Spectrally Generated Cone Programming: A Unified Operator-Induced Framework for Eigenvalue-Constrained Optimization

This paper establishes a novel optimization and geometric structure theory—the theory of spectral generating cones. Unlike the assumption of pre-fixed cone sets in traditional cone programming, we introduce a spectral structure induced by parameterized symmetric matrix operators, making the cone sets endogenously generated by the spectral minimum eigenvalue condition. We prove that spectral generating cones form closed cones under homogeneous operator conditions and provide their convexity criterion, representation theorem, and polar ray structure characterization. Furthermore, we establish its complete Lagrange duality theory and self-dual conditions, and prove that it includes general cone programming and semidefinite programming as strict special cases. Finally, we give the geometric stability and spectral continuity theorem of spectral generating cones under perturbations. This theory realizes a paradigm shift from externally given cone structures to endogenously spectrally induced cone structures.