Schur complement

1

Hi guys,
I am having an issue writing the Schur complement in CVX that keeps telling me “Disciplined convex programming error”. Here is my problem, I have the following matrix [X x; x’ 1] which is positive semidefinite. Note that X is a K \times K matrix and x column vector of size K. If I write
[[X x];[x’ 1]] == semidefinite(K+1), it works fine. For some reasons, I need to write it as
two constraints, i.e. X positive semidefinite and X-xx’ positive semidefinite. So when I write
X== semidefinite(K) and X-xx’ == semidefinite(K)
CVX says that “Disciplined convex programming error”. I am almost certain that the problem is with the product "xx’ ". Does someone have a solution to write "xx’ " in CVX?

2

You simply cannot. You will have to use the first approach. Why do you think you “need” to write it as two equations.

3

@mcg I am solving an optimization problem using SDR approach. I have provided an optimality condition (which permits to have a rank one solution) where the theoretical analysis was mainly based on the rank of the Lagrangian multipliers associated with the constraint X-x*x’. So I need to see if empirically, it is really true.
Thanks for replying anyway.
Cheers,

4

That doesn’t explain why you need the two constraints together. The first one is exactly equivalent to the two-constraint form, only it is actually convex and in a form CVX can handle.

5

Hi, can you help me?
I have a problem. when i write the picture to matlab, CVX says:
‘‘Disciplined convex programming error:
Invalid constraint: {complex affine} >= {real affine}’’

image

variable teta(3) 
variable t(1)   
variable alfa(1)

minimize( t )
subject to
[eye(2) [teta(1); teta(2)];[teta(1); teta(2)]' teta(3)]>=zeros(3,3);
 [(1-alfa)*eye(length(sx)-1)    (phi*teta-Rho)     zeros(9,3)   ; ...
     (phi*teta-Rho)'                   t                      -kesi*teta'   ; ...
     zeros(3,9)                   -kesi*teta                  alfa*eye(3) ]>=zeros(13,13);

I think the constraint is quadratic and no affine. what your idea?

6

Per the error message, your constraint is affine, not quadratic, and as must be the case presuming you have shown all the variable declarations.

My guess: one or more of the MATLAB variables (input data) are complex and you dud not specify sdp mode, i.e.,cvx_begin sdp

Use the sdp mode, and you can make the right-hand side 0, but think it is o.k. as is. Without sdp mode, the constraint is interpreted element-wise, hence the error message if there is complex data on the left-hand side.

7

Mr Mark, Thanks for your reply. cvx_begin sdp was correct and variable was complex.