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I am reading these notes from Stanford called "Optimization with uncertain data".

In section 2.2, Example 2 (page 7), the author mentions the following portfolio problem $(P)$:

$$ \max \; t $$ subject to \begin{align*} \text{Prob}\left(\sum_i R_i x_i \ge t\right) &\ge 1-\epsilon \\ \sum_i x_i &= 1 \\ x_i &\ge 0 \quad \forall i \end{align*} where $R_i$ (the random return of each asset $i$) and $\epsilon$ (the value at risk) are given parameters.

Assuming $R_i \in [\mu_i -\text{u}_i,\mu_i +\text{u}_i]$ with $\text{E}(R_i)=\mu_i$, the author shows that the problem is equivalent to maximizing $$ \sum_i \mu_i x_i -\sqrt{2 \log \frac{1}{\epsilon}} \lVert\text{diag}(\text{u})x \rVert_2 $$ subject to \begin{align*} \sum_i x_i &= 1 \\ x_i &\ge 0 \quad \forall i \end{align*}

And states:

This corresponds to the uncertainty set given by the scaled $\ell_2$ ball

$$ \mathcal{U}=\{u \in \mathbb{R}^n , \lVert\text{diag}(u)^{-1/2}u \rVert_2 \le \sqrt{2 \log \frac{1}{\epsilon}}\} $$ in the robust inequality $\sum_i (\mu_i+u_i)x_i \le t$ for all $u\in \mathcal{U}$.


I don't understand the last quoted part ($\text{diag}(u)^{-1/2}u$?) and suspect there is at least one typo ($\sum_i (\mu_i+u_i)x_i \color{red}{\ge} t$?). In the end, if I want to solve the initial problem, what formulation do I use ?

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1 Answer 1

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I believe there is a typo in the document, and perhaps some confusion between parameters $\text{u}$ and variables $u$.

My understanding is that the initial problem $(P)$ is equivalent to:

$$ \max_x \; \sum_i \mu_i x_i -\sqrt{2 \log \frac{1}{\epsilon}} \lVert\text{diag}(\text{u})x \rVert_2 $$ subject to \begin{align*} \sum_i x_i &= 1 \\ x_i &\ge 0 \quad \forall i \end{align*}

and to: $$ \max_{x,t,u} \; t $$ subject to \begin{align*} \sum_i x_i &= 1 \\ x_i &\ge 0 \quad \forall i \\ \sum_i (\mu_i + u_i\text{u}_i)x_i &\ge t \\ \lVert u \rVert_2 &\le \sqrt{2 \log \frac{1}{\epsilon}} \end{align*}

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