  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.
