A nonmonotone proximal quasi-Newton method for multiobjective optimization

Abstract: This paper proposes a nonmonotone proximal quasi-Newton algorithm for unconstrained convex multiobjective composite optimization problems. To design the search direction, we minimize the max-scalarization of the variations of the Hessian approximations and nonsmooth terms. Subsequently, a nonmonotone line search is used to determine the step size, we allow for the decrease of a convex combination of recent function values. Under the assumption of strong convexity of the objective function, we prove that the sequence generated by this method converges to a Pareto optimal. Furthermore, based on the strong convexity, Hessian continuity and Dennis-Mor\'{e} criterion, we use a basic inequality to derive the local superlinear convergence rate of the proposed algorithm. Numerical experiments results demonstrate the feasibility and effectiveness of the proposed algorithm on a set of test problems.