## 2 Materials and Methods
### Convergence Analysis  
To analyze the convergence of the mixed-integer asynchronous computing (ASYN) level decomposition (LD) described above, we rely as much as possible on Reference [31]. However, to account for the mixed-integer nature of the feasible set, we need novel developments like the ones in Theorem 3.1 below. Throughout this section, we assume \(\mathrm{tol}_{\Delta}=0\), as well as the following:  
**Hypothesis 1 (H1). **_all the workers are responsive;_  
**Hypothesis 2 (H2). **_algorithm generates only finitely many feasibility cuts;_  
**Hypothesis 3 (H3). **_the workers provide bounded subgradients._  
As for H1, the assumption H2 is a mild one: H2 holds, for instance, when \(f\) is a polyhedral function, or when \(X\) has only finitely many points. The problem of interest satisfies both these properties, and, therefore, H2 is verified. Due to convexity of \(f\), assumption H3 holds, e.g., if \(X\) is contained in an open convex set that is itself a subset of \(Dom(f)\) (in this case, no feasibility cut will be generated). H3 also holds in our setting if subgradients are computed via basic optimal dual solutions of the second-stage subproblems. Under H3, we can ensure that the parameter \(L\) in the algorithm is finite. In our analysis, we use the fact that the sequences of the optimality gap, \(\Delta_{k}\), and upper bound, \(f_{k}^{\mathrm{up}}\), are non-increasing by definition, and that the sequence of lower bound, \(f_{k}^{\mathrm{low}}\), is non-decreasing. More specifically, we update the lower bound only when the MP is infeasible. We count with \(\ell\) the number of times the gap significantly decreases, meaning that the test of line 38 is triggered, and denote by \(k(\ell)\) the corresponding iteration. We have the following by construction:  
\[\Delta_{k(\ell+1)}\leq\alpha\Delta_{k(\ell)}\leq\alpha^{2}\Delta_{k(\ell-1)} \leq\dots\leq\alpha^{\ell}\Delta_{1}\quad\forall\ell=1,2,\dots \tag{6}\]  
As in Reference [31], \(k(\ell)\) denotes a critical iteration, and \(\mathbf{x}_{k(\ell)}\) denotes a critical iterate. We introduce the set of iterates between two consecutive critical iterates by \(K^{\ell}:=[k(\ell)+1,\dots,k(\ell+1)-1]\). The proof of convergence of the ASYN LD consists in showing that the algorithm performs infinitely many critical iterations when \(\mathrm{tol}_{\Delta}=0\). We start with the following lemma, which is a particular case of Reference [31], Lemma 3, and does not depend on the structure of \(X\). **Lemma 1**. : _Fix an arbitrary \(\ell\) and let \(K^{\ell}\) be defined as above. Then, for all \(k\in K^{\ell}\), (a) the MP is feasible, and (b) the stability center is fixed: \(\hat{\mathbf{x}}_{k}=\hat{\mathbf{x}}_{k(\ell)}\)._  
Item (a) above ensures that the MP is well-defined and \(f_{k}^{\mathrm{low}}\) is fixed for all \(k\in K^{\ell}\). Note that the lower bound is updated only when the MP is found infeasible, and this fact immediately triggers the test at line 38 of the algorithm. Similarly, Algorithm 1 guarantees that the stability center remains fixed for all \(k\in K^{\ell}\), since an updated on the stability center would imply a new critical iteration. **Theorem 1**. : _Assume that \(X\) is a compact set and that H1-H3 hold. Let \(\mathrm{tol}_{\Delta}=0\) in the algorithm, and then \(\underset{k}{lim}\Delta_{k}=0\)._  
**Proof of Theorem 1. ** By (6), we only need to show that the counter \(\ell\) increases indefinitely (i.e., that there are infinitely many critical iterations). We obtain this by showing that, for any \(\ell\), the set \(K^{\ell}\) is finite; for this, suppose that \(\Delta_{k}>\Delta>0\) for all \(k\in K^{\ell}\). We proceed in two steps, showing the following: (i) The number of asynchronous iterations between two consecutive coordination steps is finite, and (ii) the number of coordination steps in \(K^{\ell}\) is finite, as well.