## 2 Materials and Methods  
Current research on feasible region analysis includes methods such as the vertex search method [9, 10, 11], multi-parametric programming [12, 13, 14], Fourier-Motzkin elimination [15, 16, 17], and the Minkowski sum approach [18, 19, 20, 21]. In [9, 10, 11], the vertex search method is utilized to construct the external boundary of the feasible region by starting from an initial point within the region and iteratively searching along possible directions for adjacent vertices. This method is mathematically precise but exhibits slow computational speed for solving systems. In [12, 13, 14], multi-parametric programming is employed to analyze the variation of the feasible region under parameterized constraints. This method explores the impact of parameters on solutions and generates a set of critical regions, thereby determining the topological structure of the feasible region under different parameter conditions. But its practical adoption is constrained by scalability issues in high-dimensional systems and a reliance on simplified models. Fourier-Motzkin elimination is an exact method for projecting high-dimensional problems onto a lower-dimensional space, which rigorously ensures the mathematical equivalence of the solution space [15, 16, 17]. The projected lower-dimensional feasible region is fully consistent with the original system. However, as the number of constraints increases, the elimination combinations grow exponentially, leading to numerous redundant constraints. In complex scenarios, this results in high computational complexity and slow solution times. The exact computation of the Minkowski sum of arbitrary convex polytopes is an NP-hard problem, and no effective solution currently exists [22]. To address this, refs. [18, 19] introduce the concept of basic polytopes, wherein the feasible region of each distributed resource is obtained by stretching and translating the basic polytope. This ensures that the feasible regions of all distributed resources share the same structure, simplifying the computational complexity of Minkowski sum operations. References [20, 21] utilize the zonotope, a special type of polytope characterized by its efficiency in Minkowski sum computations. This approach transforms the problem of the maximal internal approximation of the feasible region into an optimization problem that maximizes the sum of diameter similarities across multiple directions. However, these approaches require manually specifying the geometric characteristics of the polytope, which is subject to a high degree of subjectivity, and the solution accuracy is slightly lower. The core problem in solving the UFR of MG is the process of projecting the boundary of a high-dimensional polytope, defined by a system of linear inequalities, onto a lower-dimensional space [23]. Currently, most mainstream feasible region analysis methods adopt an inner approximation strategy, which involves starting from an initial feasible point within the region and progressively expanding outward to search for the vertices or boundaries of the polytope. The primary advantage of this method lies in its ability to ensure that the search process remains within the feasible region, effectively preventing infeasibility issues. Additionally, inner approximation methods exhibit good robustness, particularly when dealing with multiple constraints in high-dimensional spaces, and are able to approach the optimal solution without compromising stability. In [18, 19, 20, 21], inner approximation strategies rely on known geometric structures or algorithmic derivations, with each step of expansion based on feasibility constraints, thus avoiding erroneous solutions that may arise from external searches in uncertain scenarios. However, despite its many advantages, the limitations of inner approximation methods should not be ignored.