**Problem 6.**

$$\min\_{(w,\mathbb{C},a)\in\mathcal{X}\times\mathbb{R}\times\mathbb{R}^K} \mathbb{C} + \lambda \|w - w^-\|\_1 \tag{24}$$

$$\text{s.t.}\\\bar{F}^{(i)}(w, a\_k | \beta\_k) \le \mathbb{C} + \mathcal{W}\mathbb{C}\_{\beta\_k}$$

$$(i = 1, \ldots, l, \ k = 1, \ldots, K) \tag{25}$$

We can easily prove that Problem 6 is a linear programming problem similar to the usual CVaR minimization problem.

**Theorem 1.** *The Regularized Multiple β WCVaR Portfolio Optimization problem is equivalent to the following linear programming problem.*

$$\begin{aligned} \min\_{\mathbf{C}, w, t, \mathbf{u}} \mathbf{C} &\to \sum\_{j=1}^{n} u\_{j} \\ \text{s.t. } &u\_{j} \ge \lambda \left(w\_{j} - w\_{j}^{-}\right) \\ &u\_{j} \ge -\lambda \left(w\_{j} - w\_{j}^{-}\right) \\ &t\_{\text{ijk}} \ge 0 \\ &t\_{\text{ijk}} \ge -w^{\top} r^{(i)}[q] - a\_{k} \\ &\alpha\_{k} + \frac{1}{|\mathcal{N}\_{i}|(1-\beta\_{k})} \sum\_{q \in \mathcal{N}\_{i}} t\_{iqk} \le \mathcal{C} + WC\_{\beta\_{k}} \\ &(i = 1, \dots, l, \ k = 1, \dots, K) \\ &1^{\top}w = 1 \\ &w\_{j} \ge 0 \quad (j = 1, \dots, n) \end{aligned}$$

**Proof.** Using a standard approach in optimization, we can replace the absolute value term *<sup>λ</sup>w* − *<sup>w</sup>*−1 in the objective function with *u* ≥ *<sup>λ</sup>*(*w* − *w*<sup>−</sup>) and *u* ≥ −*<sup>λ</sup>*(*<sup>w</sup>* − *w*<sup>−</sup>) in the constraint. Thereafter, the objective and constraints all became linear.

The Algorithm 1 summarizes the sequential procedure of the RM-WCVaR portfolio.

**Algorithm 1** Regularized Multiple *β* WCVaR Portfolio Optimization. **Input:** *K* probability levels *βk* ∈ (0, 1) (*k* = 1, . . . , *<sup>K</sup>*), Coefficient of the regularization term *λ* ∈ R>0, Number of blocks of a partition *l* ∈ Z><sup>0</sup> and Return matrix *Rt* ∈ R*n*×*<sup>N</sup>* (*t* = 1, . . . , *T*) **Output:** Set of optimal weights W = {*wt* ∈ <sup>R</sup>*<sup>n</sup>*}*Tt*=<sup>1</sup> 1: **for** *t* = 1, . . . , *T* **do** 2: Calculate *WCβk* via solving Problem 2 (*k* = 1, . . . , *<sup>K</sup>*). 3: Randomly divide the set {1, . . . , *N*} into blocks of a partition {N*i*}*li*=<sup>1</sup> s.t. |N*i*| = *Nl* (*i* = 1, . . . , *l*) 4: *w*<sup>−</sup> ← *wt*−1 5: Solve the linear programming introduced in Theorem 1 6: Add to the output set W the solution *w*<sup>∗</sup> as *wt* 7: **end for** 8: **return** W
