Novel Reformulations and Efficient Algorithms for the Generalized Trust Region Subproblem

Abstract

We present a new solution framework to solve the generalized trust region subproblem (GTRS) of minimizing a quadratic objective over a quadratic constraint. More specifically, we derive a convex quadratic reformulation (CQR) via minimizing a linear objective over two convex quadratic constraints for the GTRS. We show that an optimal solution of the GTRS can be recovered from an optimal solution of the CQR. We further prove that this CQR is equivalent to minimizing the maximum of the two convex quadratic functions derived from the CQR for the case under investigation. Although the latter minimax problem is nonsmooth, it is well structured and convex. We thus develop two steepest descent algorithms corresponding to two different line search rules. We prove global sublinear convergence rates for both algorithms. We also obtain a local linear convergence rate of the first algorithm by estimating the Kurdyka-Łojasiewicz exponent at any optimal solution under mild conditions. We finally demonstrate the efficiency of our algorithms with numerical experiments.