# Transform LMI problem into different SDP form [duplicate]

I am trying to transform a LMI problem of the following form:

$min_x \quad c^Tx \\ s.t. \quad x_1A_1+\dots+x_nA_n \preceq R$

into another SDP formulation:

$min_x \quad c^Tx \\ s.t. \quad Ax = b, \quad X \preceq 0$

where x=vec(X) obtained by stacking the columns of X. Thank you!

Initial Idea:

1. Introduce slack variable S.
2. Replace LMI constraint by:

$x_1A_1+\dots+x_nA_n - R = S, \quad S \preceq 0$

1. Augment the decision variable x --> $\tilde{x}=[x,s]^T$.
2. Replace c by $\tilde{c}=[c,0]^T$.

But then I dont know how to continue.

• Hi, welcome to MSE. Can you show us what you have tried so far?
– ArtW
Jan 14 '18 at 17:29
• Hi, thank you. I edited the post Jan 14 '18 at 17:37
• I am certain we have answered this before, but search fails me. Jan 14 '18 at 21:03
• Ah, here we go: Convert Semidefinite program forms Jan 14 '18 at 21:05