a contradiction. Therefore, $\beta_{\ell',i'} = 0$ (and thus $[1 - \frac{a_{\ell'}}{a_{\ell'}}]^+ = 0$) for all $i' \in W^0$ and $\ell' \in [n]_0$ such that $y_{\ell',i'}^* > 0$. Transposing the above reasoning to $\ell' = 0$, we have $\beta_{0,i'} > 0$ so that $y_{0,i'}^* = 0$ for each $i' \in W^0$.

Using the above, we get that the objective value of $D(w^*)$ is given by

$$r \left( \mu^* + \sum_{\ell \in [n]} \alpha_{\ell} u_{\ell}^* \right) + \sum_{i \in [n]} \left( d_i \sigma_i^* + c_i \sum_{\ell \in [n]} y_{\ell,i}^* \right). \qquad (32)$$

The rest of the proof constructs an algorithm that iteratively modifies the dual solution without changing the variables $\sigma^*$ and the values of the sums $\mu^* + \sum_{\ell \in [n]} \alpha_{\ell} u_{\ell}^*$ and $\sum_{\ell \in [n]} y_{\ell,i}^*$. Thanks to (32), the algorithm does therefore not modify the cost of the solution.

The dual constraint of $\tilde{\xi}_i, i \in W^1$ may be rewritten as:

$$\mu^* \geq \frac{1}{a_i} \left( -\sigma_i^* + \sum_{\ell \in [n]} y_{\ell,i}^* \right) - \sum_{\ell \in [n]} \alpha_{\ell} u_{\ell}^*,$$

which must be active for at least one element of $W^1$, which we denote $j$. Recalling $1/a_j = \alpha_j$, this means in particular that

$$\mu^* \leq \alpha_j \sum_{\ell \in [n]} y_{\ell,j}^* - \sum_{\ell \in [n]} \alpha_{\ell} u_{\ell}^* \leq \sum_{\ell \in [n]} (\alpha_j - \alpha_{\ell}) u_{\ell}^*.$$

As a consequence, there is $\ell \in [n]_0$ such that $u_{\ell}^* > 0$ and $\alpha_{\ell} < \alpha_j$. Using the above, we build another optimal solution of $D(w^*)$, $(y', u', \mu', \sigma^*)$, where $\mu' = 0$, by iteratively decreasing the values of nonzero variables $u_{\ell}$, and increasing the value of $u_j$ while keeping constant the value of $\mu^* + \sum_{\ell \in [n]} \alpha_{\ell} u_{\ell}^*$. The iterative construction is formalized in Algorithm 1. At each step, one index $\ell$ such that $u_{\ell}^* > 0$ and $\alpha_{\ell} < \alpha_j$ is considered. The first computed value, $\delta$, is the largest decrease of $u_{\ell}'$ such that $u_{\ell}' \geq 0$ and $\mu' \geq 0$ at the end of the algorithm. The update of $y_{\ell}'$ and $y_{j}'$ then guarantee that $y_{\ell}' \leq u_{\ell}', \forall \ell \in [n]$, $By_{\ell}' \geq u_{\ell}'b, \forall \ell \in [n]$, and $\sum_{\ell \in [n]} y_{\ell}' = \sum_{\ell \in [n]} y_{\ell}^*$. Finally, the update of $\mu'$ is such that $\mu' + \sum_{\ell \in [n]} \alpha_{\ell} u_{\ell}' = \mu^* + \sum_{\ell \in [n]} \alpha_{\ell} u_{\ell}^*$. It is then straightforward to verify that each step keeps $(y', u', \mu', \sigma^*)$ feasible and leaves its objective value unchanged. Moreover, we can verify that either $\mu' = 0$ at the end of an iteration or $\delta = u_{\ell}'$. At the end of the algorithm, we thus have

$$\mu' \leq \mu^* - \sum_{k \in K} (\alpha_j - \alpha_k) u_k^* \\ \leq \sum_{\ell \in [n]} (\alpha_j - \alpha_{\ell}) u_{\ell}^* - \sum_{\ell \in [n]: a_{\ell} > a_j} (\alpha_j - \alpha_{\ell}) u_{\ell}^* \leq 0.$$

initialization: $y' := y^*, u' := u^*, \mu' := \mu^*, K = \{k \in [n] \mid u_k^* > 0, \alpha_k < \alpha_j\}$.

