We present an iterative proximal inertial forward-backward method with memory effects, based on recent advances in solving scalar convex optimization problems and monotone inclusions, for determining weakly efficient solutions to convex vector optimization problems consisting in vector-minimizing the sum of a differentiable vector function with a nonsmooth one, by making use of some adaptive linear scalarization techniques. During the talk, the difficulties encountered while formulating the algorithm and proving its convergence will be stressed, while the related (still unsolved) challenge of extending the celebrated FISTA method from scalar to vector optimization problems will be mentioned, too. The talk is based on joint work with Radu Ioan Boț.