Question on Rockafellar's Paper for optimisation of CVaR

Question

In Rockafellar and Uryasev's Paper about CVaR Optimisation they showed in Equation (17) that using Monte-Carlo-Simulation one can use $$\tilde F_{\beta}(x,\alpha)=\alpha+\frac{1}{q(1-\beta)}\sum_{k=1}^qmax(-x^Ty_k-\alpha,0)$$ to approximate the CVaR approximation, where $\alpha$ denotes the percentile of $\beta-$CVaR, $x\in\Bbb R^n$ the portfolio and $y_k$ the return in k-th scenario. In the following paragraph he introduced some auxiliary real variable $u_k$ for $k=1,...,r$ and claim it is equivalent to minimising the linear expression $$\alpha+\frac{1}{q(1-\beta)}\sum_{k=1}^qu_k$$ subject to $u_k\geq0$ and $x^Ty_k+\alpha+u_k\geq0$.

Question:

1. Why is the indices of $u_k$ running from 1 to r and not to q?
2. Why are the both problem equivalent? I can rewrite the last condition in the second problem as $u_k\geq-x^Ty_k-\alpha$ and together with $u_k\geq0$ I am allowing $-x^Ty_k-\alpha\leq0$ in the sum, which has value 0 in the first problem with $max(-x^Ty_k-\alpha,0)$ in the summand.

Answer 1

On 1, I suspect that is a typo and that the second formula should sum to r.

On 2, that is applying well-known techniques in how to handle piece-wise linear functions in an optimizer. For instance, see page 4 of these lecture notes. It's basically doing the same thing with a few additional complications. In CVaR optimization, there are more things to sum and also $\alpha$ is also part of the optimization (following the optimization, $\alpha$ should equal the Value-at-Risk).

Finally, there's an issue with your math: $-x^Ty_k-\alpha\leq0$ only follows if $u_k$ is always $0$.