In this paper, we propose some new proximal quasi-Newton methods with line search or without line search for a special class of nonsmooth multiobjective optimization problems, where each objective function is the sum of a twice continuously differentiable strongly convex function and a proper convex but not necessarily differentiable function. In these new proximal quasi-Newton methods, we approximate the Hessian matrices by using the well known BFGS, self-scaling BFGS, and the Huang BFGS method. We show that each accumulation point of the sequence generated by these new algorithms is a Pareto stationary point of the multiobjective optimization problem. In addition, we give their applications in robust multiobjective optimization, and we show that the subproblems of proximal quasi-Newton algorithms can be regarded as quadratic programming problems. Numerical experiments are carried out to verify the effectiveness of the proposed method.
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