approximate solution of the problem along the ellipsoidal approximation of the central path. Assuming a strictly feasible starting point is available, we show that the algorithm has the iteration complexity bound $O\left( n^{\frac{3}{4}}\log \frac{{X^{0}}\bullet {S}^{0}}{\varepsilon }\right)$ for the Nesterov–Todd direction, which is similar to that of the corresponding algorithm for linear programming. The numerical results show that our algorithm is efficient and promising.

Appendix

The following results introduced in [24, 28] are used during the analysis.

Lemma A.1

([28]) Let $\lambda _{1}$ be the smallest eigenvalue of the matrix ${\hat{X}}{\hat{S}}$. Then for any $P\in {\mathcal {P}}(X,S)$, one has

$$\begin{aligned} \rho (({\hat{E}}{\hat{F}})^{-1})=\frac{1}{4\lambda _{1}}. \end{aligned}$$

Lemma A.2

([24]) For any $u,v\in {\mathbb {R}}^{n}$ and $G\in {\mathcal {S}}^{n}_{++},$ we have

$$\begin{aligned} ||u||~||v||\le \sqrt{\textit{cond}(G)}||G^{-\frac{1}{2}}u||~||G^{\frac{1}{2}}v||\le \frac{\sqrt{\textit{cond}(G)}}{2}(||G^{-\frac{1}{2}}u||^{2}+||G^{\frac{1}{2}}v||^{2}). \end{aligned}$$

Lemma A.3

([24]) Let $P\in {\mathcal {P}}(X,S)$ be given. Then

$$\begin{aligned} ||({\hat{F}}{\hat{E}})^{-\frac{1}{2}}\mathrm {vec}(\sigma \mu {I}-H({\hat{X}}{\hat{S}}))||^{2}\le \left( 1-2\sigma +\frac{\sigma ^{2}}{\gamma }\right) n\mu . \end{aligned}$$

Lemma A.4

([24]) Suppose that $(X,y,S)\in {\mathcal {S}}^{n}_{++}\times {\mathbb {R}}^{m}\times {\mathcal {S}}^{n}_{++}$, $P\in {\mathcal {S}}^{n}_{++}$, and $Q\in {\mathcal {P}}(X,S)$. Then

$$\begin{aligned} \lambda _{\min }[H_{P}(XS)]\le \lambda _{\min }[XS]=\lambda _{\min }[H_{Q}(XS)]. \end{aligned}$$