In semidefinite optimization we optimize over a matrix variable that must be symmetric and positive semidefinite.

Assume we want to relax the assumption about symmetry. Is that an important generalization? The answer is no for the following reason. Since

(X+X')/2 is PSD

implies

X

is PSD. Observe

X = (X+X')/2+(X-X')/2

and

y'( (X-X')/2) y >= 0.

implying X is PSD.

Note (X-X')'=-(X-X') implying X-X' is skew symmetric.

Hence, any nonsymmetric semidefinite optimization problem can easily be posed as a standard symmetric semidefinite optimization problem.