A Nonmonotone Projected Gradient Method for Multiobjective Problems on Convex Sets

In this work we consider an extension of the classical scalar-valued projected gradient method for multiobjective problems on convex sets. As in Fazzio et al. (Optim Lett 13:1365–1379, 2019) a parameter which controls the step length is considered and an updating rule based on the spectral gradient method from the scalar case is proposed. In the present paper, we consider an extension of the traditional nonmonotone approach of Grippo et al. (SIAM J Numer Anal 23:707–716, 1986) based on the maximum of some previous function values as suggested in Mita et al. (J Glob Optim 75:539–559, 2019) for unconstrained multiobjective optimization problems. We prove the accumulation points of sequences generated by the proposed algorithm, if they exist, are stationary points of the original problem. Numerical experiments are reported.