  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 our 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
for both algorithms their global sublinear convergence rates. We also obtain a
local linear convergence rate of the first algorithm by estimating the Kurdyka-
Lojasiewicz exponent at any optimal solution under mild conditions. We finally
demonstrate the efficiency of our algorithms in our numerical experiments.
