# LMI-constrained least-squares problem in Mosek

I want to solve a least-squares problem of the form:

$$\begin{array}{ll} \text{minimize} & \|Ax-b\|_2^2\\ \text{subject to} & \mathcal{L}(x)\succeq0\end{array}$$

with $$\mathcal{L} : \mathbb{R}^n \to \mathbb{R}^{m \times m}$$ being a linear operator.

This paper claims that they used Mosek to solve a problem of this form. To my best knowledge, the Mosek documentation on semidefinite programming does only include examples with linear objectives.

How do I need to formulate the problem described above to solve it with Mosek?

• Mosek handles quadratic objectives with linear constraints (and certain varieties of quadratic constraints). – anomaly Nov 16 '19 at 20:27
• Do you mean that this problem is not solvable with Mosek? – Felix Crazzolara Nov 16 '19 at 20:28
• No, I mean that the problem you describe is exactly what mosek handles. – anomaly Nov 16 '19 at 20:29
• Could you please link to the corresponding page in the documentation or an example. I don't understand how this is equivalent to quadratic or linear constraints. – Felix Crazzolara Nov 16 '19 at 20:33
• Please let me know if you agree with my edits. – Rodrigo de Azevedo Nov 17 '19 at 9:43

You first write the problem as minimization of a new variable $$t$$ with the constraints $$\|Ax-b\|_2^2\leq t$$ (and all your other constraints). The quadratic constraints can then be written using a second-order cone constraint as $$\left|\left|\begin{matrix}1-t\\2(Ax-b)\end{matrix}\right|\right|\leq 1+t$$. At that point, you have a mixed second-order and semidefinite cone program.