Adaptive proximal Barzilai-Borwein methods for nonlinear composite optimization

This work introduces gradient-based optimization methods for solving nonlinear composite optimization that leverage Barzilai-Borwein (BB) stepsizes to accelerate the convergence. For convex objective functions, we propose new adaptive stepsize rules that eliminate the need for traditional line search procedures, thereby enhancing both robustness and computational efficiency. In the nonconvex setting, we naturally generalize the stepsize rules derived from the convex case and develop a novel nonmonotone line search strategy to ensure global convergence and efficiency. We establish global convergence for the proposed methods and analyze the convergence rate in the convex setting under appropriate conditions. Our extensive numerical experiments demonstrate very promising performance of the proposed methods compared with other state-of-the-art proximal gradient methods that employ adaptive stepsizes or line searches to solve both convex and nonconvex composite optimization.