DRSOM: A Dimension Reduced Second-Order Method

In this paper, we propose a Dimension-Reduced Second-Order Method (DRSOM) for convex and nonconvex (unconstrained) optimization. Under a trust-region-like framework, our method preserves the convergence of the second-order method while using only curvature information in a few directions. Consequently, the computational overhead of our method remains comparable to the first-order such as the gradient descent method. Theoretically, we show that the method has a local quadratic convergence and a global convergence rate of $O(\epsilon^{-3/2})$ to satisfy the first-order and second-order conditions if the subspace satisfies a commonly adopted approximated Hessian assumption. We further show that this assumption can be removed if we perform a corrector step using a Krylov-like method periodically at the end stage of the algorithm. The applicability and performance of DRSOM are exhibited by various computational experiments, including $L_2 - L_p$ minimization, CUTEst problems, and sensor network localization.