# Is this following SDP problem Convex?

Is the following problem convex function?

\begin{eqnarray} \begin{aligned} \underset{\mathbf{X}} {\text{minimize}}\,\,\,\, & \text{Trace}(\mathbf{RX}) \\ \text{subject to} &\\ & \text{Trace}(\mathbf{Q_{k}X}) = {0}\,\, k = 1,2...N\\ &\mathbf{AX} = \mathbf{0} \\ & \text{Trace}(\mathbf{X}) = {1} \\ & X\succeq 0,\,\, X \in \mathbb{R}^{M \times M}. \end{aligned} \end{eqnarray} Where, $\mathbf{R} \in \mathbb{C}^{M \times M}$ is a positive definite symmetric matrix.

and

$\mathbf{Q_{k}} \in \mathbb{R}^{M \times M}\,\, k = 1,2,...N$ are the symmetric Toeplitz matrices, like the matrices given below \begin{eqnarray} \begin{aligned} Q_1 &= \left[\begin{array}{cccccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array} \right]\\ \end{aligned} \end{eqnarray} and \begin{eqnarray} \begin{aligned} Q_2 &=\left[\begin{array}{cccccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array} \right] \end{aligned} \end{eqnarray} In General, $Q_k$ = Toep( [0 0 0... 1 ... 0\, 0] ), Where $1$ is at $k^{th}$ position.

and

$\mathbf{A} \in \mathbb{R}^{V \times M}$, $V < \frac{M}{2}$.

Note that $trace(A X)$ is just a linear function. So you're minimising a linear function subject to a bunch of linear equality constraints and a positive semidefiniteness constraint.
The text passage contains a mathematical problem related to convex optimization, specifically asking whether a given semidefinite programming (SDP) problem is convex. The problem is described in detail, including the objective function, constraints, and the properties of the matrices involved. No answer to the question of convexity is provided in the text.
single_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Is this following SDP problem Convex?

Is the following problem convex function?

\begin{eqnarray} \begin{aligned} \underset{\mathbf{X}} {\text{minimize}}\,\,\,\, & \text{Trace}(\mathbf{RX}) \\ \text{subject to} &\\ & \text{Trace}(\mathbf{Q_{k}X}) = {0}\,\, k = 1,2...N\\ &\mathbf{AX} = \mathbf{0} \\ & \text{Trace}(\mathbf{X}) = {1} \\ & X\succeq 0,\,\, X \in \mathbb{R}^{M \times M}. \end{aligned} \end{eqnarray} Where, $\mathbf{R} \in \mathbb{C}^{M \times M}$ is a positive definite symmetric matrix.

and

$\mathbf{Q_{k}} \in \mathbb{R}^{M \times M}\,\, k = 1,2,...N$ are the symmetric Toeplitz matrices, like the matrices given below \begin{eqnarray} \begin{aligned} Q_1 &= \left[\begin{array}{cccccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array} \right]\\ \end{aligned} \end{eqnarray} and \begin{eqnarray} \begin{aligned} Q_2 &=\left[\begin{array}{cccccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array} \right] \end{aligned} \end{eqnarray} In General, $Q_k$ = Toep( [0 0 0... 1 ... 0\, 0] ), Where $1$ is at $k^{th}$ position.

and

$\mathbf{A} \in \mathbb{R}^{V \times M}$, $V < \frac{M}{2}$.

Note that $trace(A X)$ is just a linear function. So you're minimising a linear function subject to a bunch of linear equality constraints and a positive semidefiniteness constraint.
user
The text passage contains a mathematical problem related to convex optimization, specifically asking whether a given semidefinite programming (SDP) problem is convex. The problem is described in detail, including the objective function, constraints, and the properties of the matrices involved. No answer to the question of convexity is provided in the text.

TYPE: SINGLE_QUESTION