**Optimization Problem 2**

• *Step 1. Find an optimal vector C*<sup>∗</sup> *by minimizing deviation from the CVaR quadrangle:*

$$\min\_{\mathbf{C} \in \mathbb{R}^m} \overline{\mathcal{D}}\_{\mathbf{a}} (Z\_0(\mathbf{C})) $$

*where <sup>Z</sup>*0(*C*) = *V* − *CTY*.

• *Step 2. Calculate:*

$$\mathbb{C}\_0^\* = \mathbb{C}VaR\_\alpha(Z\_0(\mathbb{C}^\*))$$

In Optimization Problem 2 in Step 2 the statistic equals *<sup>S</sup>*(*<sup>Z</sup>*0(*C*<sup>∗</sup>)) = *CVaR*α(*<sup>Z</sup>*0(*C*<sup>∗</sup>)), which is the specification of the inclusion operation in the Optimization Problem General 2 in Step 2.

Optimization Problem 2 is used in Rockafellar et al. (2014) for the construction of the linear regression algorithms for estimating CVaR.

According to the decomposition theorem, when E, D , and S are elements of a mixed-quantile quadrangle, the following Optimization Problems 3 and 4 are equivalent.
