Actually, I believe this will work:

```
cvx_begin sdp
variables x1 x2 x3 x4 y z
x4 + y^2 <= z
[ x1, x2 ; x3, z ] <= 0
```

You’ll want to confirm this for yourself. That is, *prove*, that if x_1,x_2,x_3,x_4,y satisfy your nonlinear LMI, then there must exist a z that satisfies these LMIs; and likewise, if there is a x_1,x_2,x_3,x_4,y,z satisfies these LMIs, then the nonlinear LMI is also satisfied.

It seems rather evident to me, but my grade/research/publication isn’t depending on it! So do verify this.

EDIT: Actually, it’s not difficult to prove. For a symmetric matrix X, \lambda_{\max}(X) is convex in the elements of X; and it is also *non-decreasing* in the *diagonal* elements of X. Therefore, standard composition rules of convex analysis say that convexity is preserved if the diagonal elements are *convex* functions of underlying variables.

So your nonlinear LMI is definitely convex, and the approach I’ve outlined here will indeed give you the results you seek. Of course, in *general*, nonlinear LMIs cannot be handled by CVX; but in the specific case where one or more of the diagonal elements is a convex expression, you can handle it as I have outlined here.