**Problem 3.**

$$\min\_{(w,\mathcal{C})\in X\times\mathcal{R}} \mathcal{C} \tag{17}$$

$$\text{s.t.}\\\text{WCR}(w|\beta\_k) \le \text{C} + \text{WC}\_{\beta\_k} \quad (k = 1, \dots, K) \tag{18}$$

As in Lemma 1, We define *<sup>F</sup><sup>L</sup>*(*<sup>w</sup>*, *<sup>α</sup>k*|*βk*) = max*i*∈*L <sup>F</sup>*(*i*)(*<sup>w</sup>*, *<sup>α</sup>k*|*βk*). Then, Problem 3 can be written as follows.

**Problem 4.**

$$\min\_{(w,C)\in\mathcal{X}\times\mathcal{R}} \mathbb{C} \tag{19}$$

$$\text{s.t.} \min\_{a\_k} F^L(w, a\_k | \beta\_k) \le \mathbb{C} + \mathcal{W} \mathbb{C}\_{\beta\_k} \quad (k = 1, \dots, K) \tag{20}$$

Thereafter, we consider the following Problem 5.
