An efficient algorithm for large-scale extended trust-region subproblems with non-intersecting linear constraints

In this paper, we study the extended trust-region subproblem in which the trust-region intersects the ball with m linear inequality constraints (m-eTRS). We assume that the linear constraints do not intersect inside the ball. We show that the optimal solution of m-eTRS can be found by solving one TRS, computing the local non-global minimizer of TRS if it exists and solving at most two TRSs with an additional linear equality constraint (1-eqTRS). Both TRS and (1-eqTRS) are polynomially and efficiently solvable, thus the new algorithm significantly improves over the SOCP/SDP relaxation of Burer and Yang [Math Program 149(1-2):253–264, 2015]. on two classes of test problems, the efficiency of the proposed approach is compared with the SOCP/SDP relaxation and branch and bound algorithm of Beck and Pan [J Global Optim 69(2):309–342, 2017].