1 for $k \in K$ do

2 $\delta := \min \{u_k^*, \frac{\mu'}{\alpha_j - \alpha_k}\}$

3 $u_k' \leftarrow u_k' - \delta$

4 $u_j' \leftarrow u_j' + \delta$

5 $y_k' \leftarrow y_k' - \frac{\delta}{u_k^*} y_k^*$

6 $y_j' \leftarrow y_j' + \frac{\delta}{u_k^*} y_k^*$

7 $\mu' \leftarrow \mu' - \delta(\alpha_j - \alpha_k)$

8 if $\mu' = 0$ then

9 break

10 return $(y', u', \mu')$

**Algorithm 1:** Construction of an optimal solution where $\mu' = 0$

## E Compact reformulation for selection problem with K-adaptability

The K-adaptability approximation amounts to pre-select K recourse policies and choose the best of them upon realization of the uncertain parameters. Applied to (DDID), one obtains

$$z^{K_{\text{adapt}}} = \min_{w \in W} \max_{\xi \in \Xi} \min_{k \in [K]} \max_{\xi \in \Xi(w, \xi)} \sum_{i \in [n]} (c_i + \xi_i)y_i. \qquad (K-\text{ADAPT})$$
The rest of the proof constructs an algorithm that iteratively modifies the dual solution without changing the variables $\sigma^*$ and the values of the sums $\mu^* + \sum_{\ell \in [n]} \alpha_\ell u_\ell^*$ and $\sum_{\ell \in [n]} y_{\ell,i}^*$ Thanks to (32), the algorithm does therefore not modify the cost of the solution.

The dual constraint of $\bar{\xi}_i$, $i \in \mathcal{W}^1$ may be rewritten as:

$$ \mu^* \geq \frac{1}{a_i} \left( -\sigma_i^* + \sum_{\ell \in [n]} y_{\ell,i}^* \right) - \sum_{\ell \in [n]} \alpha_\ell u_\ell^*, $$

which must be active for at least one element of $\mathcal{W}^1$, which we denote $j$. Recalling $1/a_j = \alpha_j$, this means in particular that

$$ \mu^* \leq \alpha_j \sum_{\ell \in [n]} y_{\ell,j}^* - \sum_{\ell \in [n]} \alpha_\ell u_\ell^* \leq \sum_{\ell \in [n]} (\alpha_j - \alpha_\ell) u_\ell^*. $$

As a consequence, there is $\ell \in [n]_0$ such that $u_\ell^* > 0$ and $\alpha_\ell < \alpha_j$. Using the above, we build another optimal solution of $\mathbf{D}(w^*)$, $(y', u', \mu', \sigma^*)$, where $\mu' = 0$, by iteratively decreasing the values of nonzero variables $u_\ell$, and increasing the value of $u_j$ while keeping constant the value of $\mu^* + \sum_{\ell \in [n]} \alpha_\ell u_\ell^*$. The iterative construction is formalized in Algorithm 1. At each step, one index $\ell$ such that $u_\ell^* > 0$ and $\alpha_\ell < \alpha_j$ is considered. The first computed value, $\delta$, is the largest decrease of $u'_k$ such that $u'_k \ge 0$ and $\mu' \ge 0$ at the end of the algorithm. The update of $y'_k$ and $y'_j$ then guarantee that $y'_\ell \le u'_\ell$, $\forall \ell \in [n]$, $By'_\ell \ge u'_\ell b$, $\forall \ell \in [n]$, and $\sum_{\ell \in [n]} y'_\ell = \sum_{\ell \in [n]} y_\ell^*$. Finally, the update of $\mu'$ is such that $\mu' + \sum_{\ell \in [n]} \alpha_\ell u'_\ell = \mu^* + \sum_{\ell \in [n]} \alpha_\ell u_\ell^*$. It is then straightforward to verify that each step keeps $(y', u', \mu', \sigma^*)$ feasible and leaves its objective value unchanged. Moreover, we can verify that either $\mu' = 0$ at the end of an iteration or $\delta = u_k^*$. At the end of the algorithm, we thus have

$$ \mu' \leq \mu^* - \sum_{k \in K} (\alpha_j - \alpha_k) u_k^* \\ \leq \sum_{\ell \in [n]} (\alpha_j - \alpha_\ell) u_\ell^* - \sum_{\ell \in [n]: a_\ell > a_j} (\alpha_j - \alpha_\ell) u_\ell^* \leq 0. $$

**Algorithm 1:** Construction of an optimal solution where $\mu' = 0$

initialization: $y' := y^*, u' := u^*, \mu' := \mu^*, K = \{k \in [n] \mid u_k^* > 0, \alpha_k < \alpha_j\}$.

