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I am trying to formulate following problem (with some constraint) as a semidefinite programming problem (SDP),

\begin{equation} \text{minimize } ~~ -a^T B^{-1} a \end{equation}

where $B$ is a positive definite matrix. I used auxiliary variable and Schur complement and tried to formulate it as following :

\begin{eqnarray} \text{minimize }~~ t \\ \text{s.t.} ~~~t \geq -a^T B^{-1} a \end{eqnarray}

then tried to to reformulate the constraint as a Linear Matrix inequality (LMI) using Schur complement, but I got stuck to following:

\begin{align*} \left[\begin{array}{cc} t & a' \\ a & -B^{-1} \end{array}\right] \overset{\Large\text{??}}\geq 0 \end{align*}

but the LMI is not positive definite because $-B^{-1}$ is not positive definite.

Is there another way /scape that I can formulate the original problem as SDP?

Rodrigo de Azevedo
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Alireza
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  • I suggest to double check the original problem. It's a convex relaxation of an Np-hard combinatorial problem? – user85361 Sep 23 '15 at 22:10
  • no it's not , actually it is Mean Square Error minimization of an LMMSE estimator as var[unknown] - $a'B^{-1}a$, so I wanted to minimize it – Alireza Sep 23 '15 at 22:13
  • i know, that is the problem – Alireza Sep 23 '15 at 22:40
  • This is equal to $\text{maximize} a^T B^{-1} a$. Since $B$ is positive semidefinite, $B^{-1}$ is psd. So it's maximum is infinity. Because it's a quadratic form. – user85361 Sep 23 '15 at 22:41
  • I know :) , that's why I have mentioned with some constraints , if there is no constraint you are right , but I have some constraint on that – Alireza Sep 23 '15 at 22:46
  • you know, you miss something. Schur complement is for semidefinite lower bound . But you want upper bound which is determined by your constraints , not mentioned here and not by semidefinite constraint. – user85361 Sep 23 '15 at 22:55
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    that is the original problem http://math.stackexchange.com/questions/1445731/maximizing-a-convex-function-with-special-structure – Alireza Sep 23 '15 at 22:57
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    It's not convex, so it cannot be cast as an SDP. End of story, I'm afraid. – Michael Grant Sep 24 '15 at 15:10
  • This is effectively a duplicate of your other problem---granted, you've simplified it somewhat, but it's still a duplicate. That's not good form. – Michael Grant Sep 24 '15 at 15:13

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