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I'm reading through this paper and on page 8 (PDF page 9), Step 2 in Algorithm 1 reads

$$ W \leftarrow \text{argmin} \{\text{tr}(WQW^T) + \alpha\, \text{tr}(Q^{−1}): Q \succeq \epsilon\, I_n , Q_{\text{off}} \leq 0, Q \,\mathbb{1}_n = \epsilon\, \mathbb{1}_n \} $$

where $Q$ is symmetric and $W$ is a $d\times n$ matrix. The following paragraphs claim this is an SDP,

since $\operatorname{tr} (Q^{-1})\leq\beta$ can be rewritten as $$ \text{tr}(R)\leq\beta, \qquad \begin{bmatrix}R & I_n \\ I_n & Q\end{bmatrix} \succeq 0.$$

How are the two conditions equivalent? More importantly, how would this be implemented in an SDP solver like CVX?

Rodrigo de Azevedo
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BranchedOut
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1 Answers1

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You introduce a new variable $R$ for which it holds $R \succeq Q^{-1}$ from which you know $\text{tr}(R) \geq \text{tr}(Q^{-1})$. If you now add a cost in the objective of $\text{tr}(R)$, at optimality it would be optimal to select $R=Q^{-1}$ (otherwise you could decrease the objective by making $R$ smaller). The inequality $R \succeq Q^{-1}$ is then converted to an LMI using a Schur complement.

The reformulation introducing yet another variable $\beta$ to upper bound $\text{tr}(R)$ is just silly and adds nothing. Simply use $\text{tr}(R)$ in the objective.

The model is then an absolutely bog standard LMI and would be immediately implemented in CVX or YALMIP or your language of choice.

Johan Löfberg
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    To fill in the details re: Schur complement a little: Since $Q\succeq \epsilon I_n\succ 0$, the Schur complement condition tells us that $X=\begin{bmatrix}R & I_n \ I_n & Q\end{bmatrix} \succeq 0$ if and only if $X/Q=R-Q^{-1}\succeq 0$ i.e. $R\succeq Q^{-1}$. – Semiclassical Aug 19 '19 at 18:13