While working on optimization issues I encountered the following problem: $$\left\{\begin{array}{ll} {\displaystyle \sup_{z\in\mathbb{R}^{m}}} &\frac{ \langle c,z \rangle + \rho}{ \left\|B z\right\|} \\ \mbox{s. t} & Az\leq b\\ & \langle f,z\rangle =1. \end{array} \right.$$ where $\left\|\cdot\right\|$ is the Euclidean norm. I know that this is a fractional problem, and I know that the easiest fractional problems to solve are those that are a quotient of two affine functions. However, this problem is not of that type. I have tried to solve it in Matlab with fmincon, but it takes too long to find a solution. My intention is to know if there is a more efficient way to solve this problem. The reason for this posting is to ask you for suggestions on how to deal with this problem.
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$\begingroup$ Perhaps fix $||Bz||=c$ which might be easier to solve for fixed c, and then do a line search over c. $\endgroup$– ericfMar 22 at 2:24
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$\begingroup$ Cross-posted: math.stackexchange.com/questions/4663551/… $\endgroup$– RobPrattMar 22 at 2:50
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3$\begingroup$ It is a Sharpe-ratio type issue, you can formulate it as SOCP by homogenizing, for instance docs.mosek.com/modeling-cookbook/… and themosekblog.blogspot.com/2020/10/… $\endgroup$– Michal AdamaszekMar 22 at 10:09
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$\begingroup$ @Michal Adamaszekk You should make your comment into an answer. $\endgroup$– Mark L. StoneMar 23 at 11:47
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$\begingroup$ @Michal Adamaszek Some people on this forum and math SE don't like link only. answers because they rot, whereas on mathoverflow SE they are acceptable and common. I suggest just copy and pasting material from the links, which I presume you wrote anyway, and provide the links. $\endgroup$– Mark L. StoneMar 24 at 10:33
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