A Unified Spectral Cone Programming Framework: Integrating General Conic Optimization with Schatten p-Norm Geometry

This paper proposes a new geometrically unified framework—spectral cone programming theory—based on general cone programming and Schatten p-norm theory. Unlike the traditional "regularized superposition" approach, this study reconstructs the constraint set at the geometric structural level, embedding convex cone structures in singular value space into matrix space through spectral mapping, thus forming a set of spectral cones. The introduction systematically analyzes the structural differences and integration challenges of the two theories. The second part rigorously defines the spectral cone and proves its closure and convexity conditions. The third part constructs a unified model of spectral cone programming and reveals its two-layer geometric structure. This paper emphasizes structural integration rather than formal superposition, laying the foundation for subsequent development of spectral duality theory and the spectral KKT system.