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$$\begin{array}{ll} \text{minimize} & \dfrac{(c^T x)^2}{(d^Tx)}\\ \text{subject to} & Ax \leq b\\ & d^T x > 0\end{array}$$

I have been stuck on this question for a couple days. I am sharing with you what I tried, although I am pretty sure it's wrong. Please help.

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Rodrigo de Azevedo
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Adar
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Let $r=d^Tx > 0$ and $s=c^T x$. Your objective $\min t$ subject to $t \geq \frac{s^2}{r}$ is conic representable via the rotated quadratic cone (best for practical optimization):

$(t,r,s) \in \mathcal{Q}_r^3$,

or alternatively via the semidefinite cone:

$\left(\begin{array}{ll} t & s\\ s & r \end{array}\right) \in \mathcal{S}^2_+$

If you for some reason need a standard form SDP you need to reformulate $r=d^Tx$ and $s=c^T x$ with two inequalities each and append them to your system of inequalities $Ax \leq b$.

Henrik Alsing Friberg
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  • I would add that $t \geq 0$ because of the strict inequality constraint. – Rodrigo de Azevedo Jul 11 '20 at 20:44
  • First of all thanks! I appriciate your help! Now I actually do need a standard form of SDP but i went to the duality way because i had no idea how to.. can u explain more how can u go for a standard sdp form? btw, where is the condition on Ax<=b in your solution? I will be thankful if you can write the full solution cause i feel like I don't fully understand it.. like what is the final SDP form? – Adar Jul 13 '20 at 11:48
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    @AdarCohen Linear inequality constraints can also be written in LMI form. Take a look at this. – Rodrigo de Azevedo Jul 13 '20 at 12:00