New Globalized Newton-Type Methods for Nonconvex Optimization Problems

Abstract

Newton's method is one of the most effective second-order algorithms for smooth optimization because of its fast local convergence. However, existing globally convergent Newton-type methods typically require convexity or strong convexity of the objective function, while approaches for nonconvex optimization often rely on Hessian regularization at every iteration. In this paper, we propose a general line-search Newton framework for unconstrained optimization that avoids repeated Hessian regularization by exploiting the Newton direction only when it is well-defined and suitable. The proposed framework encompasses several existing hybrid gradient--Newton methods as special cases and naturally yields a new extragradient Newton method. We establish global convergence under mild assumptions, including the Polyak--Lojasiewicz--Kurdyka (PLK) condition, allowing both isolated and nonisolated accumulation points. We further prove local superlinear and quadratic convergence under appropriate regularity assumptions. Finally, we apply the proposed framework to strongly quasiconvex optimization and provide, to the best of our knowledge, the first Newton-type algorithm together with a comprehensive convergence analysis for this important class of nonconvex optimization problems. Numerical experiments demonstrate the effectiveness of the proposed methods.