## 5 Conclusions  
This work proposes a GNN-LP hybrid learning framework to accelerate UCP optimization through stability-driven variable fixation and adaptive search strategies. Key contributions include the following:  
1. Variable Stability Classification: We design a variable fixation strategy that integrates GNN-predicted stability and LP root relaxation to guide the hard and soft fixation of binary variables. Stable variables are fixed to shrink the search space, while temporal flexibility is preserved for variables with uncertain stability. This layered fixation  
Figure 5: The primal gap as a function of runtime, averaged over 36 instances. From left to right, the units are 50, 100, and 1080.  
Figure 6: The primal gap as a function of runtime, averaged over 36 instances. From left to right, the units are 50, 100, and 1080.  
mechanism significantly reduces suboptimality, as confirmed by ablation studies across 50-, 100-, and 1080-unit systems.
2. Utilization of Multi-Type Binary Variables: Instead of focusing solely on commitment variables, we incorporate startup and shutdown variables into both the MIP formulation and learning framework. These auxiliary variables enhance the stability inference of commitment variables to enable safer and more accurate hard fixation, thereby reducing suboptimality and introducing additional stable candidates for soft fixation, which accelerates the solution process by further reducing the search space. Experiments show that this joint modeling approach consistently improves primal gaps and solving times over single-variable baselines, particularly in large-scale settings.
3. Stability-Aware Branching Prioritization: We leverage predicted variable stability to guide the branching process in branch-and-bound trajectories. By assigning a higher branching priority to unstable variables, the solver is able to explore critical decisions earlier, tighten bounds faster, and accelerate convergence. This strategy achieved faster gap reduction and earlier convergence.  
While the proposed branching prioritization strategy based on variable stability has demonstrated significant efficiency gains, it remains relatively simple in its current form. Additionally, the success of incorporating auxiliary binary variables into our learning and optimization framework suggests broader applicability. Future work could investigate which types or structures of multi-variable models yield the best performance, and how such strategies can be generalized to other UCP acceleration frameworks, for example, by leveraging power system topology via graph-based representations to enhance prediction accuracy and generalization.  
Conceptualization, L.Y. and H.Z.; methodology, L.Y.; software, P.L.; validation, L.Y., P.L. and S.C.; formal analysis, L.Y.; investigation, L.Y.; resources, H.Z.; data curation, L.Y.; writing--original draft preparation, L.Y.; writing--review and editing, P.L. and S.C.; visualization, P.L.; supervision, H.Z.; project administration, H.Z.; funding acquisition, H.Z. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Natural Science Foundation of China (72361003) and the Key Research and Development Program of Guangxi (GuiKe-AB23075199).  
Not applicable.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflict of interest.