In the paper written by Rockafellar about C-VaR (https://www.ise.ufl.edu/uryasev/files/2011/11/CVaR1_JOR.pdf), it is explained that this quantity can be approximated using the following problem (problem 17 in their paper):
\begin{equation} \min_{(x, \alpha)\in \mathbb R^n \times \mathbb R}\quad \ c(x,\alpha):=\alpha+\frac{1}{q(1-\beta)} \sum_{k=1}^q(-x^Ty_k-\alpha)^+, \end{equation} where $\beta\in (0,1)$ and $q\in \mathbb N$. I am wondering if the solution set of this problem is bounded or not; especially if $\alpha^*$ is bounded or not?
I believe that the paper has the additional constraint that $x \ge 0$ and $\sum_k x_k = 1$, so the set of $x$ values is compact.
Since $\beta \in (0,1)$ we have $1-\beta \in (0,1)$ and so ${1 \over 1-\beta} > 1$ and so $1-{1 \over 1-\beta} < 0$.
\begin{eqnarray} c(\alpha, x) &=& \alpha + {1 \over q(1-\beta)} \sum_k \max(0, -x^T y_k-\alpha) \\ &=& \alpha + \sum_k \max(0, -{1 \over q(1-\beta)} x^T y_k-{1 \over q(1-\beta)} \alpha) \\ &=& \sum_k \max({\alpha \over q}, -{1 \over q(1-\beta)} x^T y_k + {1 \over q}(1-{1 \over 1-\beta}) \alpha) \\ \end{eqnarray}
As an aside, note that if $\gamma>0$ then $\max(t, u-\gamma t) \ge {u \over 1 + \gamma}$ and so it follows that \begin{eqnarray} c(\alpha,x) &\ge& \sum_k {1 \over {1 \over q} ( { \beta \over 1-\beta })} (-{1 \over q(1-\beta)} y_k^T x ) \\ &=& -{ 1\over \beta} (\sum_k y_k)^T x \end{eqnarray} Since $x$ is in a compact set it follows that the right hand side is bounded below.