A convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid (Q1016108)

The authors study a class of nonconvex optimization problems which possesses a hidden convexity property, that is, which can be shown to be equivalent to a convex problem and thus can be efficiently solved. The nonconvex minimization problem (RQ) considered in this paper involves an objective function which is the ratio of two nonconvex quadratic functions over a possibly degenerate ellipsoid. This formulation is a generalization of the so-called Regularized Total Least Squares problem (RTLS). This problem RQ can be recast as a semidefinite programming problem with no gap, and its global optimal solution can be extracted from the optimal solution of this semidefinite formulation. Then an extension of an iterative procedure used for the problem RTLS is proposed for solving the problem RQ. The corresponding algorithm is proven to be superlinearly convergent. Furthermore it produces an \(\epsilon\)-global solution after at most \(O(\sqrt{\ln\varepsilon^{-1}})\) iterations.