This paper proposes an efficient hypergraph partitioning framework based on a novel multi-objective non-convex constrained relaxation model. A modified accelerated proximal gradient algorithm is employed to generate diverse $k$-dimensional vertex features to avoid local optima and enhance partition quality. Two MST-based strategies are designed for different data scales: for small-scale data, the Prim algorithm constructs a minimum spanning tree followed by pruning and clustering; for large-scale data, a subset of representative nodes is selected to build a smaller MST, while the remaining nodes are assigned accordingly to reduce complexity. To further improve partitioning results, refinement strategies including greedy migration, swapping, and recursive MST-based clustering are introduced for partitions. Experimental results on public benchmark sets demonstrate that the proposed algorithm achieves reductions in cut size of approximately 2\%--5\% on average compared to KaHyPar in 2, 3, and 4-way partitioning, with improvements of up to 35\% on specific instances. Particularly on weighted vertex sets, our algorithm outperforms state-of-the-art partitioners including KaHyPar, hMetis, Mt-KaHyPar, and K-SpecPart, highlighting its superior partitioning quality and competitiveness. Furthermore, the proposed refinement strategy improves hMetis partitions by up to 16\%. A comprehensive evaluation based on virtual instance methodology and parameter sensitivity analysis validates the algorithm's competitiveness and characterizes its performance trade-offs.

State-of-the-art hypergraph partitioners follow the multilevel paradigm that constructs multiple levels of progressively coarser hypergraphs that are used to drive cut refinement on each level of the hierarchy. Multilevel partitioners are subject to two limitations: (i) hypergraph coarsening processes rely on local neighborhood structure without fully considering the global structure of the hypergraph; and (ii) refinement heuristics risk entrapment in local minima. In this paper, we describe K-SpecPart, a supervised spectral framework for multi-way partitioning that directly tackles these two limitations. K-SpecPart relies on the computation of generalized eigenvectors and supervised dimensionality reduction techniques to generate vertex embeddings. These are computational primitives that are fast and capture global structural properties of the hypergraph that are not explicitly considered by existing partitioners. K-SpecPart then converts the vertex embeddings into multiple partitioning solutions. K-SpecPart introduces the idea of ''ensembling'' multiple solutions via a cut-overlay clustering technique that often enables the use of computationally demanding partitioning methods such as ILP (integer linear programming). Using the output of a standard partitioner as a supervision hint, K-SpecPart effectively combines the strengths of established multilevel partitioning techniques with the benefits of spectral graph theory and other combinatorial algorithms. K-SpecPart significantly extends ideas and algorithms that first appeared in our previous work on the bipartitioner SpecPart. Our experiments demonstrate the effectiveness of K-SpecPart. For bipartitioning, K-SpecPart produces solutions with up to 15% cutsize improvement over SpecPart. For multi-way partitioning, K-SpecPart produces solutions with up to 20% cutsize improvement over leading partitioners hMETIS and KaHyPar.