# Need help understanding robust optimization formulation

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 ?

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*}