**Lemma 2.**


**Proof.** Assume that (*w*<sup>∗</sup>, *C*∗) is the optimal value for Problem 4. Given that (*w*<sup>∗</sup>, *C*∗) is a feasible solution of Problem 4, min*<sup>α</sup>k <sup>F</sup><sup>L</sup>*(*w*<sup>∗</sup>, *<sup>α</sup>k*|*βk*) ≤ *C*∗ + *WCβk* holds. We defined *α*∗*k* = (*α*<sup>∗</sup>1, ... , *<sup>α</sup>*<sup>∗</sup>*K*) as *α*∗*k* := argmin*<sup>α</sup>k*∈*RF<sup>L</sup>*(*w*<sup>∗</sup>, *<sup>α</sup>k*|*βk*)(*k* = 1, ... , *<sup>K</sup>*). Then, (*w*<sup>∗</sup>, *C*<sup>∗</sup>, *α*<sup>∗</sup>) became a feasible solution of Problem 5 since *<sup>F</sup><sup>L</sup><sup>w</sup>*<sup>∗</sup>, *α*∗*k βk* <sup>≤</sup> *C*∗ + *WCβk* holds. If (*w*<sup>∗</sup>, *C*<sup>∗</sup>, *α*<sup>∗</sup>) is not the optimal solution of Problem 5, there exists a feasible solution (*w*ˆ, *C*ˆ, *α*ˆ) satisfying *C* ˆ < *C*<sup>∗</sup>. Then, min*<sup>α</sup>k*∈*RF<sup>L</sup>*(*w*ˆ, *<sup>α</sup>k*|*βk*) ≤ *<sup>F</sup><sup>L</sup>*(*w*ˆ, *<sup>α</sup>*<sup>ˆ</sup>*k*|*βk*) ≤ *C*ˆ + *WCβk* (*k* = 1, ... , *K*) holds. Therefore, (*w*ˆ, *C*ˆ) is a feasible solution of Problem 4, thereby contradicting that *C*∗ is the optimal solution of Problem 4. Therefore, (*w*<sup>∗</sup>, *C*<sup>∗</sup>, *α*<sup>∗</sup>) is the optimal solution. Assume that (*w*∗∗, *C*∗∗, *α*∗∗) is the optimal value for Problem 5. Then, because (*w*∗∗, *C*∗∗, *α*∗∗) is a feasible solution of Problem 5, *<sup>F</sup><sup>L</sup><sup>w</sup>*∗∗, *<sup>α</sup>*∗∗*k βk* <sup>≤</sup> *C*∗∗ + *WCβk* (*k* = 1, ... , *K*) holds. (*w*∗∗, *C*∗∗) is a feasible solution for Problem 4 given that min*<sup>α</sup>k*∈*RF<sup>L</sup>*(*w*∗∗, *<sup>α</sup>k*|*βk*) ≤ *<sup>F</sup><sup>L</sup><sup>w</sup>*∗∗, *<sup>α</sup>*∗∗*k βk* <sup>≤</sup> *C*∗∗ + *WCβk* (*k* = 1, ... , *K*) holds. If (*w*∗∗, *C*∗∗) is not the optimal solution of Problem 4, there exists a feasible solution (*w*ˆ, *C*ˆ) satisfying *C*ˆ < *C*∗∗. We defined *α*ˆ = (*α*ˆ 1,..., *α*ˆ *K*) as *α* ˆ*k* := arg min *<sup>α</sup>k FLβk* (*w*ˆ, *<sup>α</sup>k*). Then, (*w*ˆ, *C*ˆ, *α*ˆ) became a feasible solution of Problem 5, thereby contradicting that *C*∗∗ is the optimal solution of Problem 5. Therefore, (*w*∗∗, *C*∗∗) is the optimalsolution.

Here, *r*(*i*)[*q*] is the *q*-th sample with respect to the *i*-th density distribution *<sup>p</sup>*(*i*)(*r*), and N*i* denotes the set of corresponding samples. The function *<sup>F</sup>*(*i*)(*<sup>w</sup>*, *<sup>α</sup>k*|*βk*) is approximated as follows.

$$\bar{F}^{(i)}(w, a\_k | \beta\_k) \simeq a\_k + \frac{1}{|\mathcal{N}\_i^{\prime}|(1 - \beta\_k)} \sum\_{q \in \mathcal{N}\_i^{\prime}} [-w^\top r^{(i)}[q] - a\_k]^+ \tag{23}$$

Finally, we derive the RM-WCVaR Portfolio, where the objectives are minimizing multiple WCVaR values and also controlling the portfolio turnover. Controlling the portfolio turnover is realized through imposing the *L*1-regularization term as *w* − *<sup>w</sup>*−1 = ∑*nj*=<sup>1</sup> |*wj* − *<sup>w</sup>*<sup>−</sup>*j* | where *<sup>w</sup>*<sup>−</sup>*i* denotes the portfolio weight before rebalancing.

Based on the above discussion, the RM-WCVaR Portfolio Optimization problem can be formulated as follows:
