Spectral–Conic Coupled Optimization: A Unified Framework for Cone Geometry and Constraint Eigen-Decomposition

Cone programming is a fundamental concept in modern optimization theory, widely applied to semidefinite programming, second-order cone programming, and convex optimization problems in machine learning. However, traditional cone programming only utilizes the geometric structure of variables in the cone space, neglecting the latent spectral structure within the constraint matrix or variables. Constrained eigenvalue decomposition optimization methods achieve dimensionality reduction and structured solutions through matrix eigenvalue analysis, but lack a unified feasible region framework, limiting efficiency in high-dimensional problems. This paper proposes Spectral–Conic Coupled Optimization (SCCO), which achieves coupled optimization of cone geometry and spectral structure by unifying the modeling of the cone feasible region and the eigenspectral structure of the constraint matrix, and introducing a spectral cone mapping operator. This model derives new optimization equations and provides optimality conditions, duality theory, convergence properties, and degree-of-freedom analysis. Theoretical proofs demonstrate that this model significantly reduces the effective degrees of freedom in high-dimensional problems while maintaining convex optimization stability, providing a new theoretical tool for large-scale optimization. This paper also designs an algorithmic framework based on spectral alternation optimization and proposes potential applications in neural network weight optimization, graph embedding, and semidefinite programming acceleration.