Semdefinite constraint with cubic entry

Nonconvex
1

Hi all,
I am trying to solve the following optimization problem:

image

where Q \in \mathbb{S}^2_{++} is a known constant matrix and K is another known constant matrix. The optimization variable x \in \mathbb{R}_{+} is nonnegative. I started by setting K = \mathbf{0} i.e. a zero matrix. In this case, there is a trivial solution x = 0, however, the presence of the cubic and quadratic terms seems to violate the Disciplined Convex Programming rules. Since x is nonnegative, the constraint should be a convex constraint. Is there a way to formulate this problem in a manner that is consistent with DCP rules?

I am reading up on similar posts here to see if power cones would be applicable. Any suggestions would be appreciated. Thanks.

2

That is a Polynomial Matrix Inequality, which can be lifted to a Bilinear Matrix Inequality (BMI), and is non-convex, unless there is some transformation I haven’t considered. If I am overlooking something, other people should feel free to provide an appropriate reformulation.

You can try using YALMIP (which I believe will automatically reformulate this as a BMI) on this with either a local solver or the BMIBNB global solver. It is only a one variable problem, so perhaps it is rather easy to solve to global optimality.

As for convexity, power cones, etc, keep in kind this is an SDP constraints, so those other posts are not applicable.