1. for $k \in K$ do

2. &nbsp;&nbsp;&nbsp;&nbsp; $\delta := \min \left\{ u_k^*, \frac{\mu'}{\alpha_j - \alpha_k} \right\}$

3. &nbsp;&nbsp;&nbsp;&nbsp; $u'_k \leftarrow u'_k - \delta$

4. &nbsp;&nbsp;&nbsp;&nbsp; $u'_j \leftarrow u'_j + \delta$

5. &nbsp;&nbsp;&nbsp;&nbsp; $y'_k \leftarrow y'_k - \frac{\delta}{u_k^*} y_k^*$

6. &nbsp;&nbsp;&nbsp;&nbsp; $y'_j \leftarrow y'_j + \frac{\delta}{u_k^*} y_k^*$

7. &nbsp;&nbsp;&nbsp;&nbsp; $\mu' \leftarrow \mu' - \delta (\alpha_j - \alpha_k)$

8. &nbsp;&nbsp;&nbsp;&nbsp; if $\mu' = 0$ then

9. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; break

10. return $(y', u', \mu')$

## E Compact reformulation for selection problem with K-adaptability

The K-adaptability approximation amounts to pre-select K recourse policies and choose the best of them upon realization of the uncertain parameters. Applied to (DDID), one obtains

$$ z^{\text{Kadapt}} = \min_{w \in \mathcal{W}} \max_{\xi \in \Xi} \min_{k \in [K]} \max_{\xi \in \Xi(w, \xi)} \sum_{i \in [n]} (c_i + \xi_i)y_i. \quad (\text{K-ADAPT}) $$

Consider the sets $\mathcal{Y}^{\text{sel}}$, $\mathcal{W}^{\text{sel}}$ and the budget uncertainty polytope

$$ \Xi = \left\{ \xi \in \mathbb{R}^n \mid \sum_{i \in [n]} \frac{\xi_i - c_i}{d_i} \leq \Gamma, \xi \leq c + d, -\xi \leq -c \right\}. $$
Consider the sets $\mathcal{Y}^{\text{sel}}$, $\mathcal{W}^{\text{sel}}$ and the budget uncertainty polytope

$$ \Xi = \left\{ \xi \in \mathbb{R}^n \mid \sum_{i \in [n]} \frac{\xi_i - c_i}{d_i} \le \Gamma, \xi \le c+d, -\xi \le -c \right\}. $$

Applying [44, Corollary 1] to (K-ADAPT) together with the symmetry breaking constraints detailed in Section EC.3.1. of [44], leads to the formulation presented below. For readability, we subdivide the dual variables $\beta$ into $\beta^\Gamma$, $\beta^{ub}$ (for the upper bounds on $\xi$) and $\beta^{lb}$ (for the lower bounds on $\xi$) and similarly for $\beta^k$. We model the constraints $\bar{\gamma}_i^k = w_i \gamma_i^k$ with indicator constraints to avoid the burden of computing tight big M. Furthermore, we define $\Gamma' = \Gamma + \sum_{i \in [n]} c_i / d_i$.

$$ \min \quad \Gamma' \left( \beta^\Gamma + \sum_{k \in [K]} \beta^{k, \Gamma} \right) + \sum_{i \in [n]} (c_i + d_i) \left( \beta_i^{ub} + \sum_{k \in [K]} \beta_i^{k, ub} \right) - \sum_{i \in [n]} c_i \left( \beta_i^{lb} + \sum_{k \in [K]} \beta_i^{k, lb} \right) $$

$$ \text{s.t.} \quad \sum_{i \in [n]} w_i = q $$

$$ \sum_{i \in [n]} y_i^k = p, \qquad \forall k \in [K] $$

$$ \sum_{k \in [K]} \alpha_k = 1 $$

$$ \frac{\beta^{k, \Gamma}}{d_i} + \beta_i^{k, ub} - \beta_i^{k, lb} + \bar{\gamma}_i^k = \bar{y}_i^k, \qquad \forall i \in [n], k \in [K] $$

$$ \frac{\beta^\Gamma}{d_i} + \beta_i^{ub} - \beta_i^{lb} = \sum_{k \in [K]} \bar{\gamma}_i^k, \qquad \forall i \in [n] $$

