# How can a semidefinite program be written as a linear program?

I see that they have the same objective function and equality constraint $$Ax = b$$, but I am having trouble with showing that $$Gx \preceq h$$ can be written as $$x_1F_1 + \dots + x_nF_n + K \preceq 0$$ where $$F_i, K \in S^n$$ are symmetric matrices. I have been told that

If the matricies $$F_i, K$$ are diagonal, then $$x_1F_1 + \dots + x_nF_n + K$$ reduces to a set of linear inequalities.

but I don't see how this is. I see how the matrix $$x_1F_1 + \dots + x_nF_n + K$$ reduces to a set of linear systems $$x_1F_{1i} + \dots + x_nF_{ni} + K_i$$ for $$i = 1, \dots, n$$ (where $$F_{ki}$$ is diagonal matrix $$F_k$$'s $$i$$-th diagonal entry), but the inequality symbol is still in terms of positive-semi definiteness $$\preceq 0$$.

Does the semi-definite inequality symbol just become a component-wise inequality for vectors? If so, can someone show how this is?

• Your question is very unclear. It appears that you're interested in what happens in SDP when the matrix variables are restricted to being diagonal. Is that correct? – Brian Borchers May 6 at 2:16
• – Rodrigo de Azevedo May 6 at 16:58