A hybrid algorithm for the two-trust-region subproblem

Abstract: Two-trust-region subproblem (TTRS), which is the minimization of a general quadratic function over the intersection of two full-dimensional ellipsoids, has been the subject of several recent research. In this paper, to solve TTRS, a hybrid of efficient algorithms for finding global and local-nonglobal minimizers of trust-region subproblem and the alternating direction method of multipliers (ADMM) is proposed. The convergence of the ADMM steps to the first order stationary condition is proved under certain conditions. On several classes of test problems, we compare the new algorithm with the recent algorithm of Sakaue et. al's \cite{SakaueNakat:16} and Snopt software.