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I have the optimization program

\begin{align} \min_{x \in \mathbb{R}^n, Q \in \mathbb{S}^n} &\bigg(\frac{1}{2}\Vert x \Vert ^2 + b^T x\bigg)\\ \text{ subject to }&\text{tr}(Q \cdot A_1) + a_1^Tx = c_1,\\ &\text{tr}(Q \cdot A_2) + a_2^Tx = c_2,\\ &\qquad\qquad\vdots \\ &\text{tr}(Q \cdot A_m) + a_n^Tx = c_n.\\ &Q \succeq 0 \end{align}

Is there any way I can turn this into a primal semidefinite program?

As an attempt I have replaced the quadratic term in the objective with $\theta$ and then added the constraint $\frac{1}{2}\Vert x \Vert ^2 \leq \theta$. However I'm not sure how to deal with inequality constraints in an SDP. Thanks.

Rodrigo de Azevedo
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Rishi
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  • Take a look at this. – Rodrigo de Azevedo Mar 24 '21 at 14:48
  • @RodrigodeAzevedo I've had a look at your post. However my problem also has a semidefinite constraint at the end. – Rishi Mar 24 '21 at 14:52
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    Introduce optimization variable $y$ and try to transform the objective into an inequality that is LMI-representable. – Rodrigo de Azevedo Mar 24 '21 at 15:04
  • @RodrigodeAzevedo Cheers - got it. – Rishi Mar 24 '21 at 16:17
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    Just a slight warning: If your task simply is to solve this problem using some available solver, you would not rewrite the quadratic terms to semidefinite constraints, but write it using second-order cones if the solver supports that (most SDP solvers do). And if you use a modelling layer (which you should if you just want to solve it), you never do these reformulations at all, as the modelling language takes care of everything. – Johan Löfberg Mar 26 '21 at 11:07
  • @JohanLöfberg I see, but how do I write the condition that the matrix is positive semidefinite as a quadratic inequality? – Rishi Mar 26 '21 at 13:24
  • @Rishi You don't. Why would you do that? – Rodrigo de Azevedo Mar 26 '21 at 13:26
  • To write it as a second order cone program wouldn't you need to have all constraints as a quadratic inequality? Of the form $\Vert Ax + b \Vert^2 \leq c^T x +d $ – Rishi Mar 26 '21 at 13:28
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    I'm talking about solvers supporting both semidefinite and second-order cones. – Johan Löfberg Mar 26 '21 at 13:30
  • @JohanLöfberg Oh ok I got you. I'll have a look at that – Rishi Mar 26 '21 at 13:31
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    What is your goal? If your goal is to obtain a numerical solution, then something like CVXPY can take your problem as it is and solve it. No need to rewrite in some delicate form. – Rodrigo de Azevedo Mar 26 '21 at 13:31
  • @RodrigodeAzevedo this is for a thesis - so more on the theory side. But I will have a look at these SDP solvers when I start to implement my model. – Rishi Mar 26 '21 at 14:09
  • Note that $Q \succeq 0$ is a very succinct way of writing $2^n - 1$ inequality constraints. Only a few of those are quadratic. – Rodrigo de Azevedo Mar 26 '21 at 14:20

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