  We propose a first-order method to solve the cubic regularization subproblem
(CRS) based on a novel reformulation. The reformulation is a constrained convex
optimization problem whose feasible region admits an easily computable
projection. Our reformulation requires computing the minimum eigenvalue of the
Hessian. To avoid the expensive computation of the exact minimum eigenvalue, we
develop a surrogate problem to the reformulation where the exact minimum
eigenvalue is replaced with an approximate one. We then apply first-order
methods such as the Nesterov's accelerated projected gradient method (APG) and
projected Barzilai-Borwein method to solve the surrogate problem. As our main
theoretical contribution, we show that when an $\epsilon$-approximate minimum
eigenvalue is computed by the Lanczos method and the surrogate problem is
approximately solved by APG, our approach returns an $\epsilon$-approximate
solution to CRS in $\tilde O(\epsilon^{-1/2})$ matrix-vector multiplications
(where $\tilde O(\cdot)$ hides the logarithmic factors). Numerical experiments
show that our methods are comparable to and outperform the Krylov subspace
method in the easy and hard cases, respectively. We further implement our
methods as subproblem solvers of adaptive cubic regularization methods, and
numerical results show that our algorithms are comparable to the
state-of-the-art algorithms.
