text: # 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.
explanation: The document presents a mathematical optimization problem and asks whether the given problem is a convex function. The problem is formulated as a semidefinite program (SDP) with a linear objective function and several linear and positive semidefinite constraints. The document also provides examples of the matrices involved and additional notes on the nature of the constraints.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # 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.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document presents a mathematical optimization problem and asks whether the given problem is a convex function. The problem is formulated as a semidefinite program (SDP) with a linear objective function and several linear and positive semidefinite constraints. The document also provides examples of the matrices involved and additional notes on the nature of the constraints.
  
  Label: question