## 2 Materials and Methods  
We address the problem of an Independent System Operator (ISO) in a hydro-dominated system with a loose-pool market framework. The ISO decides the day-ahead commitment considering operation costs, forecast errors in wind generation, and inflows; and the usual generation and system-wideconstraints. The uncertainties in wind and inflows are represented by a finite set of scenarios, \(\mathcal{S}\), and the decisions are made in two stages. At the first stage, the ISO decides on the commitment of units, whereas, at the second stage, the operator determines the dispatch according to the random-variable realization. Full details on the considered stochastic hydrothermal unit-commitment (SHTUC) are given shortly. For presenting our approach, which is not limited to (stochastic) unit-commitment (UC) problems, we adopt the following generic formulation. \[f_{\star}:=\min_{x,y}\left\{\mathbf{c}^{\mathrm{T}}x+\sum_{s\in\mathcal{S}} \mathbf{q}_{\mathrm{s}}^{\mathrm{T}}y_{s}\right| \begin{array}{l}x\in \mathcal{X},\ \mathbf{T}x+\mathbf{W}y_{\mathrm{s}}\leq\mathbf{h}_{\mathrm{s}},\\ y_{\mathrm{s}}\in\mathcal{Y}_{\mathrm{s}},\ \mathbf{s}\in\mathcal{S}\end{array}  \right\}. \tag{1}\]  
In this formulation, the n-dimensional vector \(x\) represents the first-stage variables with associated cost-vector, \(\mathbf{c}\). The second-stage variables, \(y_{\mathrm{s}}\), and their associated costs, \(\mathbf{q}_{\mathrm{s}}\), depend on the scenario, \(\mathbf{s}\in\mathcal{S}\). The cost vector, \(\mathbf{q}_{\mathrm{s}}\), is assumed to incorporate the positive probability of scenario \(\mathrm{s}\). The first- and second-stage variables are coupled by constraints \(\mathbf{T}x+\mathbf{W}y_{\mathrm{s}}\leq\mathbf{h}_{\mathrm{s}}\): \(\mathbf{T}\) is the technology matrix; and \(\mathbf{W}\) and \(\mathbf{h}_{\mathrm{s}}\) are, respectively, the recourse matrix and a vector of appropriate dimensions. While \(\mathcal{X}\
eq\varnothing\) is a compact possibly nonconvex, the scenario-dependent set \(\mathcal{Y}_{\mathrm{s}}\) is a convex polyhedron. As previously mentioned, depending on the UC problem and number of scenarios, the mixed-integer linear programming (MILP) Problem (1) cannot be solved directly by an off-the-shelf solver. The problem is thus decomposed by making use of the recourse functions. \[Q_{\mathrm{s}}(\mathbf{x}):=\min_{y\in\mathcal{Y}_{\mathrm{s}}}\mathbf{s}^{ \mathrm{T}}y\ \mathrm{s.t.}\ \mathbf{W}_{\mathrm{s}}y\leq\mathbf{h}_{\mathrm{s}}-\mathbf{T}_{ \mathrm{s}}\mathbf{x}. \tag{2}\]  
It is well-known that \(\mathbf{x}\mapsto Q_{\mathrm{s}}(\mathbf{x})\) is a non-smooth convex function of \(\mathbf{x}\). If the above subproblem has a solution, then a subgradient of \(Q_{\mathrm{s}}\) at \(\mathbf{x}\) can be computed by making use of a Lagrange multiplier, \(\pi_{\mathrm{s}}\), associated with a constraint, \(\mathbf{W}_{\mathrm{s}}y_{\mathrm{s}}\leq\mathbf{h}_{\mathrm{s}}-\mathbf{T}_{ \mathrm{s}}\mathbf{x}\): \(-\mathbf{T}_{\mathrm{s}}^{\mathrm{T}}\pi_{\mathrm{s}}\in\partial Q_{\mathrm{s} }(\mathbf{x})\). On the other hand, if the recourse function \(Q_{\mathrm{s}}\) is infeasible, then the point \(\mathbf{x}\) can be cutoff by adding a feasibility cut [5]. Let \(\mathcal{P}\) be a partition of \(\mathcal{S}\) into \(\mathrm{w}\) subsets: \(\mathcal{P}=\{P_{1},\ldots,P_{\mathrm{w}}\}\), with \(P_{\mathrm{j}}\
eq\varnothing\) for all \(\mathrm{j}\in\{1,\,\ldots,\mathrm{w}\}\), and \(P_{\mathrm{j}}\cap P_{\mathrm{i}}=\varnothing\) for \(\mathrm{i}\
eq\mathrm{j}\). By defining \(f^{\mathrm{j}}(\mathbf{x}):=\sum\limits_{\mathrm{s}\in P_{\mathrm{j}}}Q_{ \mathrm{s}}(\mathbf{x})\), Problem (1) can be rewritten as  
\[f_{\star}=\min_{x\in\mathcal{X}}\mathbf{c}^{\mathrm{T}}x+f^{\mathrm{1}}(x)+ \ldots+f^{\mathrm{w}}(x). \tag{3}\]  
In our notation, \(\mathrm{w}\) stands for the number of workers evaluating the recourse functions. The workers \(\mathrm{j}\in\{1,\ldots,\mathrm{w}\}\) are processes running on a single machine or multiple machines.