Quadratically constrained convex optimization

Question

I have tried to convert the following constraint into SOCP form but can't figure it out. My issue is that whenever I create the norm, the other side has to be square rooted which invalidates the SOCP form.

The directional parameters ${\boldsymbol w}\in\Bbb R^d$ satisfies the following shaping constraint: $$\boldsymbol w^\top\boldsymbol\Sigma\boldsymbol w+\boldsymbol c^\top\boldsymbol w\le1,$$ where $\boldsymbol\Sigma\in\Bbb R^{d\times d}$ is a given symmetric, positive definite matrix and $\boldsymbol c\in\Bbb R^d$ is a given vector.

Answer 1

For all things convex, Mosek's modeling cookbook and conic modeling cheatsheet have you covered.

In your case, see the section on convex quadratic sets.

Answer 2

Since $\boldsymbol\Sigma$ is PSD, there exists $\boldsymbol Q$ s.t. $\boldsymbol Q^\top\boldsymbol Q=\boldsymbol \Sigma$ (by Cholesky, or eigendecomposition).

Then $\boldsymbol w^\top\boldsymbol \Sigma\boldsymbol w =\boldsymbol w^\top\boldsymbol Q^\top\boldsymbol Q\boldsymbol w = \|\boldsymbol Q\boldsymbol w\|_2^2.$

Then introduce new variable $y$ as epigraph of $\|\boldsymbol Q\boldsymbol w\|$ as follows:

\begin{align}y+\boldsymbol c^\top\boldsymbol w&=1\tag{linear constraint}\\\|\boldsymbol Q\boldsymbol w\|_2^2&\le y\tag{rotated quadratic cone constraint}\end{align}

As @mtanneau posted, these modeling tricks(?) are well-explained in the MOSEK tutorial.

If you use MOSEK, you can just directly use rotated quadratic cone to model second constraint. If you want to convert into second order conic constraints,

\begin{align}&\|\boldsymbol{Qw}\|_{2}^{2} \leq y \\\iff&(\boldsymbol{Qw})^\intercal (\boldsymbol{Qw}) \leq y \\ \iff& (\boldsymbol{Qw})^\intercal (\boldsymbol{Qw})+\frac{(y-1)^2}{4} \leq \frac{(y+1)^2}{4}\\ \iff& \left\|\begin{pmatrix}\boldsymbol{Qw}\\ \frac{y-1}{2} \end{pmatrix}\right\|_{2} \leq \frac{y+1}{2}\end{align}

Hope this is what you're looking for.
*Converting quadratic constraint into soc constraint is well-known tricks, you can find in https://www2.isye.gatech.edu/~nemirovs/LMCOLN2021WithSol.pdf