Change a constrin to a LMI

[mas tab]

Hi,

Can we change the following constrain to a LMI(linear matrix inequality) problem(realtion)?

(X~)X=I

X is a mxn complex matrix ; X~:transposed conjugated of X; I is nxn Identity matrix;

[Johan Löfberg]

As I said before, quadratic equality represent a non-convex set, and is thus not something you can reformulate to an LMI

[mas tab]

And if X has a specific form(tall) as:

X=[uS;V](I+S)^(-1/2)

1)u and V are hermitian(u*(u~)=I)

2)S=diag(s1,...,sn); si>=0 and si is real.

[Johan Löfberg]

You seem to fail to appreciate the fact that immediately when you introduce a quadratic equality, you leave the field of convex programming.

Consider the absolutely most simple case possible. Let x be real scalar and look at x*x == 1. What is the feasible set? Is that a convex set? Now look at x*x + y*y == 1. What is it geometrically? Convex?