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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.

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  • $\begingroup$ Hi, welcome to MSE. Can you show us what you have tried so far? $\endgroup$
    – ArtW
    Jan 14 '18 at 17:29
  • $\begingroup$ Hi, thank you. I edited the post $\endgroup$
    – miga89
    Jan 14 '18 at 17:37
  • $\begingroup$ I am certain we have answered this before, but search fails me. $\endgroup$ Jan 14 '18 at 21:03
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    $\begingroup$ Ah, here we go: Convert Semidefinite program forms $\endgroup$ Jan 14 '18 at 21:05