## 3 Framework
### Online Phase
#### 3.3.1 Learning-Guided Fixing  
The GNN is designed to output predictions exclusively for binary variables, where each value is constrained to the interval [0, 1] via a sigmoid activation function. These values represent stability confidence scores, with higher values indicating greater stability of the corresponding integer variable. To identify stable variables to be fixed, a dual-threshold criterion is applied. The set of binary variables \(x_{g,t}\) for \(x\in\{u,s,d\}\) and \(\text{g}\in\hat{G},\ \ \text{t}\in\ \hat{T}\) is considered, where  
* \(u_{g,t}\), \(s_{g,t}\), and \(d_{g,t}\) represent commitment variables, startup variables, and shutdown variables of unit \(g\) in time \(t\);
* \(\hat{G}\) and \(\hat{T}\) denote the set of units and time periods, respectively. We define a stable set \(\mathbb{S}^{stable}\) as follows:  
\[\mathbb{S}^{stable}=\Big{\{}(x,g,t)\Big{|}x\in\{u,s,d\},\text{g}\in\hat{G},\ \ \text{t}\in\ \hat{T},\mathbb{X}^{x}_{g,t}\geq\ \tau^{stable}\wedge\Big{(}r^{x}_{g,t}\leq\tau^{v}\lor r^{x}_{g,t}\geq 1 -\tau^{v}\Big{)}\Big{\}} \tag{18}\]  
where  
* \(\mathbb{X}^{x}_{g,t}\) represents the stable predictor output (stability confidence score) for the \((g,t)\)-th index and the variable type \(x\);
* \(r^{x}_{g,t}\) represents the LP relaxation value for the \((g,t)\)-th index and the variable type \(x\);
* \(\tau^{stable}\) is the stability threshold, such that variables with \(\mathbb{X}^{x}_{g,t}\geq\tau^{stable}\) are considered stable;
* \(\tau^{v}\) is the integrality tolerance threshold, ensuring that \(r^{x}_{g,t}\) is sufficiently close to a binary value. Only variables fulfilling both conditions are provisionally classified as stable. This joint criterion ensures that stability predictions are both statistically confident and theoretically aligned with the problem's LP relaxation properties, thereby reducing false positives caused by spurious correlations in the GNN's latent space. The threshold parameters are tunable to balance solution quality and computational efficiency across varying UCP instances. While fixing the generator state-transition variables \(s_{g,t}\) (startup) and \(d_{g,t}\) (shutdown) to 1 at specific time steps risks suboptimality due to the dynamic adjustability of power outputs, we propose a temporally flexible fixing strategy to balance solution quality and computational efficiency. Generators often exhibit flexibility in their startup/shutdown timing, and delaying or advancing commitments by several time steps may still yield feasible and near-optimal solutions. The fixation strategy can be classified into hard fix and soft fix, depending on the degree of flexibility allowed. Rigid fixation to a single time step may exclude such solutions, while allowing a small temporal window \(\Delta\) to preserve optimality while reducing branching. For each unit \(g\) and time step \(t\), the following fixation rules are applied:* Hard fix: \(\mathbb{S}^{hard}\) is denoted as the set of indices that are hard-fixed. These variables are forced into a fixed state, ensuring that the solution adheres to the rounded LP relaxation value for each time step. We define \(\mathbb{S}^{hard}\) as follows: \[\mathbb{S}^{hard}=\left\{\forall x\in\left\{u,s,d\right\},\left(x,g,t\right\} \in\mathbb{S}^{stable}\left|\sum_{j=max(0,t-\Delta)}^{min(T,t+\Delta)}r_{g,j}^ {s}+r_{g,j}^{d}=0\right\}\right.\] (19) For a given time \(t\) of unit \(g\), if all three binary variables (commitment, startup, and shutdown) are stable, and there are no startup or shutdown actions in the neighboring time steps, then these variables will be hard-fixed to their rounded LP relaxation value. It is worth noting that a partial binary solution from the rounded LP relaxation value does not guarantee feasibility. * Soft fix: \(\mathbb{S}^{soft}\) is denoted as the set of indices that are soft-fixed, \(\mathbb{S}^{soft}=\mathbb{S}^{stable}-\mathbb{S}^{hard}\).