Hi Mark,

Thanks for getting back to me. I was able to get some results with the combination of YALMIP + BNIB. In some cases, there were numerical warnings but in the end, it worked out. However, I would like to have some theoretical guarantees.

And so I am trying another approach based on relaxing the problem to SDP. I am able to reformulate the problem into the following form:

\begin{equation}
\begin{aligned}
\min_{\mathbf{X}\in \mathbb{R}^{3 \times 3}} \quad & \mathrm{tr}(\mathbf{A}\mathbf{X}) \\
\text{s.t.} \quad & \mathrm{tr}(\mathbf{B}_{0} \mathbf{X}) = 1 \\
\quad & \mathrm{tr}(\mathbf{B}_{i} \mathbf{X}) = 0, \quad i \in \{ 1,2,3 \} \\
\quad & \mathbf{X} \succeq 0 \\
\quad & \mathbf{J} \succeq 0
\end{aligned}
\end{equation}

\begin{align*}
\mathbf{A} = \begin{bmatrix}
0 & 0.5 & 0 & 0\\
0.5 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}, ~
\mathbf{B}_{0} \begin{bmatrix}
1 & 0 & 0 & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}, ~
\mathbf{B}_{1} = \begin{bmatrix}
0 & 0 & 1 & 0\\
0 & -2 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0
\end{bmatrix}
\end{align*}

\begin{align*}
\mathbf{B}_{2} = \begin{bmatrix}
0 & 0 & 0 & 1\\
0 & 0 & -1 & 0 \\
0 & -1 & 0 & 0 \\
1 & 0 & 0 & 0
\end{bmatrix}
\mathbf{B}_{3} = \begin{bmatrix}
0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 \\
0 & 0 & -2 & 0 \\
0 & 1 & 0 & 0
\end{bmatrix}
\end{align*}

J = \begin{bmatrix}
12 \boldsymbol{Q} \, \mathbf{X}(1,2) & 6 \boldsymbol{Q} \, \mathbf{X}(1,1) \\
6 \boldsymbol{Q} \, \mathbf{X}(1,1) & 6 \boldsymbol{Q} \, \mathbf{X}(0,1) \\
\end{bmatrix}

I drop the rank constraint \textrm{rank}(\mathbf{X}) = 1. I am following the idea of sum-of-squares relaxation (https://www.princeton.edu/~aaa/Public/Teaching/ORF523/ORF523_Lec15.pdf) approach to obtain the SDP relaxation. The additional constraints (\mathbf{B}_i, i=(1,2,3)) are necessary for tightness (to *potentitally* obtain rank-1 solution).

However, I am getting a new error this time while solving using MOSEK solver:

CVXPY) May 10 04:44:41 PM: Optimizer terminated. Time: 0.01

(CVXPY) May 10 04:44:41 PM:

(CVXPY) May 10 04:44:41 PM:

(CVXPY) May 10 04:44:41 PM: Interior-point solution summary

(CVXPY) May 10 04:44:41 PM: Problem status : UNKNOWN

(CVXPY) May 10 04:44:41 PM: Solution status : UNKNOWN

(CVXPY) May 10 04:44:41 PM: Primal. obj: 5.5498901606e-01 nrm: 1e+00

I am new to CVXPY with MOSEK and don’t quite know how to go about debugging this issue.

Also, I am not sure if I should start a different thread for this since I reformulated the problem and the solution proposed by Mark works with the original formulation. If so the case, I can start a new thread.