Complex semidefinite programming

[peterlam]

For semidefinite programming (SDP), we have the primal and dual forms as:

primal

min <C,X>
s.t. <A_i,X> = b_i, i=1,...,m
X>=0

dual

max <b,y>
s.t. y_1*A_1 + ... + y_m*A_m <=C

where the data A_i and C are assumed to be real symmetric matrices in many textbooks and online materials.

If we consider complex SDP where A_i and C are Hermitian, will all the results about real SDP be correct by replacing the real matrices to Hermitian matrices? To be precious, will the primal and dual forms be still the same? Do the interior-point methods for real SDP work for complex SDP?

 

[AlephZero]

You have slightly lost me in a forest of unfamiliar notation and jargon, but mostly in numerical analysis anything that works for real symmetric matrices will also work for complex Hermitian matrices.

You can often rewrite algorithms that use Hermitian matrices using double-sized real matrices, replacing every matrix $z = x + iy$ with

$$\bmatrix{x & y \cr -y & x}$$

and the obvious corresponding thing for vectors.

So my advce would be just try it on a problem where you can verify the answer some other way, and I would be happy to bet a few dollars it will work fine.

(BTW I can't figure out why my matrix doesn't have a closing bracket!)