Root-max Problems, Hybrid Expansion-Contraction, and Quadratically Convergent Optimization of Passive Systems

The paper’s Theorem 4.9 establishes Q‑quadratic local convergence of the HEC-based Algorithm 2 under assumptions (i)–(v), by translating the general HEC result (Theorem 3.8) and invoking the implicit function theorem to guarantee smoothness when the relevant eigenvalue is simple. The candidate solution proves the same claim via a different route: it defines the contraction as an implicit fixed-point map ξ̂=M(ξ), shows M'(ξ̃)=0 at the pseudoroot, and concludes Q‑quadratic convergence. It further derives an explicit formula for M''(ξ̃) and justifies the smoothness of γ and of the stationary path using standard Hermitian eigenvalue perturbation theory and the implicit function theorem. Assumptions and conclusions align with the paper, but the proof strategies differ.