# Translating a nuclear norm constraint to an LMI constraint

I'm attempting to solve a convex optimization problem where one of the constraints is

$$\|M\|_* \leq a$$

where $\|M\|_*$ denotes the nuclear norm of matrix $M$. I'm using CVXOPT in Python to solve this as a linear program (LP), but the nuclear norm (sum of singular values) isn't a nice linear constraint. How can I formulate this?

• Your link is not really a linear-programming approach but still a more generalized optimization of some convex problem (therefore IPM-based). I also don't think there is nice-working approach (in practice) to do this within linear-programming. What is the motivation? Bad performance because of the SDP-formulation? Or other reasons? If it's about bad performance, you can always try the SCS solver. (side-note: i think cvxpy is much nicer to use than cvxopt; but maybe you need the more accessible customization possible within cvxopt) – sascha Apr 21 '17 at 16:40
• We're attempting to implement Algorithm 1 from this paper. I haven't done a lot with convex optimization, so my apologies if I'm not thinking about this the right way. Everything else is a linear constraint, so I was hoping to fit it within that framework. – Pterosaur Apr 21 '17 at 17:12
• This work is not really ready for large-scale, at least without complex customized optimization algorithms. I think it's easy to implement using cvxpy and you can solve it with ecos or scs. But don't expect much performance. But the paper is more about the theory and ignores the real-world performance it seems. (It's very very far away from LP; as i indicated above; and the convex, but non-linear constraint is the core of this approach) – sascha Apr 21 '17 at 17:57
• We're not too worried about optimization here; this is a course project. Now it seems like we can get the nuclear norm in the constraints but can't figure out how to set the objective function as the inner product within cvxpy. – Pterosaur Apr 21 '17 at 18:15
• Is it not just trace(AB^t). Well... cvxpy's docs are good and there is also the issue-tracker on github with some questions already answered. – sascha Apr 21 '17 at 18:36

The nuclear norm inequality $$\|X\|_*\leq y$$ is satisfied if and only if there exist symmetric matrices $$W_1$$, $$W_2$$ satisfying
$$\begin{bmatrix} W_1 & X \\ X^T & W_2 \end{bmatrix} \succeq 0, ~ \mathop{\textrm{Tr}}W_1 + \mathop{\textrm{Tr}}W_2 \leq 2 y$$
Here, $$\succeq 0$$ should be interpreted to mean that the $$2\times 2$$ block matrix is positive semidefinite. Because of this transformation, you can handle nuclear norm minimization or upper bounds on the nuclear norm in any semidefinite programming setting.