Alternative extension of the Hager–Zhang conjugate gradient method for vector optimization

Recently, Gonçalves and Prudente proposed an extension of the Hager–Zhang nonlinear conjugate gradient method for vector optimization (Comput Optim Appl 76:889–916, 2020). They initially demonstrated that directly extending the Hager–Zhang method for vector optimization may not result in descent in the vector sense, even when employing an exact line search. By utilizing a sufficiently accurate line search, they subsequently introduced a self-adjusting Hager–Zhang conjugate gradient method in the vector sense. The global convergence of this new scheme was proven without requiring regular restarts or any convex assumptions. In this paper, we propose an alternative extension of the Hager–Zhang nonlinear conjugate gradient method for vector optimization that preserves its desirable scalar property, i.e., ensuring sufficiently descent without relying on any line search or convexity assumption. Furthermore, we investigate its global convergence with the Wolfe line search under mild assumptions. Finally, numerical experiments are presented to illustrate the practical behavior of our proposed method.