A non-monotone proximal gradient algorithm for solving nonsmooth multiobjective optimization problems with an extending application to robust multiobjective optimization

In this paper, a new non-monotone proximal gradient method (NPGM) for solving nonsmooth multiobjective optimization problems (MOPs) is introduced by the non-monotone backtracking linear search technique allowing the increase of objective function values in some iterative processes. The level-bounded notion of vector-valued functions is also presented. The asymptotical characterizations of the corresponding sequences generated by NPGM are obtained under some mild conditions. We establish the global convergence of the iterative sequence generated by NPGM and the sufficient condition for the existence of Pareto critical point which is the accumulation point of the iterative sequence. The linear convergence of NPGM is also obtained under the strong convexity and smoothness of the objective functions. As an application, we present a tractable second-order cone programming formulation of robust multiobjective optimization problem (RMOP) by the constraint scalarization method, and then the RMOP is also solved by NPGM. Finally, some numerical experiments are reported to demonstrate the feasibility and efficiency of NPGM for solving MOPs.