Global optimization of quadratic root-difference minimization under elliptic annulus constraints

This paper studies the nonconvex quadratic root-difference minimization under elliptic annulus constraints {\rm (QR)}. We first establish the Annulus Brickman theorem and equivalently reformulate {\rm (QR)} as a 2-dimensional convex problem {\rm (HP)} with hidden variables. We employ the Frank-Wolfe algorithm to globally solve {\rm (HP)}. A key finding is that the solutions of the Frank-Wolfe subproblems, which are traditionally viewed as mere auxiliary updates, are proven to be $O(1/\sqrt{k})$-approximate solutions of the original problem {\rm (QR)}. This transforms an algorithmic by-product into the primary output and completely bypasses the need to solve the computationally expensive quadratic system required for solution recovery. Leveraging this recovery-free property, we develop the efficient Iterative Minimum Generalized Eigenpair (IMGE) algorithm for globally solving {\rm (QR)}. Numerical experiments confirm that IMGE converges rapidly and significantly outperforms conventional methods, especially for large-scale problems.