[y z;z 1]==0 constrains all elements of the matrix to zero, which is not what you want, and is infeasible because 1 \ne 0.

You can formulate the first constraint in CVX using rotated second order cones, using rotated_lorentz , or less efficiently, an LMI as you have done.

As for the second constraint, that could also be handled by a rotated second order cone constraint, this time in terms of change of variable to y^2 rather than y, but that would result in a non-convexity when applied to one of the constraints in the last row.

I’ll defer to someone else to further advise on this, but I think the whole problem may be non-convex, unless relaxed, which will then be a different problem, and I think is basically where you confronted difficulty.