An arc-search infeasible interior-point method for semidefinite optimization with the negative infinity neighborhood

Abstract

We present an arc-search infeasible interior-point algorithm for semidefinite optimization using the Nesterov-Todd search directions. The algorithm is based on the negative infinity neighborhood of the central path. The algorithm searches an ε-approximate solution of the problem along the ellipsoidal approximation of the entire central path. The convergence analysis of the algorithm is presented and shows that the algorithm has the iteration complexity bound $\mathcal {O}\big (n^{3/2}\log {\varepsilon }^{-1}\big )$. Here, n is the dimension of the problem and ε is the required precision. The numerical results show that our algorithm is efficient and promising.

Appendix

In this section, we describe the details of the test problems in Section 4 [24].

• 1. Max-cut problem (Mc):

  $$ \begin{array}{@{}rcl@{}} \begin{array}{ccc} & \multicolumn{2}{l}{ \min~\text{Tr}(LX )} \\ \qquad \qquad&& s.t.~~ \text{diag}(X)=\frac{e}{4} \quad\\ \qquad \qquad\qquad&& X\succeq 0, \quad \end{array} \end{array} $$

  where L = B −Diag(Be) and B is the weighted adjacency matrix of a graph [9] and e is a vector of all ones.

• 2. Norm-min problem (NM):

  $$ \min_{x\in\mathbb{R}^{m}} \big\|B_{0}+x_{1}B_{1}+\cdots+x_{m}B_{m}\big\|_{2}, $$

  where B_k, for k = 0, 1,⋯ ,m is an p × q matrix. For this problem, we consider p = q = n and randomly generate the matrices as B_k = rand(n) for k = 0, 1,⋯ ,m.

• 3. Control problem(C):

  $$ \begin{array}{@{}rcl@{}} \begin{array}{ccc} & \multicolumn{2}{l}{ \max_{(P,t)} t} \\ \qquad \qquad&& s.t.~ -\text{Tr}((B_{k})^{T} P)-\text{Tr}(P(B_{k}))\succeq 0\quad k=1, \cdots, L \\ \qquad \qquad&& I-P\succeq 0, \ P-tI\succeq 0,\ P=P^{T},~~~~~~~ \end{array} \end{array} $$

  where B_k, for k = 0, 1,⋯ ,L, are the square real matrices n dimension. For this problem, we generate the matrices as B_k = rand(n), for k = 0, 1,⋯ ,m.

• 4. Graph partitioning problem (Gp):

  $$ \begin{array}{ccc} & \multicolumn{2}{l}{ \min~\text{Tr}(CX)} \\ \qquad \qquad&& s.t.~\text{Tr}((ee^{T})X)=1~~~~~~~~~~~~~~~~ \\ \qquad \qquad&& X_{ii}=1\quad i=1, \cdots, n. \end{array} $$

  where C = −(diag(Be) − B) and B is the weighted adjacency matrix of a graph with n nodes [9] and e is a vector of all ones.

• 5. Lovasz theta number problem (Ltn):

  $$ \begin{array}{ccc} & \multicolumn{2}{l}{\min~\text{Tr}(CX)} \\ \qquad \qquad&& s.t.~\text{Tr}(X)=1~~~~~~~~~~~~~~~~~~~~~ \\ \qquad \qquad&& ~~~~ \text{Tr}(A_{ij}X)=0\quad (i, j)\in E\\ \qquad \qquad&& X\succeq 0. \end{array} $$

  where:

  • C is a matrix of all minus ones.

  • B is the adjacency matrix of the given graph [9].

  • E is the set of edges of the given graph.

  • $A_{ij}=(e_{i}{e_{j}^{T}}+e_{j}{e_{i}^{T}})/\sqrt {2}$.

  • e_i denotes the i th column of B.