I have a Semidefinite problem of the form \begin{align} \min_{t\epsilon R,A \epsilon S_{+}^n}~& 1/t \\ &t>0\\ &A>=t C\\ &A\odot C_M=T_M\\ &\left\lvert\lvert A \right\rvert\rvert _F^2 <=r \end{align} or a more general form \begin{align} \min_{t\epsilon R,A \epsilon S_{+}^n}~& 1/t+b t \\ &t>0\\ &A>=t C\\ &A\odot C_M=T_M\\ &\left\lvert\lvert A \right\rvert\rvert _F^2 <=r \end{align} Which matrices $C_M , T_M$ have many zeros and $ C >=0$ is a constant matrix. How to solve this without adding any other Semidefinite constraint to the problem?(Because it makes solving it slover). How to solve it efficiently?
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You could just rewrite $\min 1/t$ into $\min -t,$ avoiding the fractional cost function. – The Pheromone Kid Apr 27 '15 at 21:30
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Thank you. I want to know that is there any general method for solving fractional problems when there is a semidefinite constraints. Of course your suggestion works for simple cases like this. – user85361 May 02 '15 at 16:06
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Could you formulate the more general and more complicated case? I think this would be helpful. – The Pheromone Kid May 03 '15 at 10:33
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I edit the question. But even in this simple case, could you go further in the method of substituting $-t$ instead of $1/t$. I have worked on this but unfortunately, I don't know how to proceed. – user85361 May 03 '15 at 16:32