**Problem 1.**

$$\min\_{(w,a,C)\in X\times R\times R} \mathbb{C} \tag{10}$$

$$\text{s.t.} \quad \mathfrak{a} + (1 - \beta)^{-1} \int\_{R^n} [-w^\top r - a]^+ p^{(i)}(r) dr \le \mathbb{C}, \ (i = 1, \dots, l) \tag{11}$$

We approximate the function *<sup>F</sup>*(*i*)(*<sup>w</sup>*, *α*|*β*) by sampling a random variable *<sup>r</sup>*(*i*), *i* = 1, ... , *l* from the density function *<sup>p</sup>*(*i*)(·). *r*(*i*)[*q*] is the *q*-th sample with respect to the *i*-th density distribution *<sup>p</sup>*(*i*)(·), and N*i* ⊆ {1, . . . , *N*} denotes the set of corresponding samples. The auxiliary function *<sup>F</sup>*(*i*)(*<sup>w</sup>*, *α*|*β*) is approximated as follows.

$$F^{(i)}(w, \boldsymbol{\alpha}|\boldsymbol{\beta}) \simeq \boldsymbol{\alpha} + (|\mathcal{N}\_i|(1-\boldsymbol{\beta}))^{-1} \sum\_{q \in \mathcal{N}\_i} [-w^\top r^{(i)}[q] - \boldsymbol{\alpha}]^+ \tag{12}$$

Finally, we can formulate the minimum WCVaR portfolio as a linear programming problem, as shown below.