$$ w_i = 1 \implies \bar{\gamma}_i^k = \gamma_i^k, \qquad \forall i \in [n], k \in [K] $$

$$ w_i = 0 \implies \bar{\gamma}_i^k = 0, \qquad \forall i \in [n], k \in [K] $$

$$ \bar{y}_i^k \le y_i^k, \quad \bar{y}_i^k \le \alpha_k, \quad \bar{y}_i^k \ge \alpha_k - 1 + y_i^k \qquad \forall i \in [n], k \in [K] $$

$$ z_i^{k, k+1} \le y_i^k + y_i^{k+1}, \quad z_i^{k, k+1} \le 2 - y_i^k - y_i^{k+1} \qquad \forall i \in [n], k \in [K-1] $$

$$ z_i^{k, k+1} \ge y_i^k - y_i^{k+1}, \quad z_i^{k, k+1} \ge y_i^{k+1} - y_i^k \qquad \forall i \in [n], k \in [K-1] $$

$$ y_i^k \ge y_i^{k+1} - \sum_{j=1}^{i-1} z_j^{k, k+1} \qquad \forall i \in [n], k \in [K-1] $$

$$ \beta \ge 0, \beta^k \ge 0, \alpha \ge 0, y, z \text{ binary} $$
Applying [45, Corollary 1] to (K-ADAPT) together with the symmetry breaking constraints detailed in Section EC.3.1. of [45], leads to the formulation presented below. For readability, we subdivide the dual variables $\beta$ into $\beta^\Gamma$, $\beta^{ub}$ (for the upper bounds on $\xi$) and $\beta^{lb}$ (for the lower bounds on $\xi$) and similarly for $\beta^k$. We model the constraints $\bar{\gamma}_i^k = w_i \gamma_i^k$ with indicator constraints to avoid the burden of computing tight big M. Furthermore, we define $\Gamma' = \Gamma + \sum_{i \in [n]} c_i / d_i$.

$$ \min \quad \Gamma' \left( \beta^\Gamma + \sum_{k \in [K]} \beta^{k, \Gamma} \right) + \sum_{i \in [n]} (c_i + d_i) \left( \beta_i^{ub} + \sum_{k \in [K]} \beta_i^{k, ub} \right) - \sum_{i \in [n]} c_i \left( \beta_i^{lb} + \sum_{k \in [K]} \beta_i^{k, lb} \right) $$

$$ \text{s.t.} \quad \sum_{i \in [n]} w_i = q $$

$$ \sum_{i \in [n]} y_i^k = p, \qquad \forall k \in [K] $$

$$ \sum_{k \in [K]} \alpha_k = 1 $$

$$ \frac{\beta^{k, \Gamma}}{d_i} + \beta_i^{k, ub} - \beta_i^{k, lb} + \bar{\gamma}_i^k = \bar{y}_i^k, \qquad \forall i \in [n], k \in [K] $$

$$ \frac{\beta^\Gamma}{d_i} + \beta_i^{ub} - \beta_i^{lb} = \sum_{k \in [K]} \bar{\gamma}_i^k, \qquad \forall i \in [n] $$

$$ w_i = 1 \implies \bar{\gamma}_i^k = \gamma_i^k, \qquad \forall i \in [n], k \in [K] $$

$$ w_i = 0 \implies \bar{\gamma}_i^k = 0, \qquad \forall i \in [n], k \in [K] $$

$$ \bar{y}_i^k \le y_i^k, \quad \bar{y}_i^k \le \alpha_k, \quad \bar{y}_i^k \ge \alpha_k - 1 + y_i^k \qquad \forall i \in [n], k \in [K] $$

$$ z_i^{k,k+1} \le y_i^k + y_i^{k+1}, \quad z_i^{k,k+1} \le 2 - y_i^k - y_i^{k+1} \qquad \forall i \in [n], k \in [K-1] $$

$$ z_i^{k,k+1} \ge y_i^k - y_i^{k+1}, \quad z_i^{k,k+1} \ge y_i^{k+1} - y_i^k \qquad \forall i \in [n], k \in [K-1] $$

$$ y_i^k \ge y_i^{k+1} - \sum_{j=1}^{i-1} z_j^{k,k+1} \qquad \forall i \in [n], k \in [K-1] $$

$\beta \ge 0, \beta^k \ge 0, \alpha \ge 0, y, z \text{ binary}$

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