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Given is a positive definite matrix $\Sigma$ and vectors $\mu$. I'm considering the convex optimisation problem

$$ f_{\mu, \Sigma}(\sigma):= \sup_{w\in W, \sqrt{w^T\Sigma w}\le \sigma}\mu^Tw$$ where $W$ is a closed convex set in $\mathbb{R}^n$. it can be easily seen that $f_{\mu, \Sigma}$ is concave and increasing over $\sigma \ge \inf_w\sqrt{w^T\Sigma w}$. I would like to formally prove the following

$$ \sup_{w\in W}\frac{\mu^Tw- r}{\sqrt{w^T\Sigma w}} = \sup_{\sigma >0}\frac{f_{\mu, \Sigma}(\sigma)-r}{\sigma}$$ for $r<\sup_{w\in W}\mu^T w$.

math
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1 Answers1

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$$ \begin{align} \sup_{\sigma >0}\frac{f_{\mu, \Sigma}(\sigma)-r}{\sigma} &= \sup_{\sigma >0} \frac{\sup_{w\in W : \sqrt{w^T\Sigma w}\le \sigma} \mu^T w-r}{\sigma} \\ &= \sup_{\sigma >0,w\in W : \sqrt{w^T\Sigma w}\le \sigma} \frac{\mu^T w-r}{\sigma} \\ &= \sup_{w\in W} \sup_{\sigma >0 : \sqrt{w^T\Sigma w}\le \sigma} \frac{\mu^T w-r}{\sigma} \\ &= \sup_{w\in W} \frac{\mu^T w-r}{\sqrt{w^T\Sigma w}} \end{align}$$ In the second step I used $\sigma>0$, while in the last step I used that the numerator is nonnegative ($r \leq \sup_{w\in W}\mu^T w$).

LinAlg
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  • thanks for your answer. two follow up question. Why does $\sigma>0$ ensures that I can take the combined supremum? Same applies to the last step. Could you elaborate a bit on these points? – math Oct 02 '18 at 13:37
  • @math Since $\sigma>0$, you can put the supremum in front of the fraction (it would have become an infimum if $\sigma<0$). You can always combine/split supremums. In the last step, it is obvious at which value of $\sigma$ the supremum is attained, namely the smallest feasible one. – LinAlg Oct 02 '18 at 14:19
  • thanks for your quick reply. Maybe I'm missing the point here, but taking first the supremum over something and then another one is not necessarily the same as taking the supremum jointly, no? – math Oct 02 '18 at 14:21
  • It is the same, see https://math.stackexchange.com/questions/53794/simple-question-the-double-supremum – LinAlg Oct 02 '18 at 14:22
  • actually....this makes my day. I really thought it isnt. Many thanks for the link. I strongly believed this is not the same. I will even assign a bounty and give you the reward for it. (need to wait another 24 hours) – math Oct 02 '18 at 14:25
  • haha, thank you – LinAlg Oct 02 '18 at 14:31