In the last two decades, many descent methods for multiobjective optimization problems were proposed. In particular, the steepest descent and the Newton methods were studied for the unconstrained case. In both methods, the search directions are computed by solving convex subproblems, and the stepsizes are obtained by an Armijo-type line search. As a consequence, the objective functions values decrease at each iteration of the algorithms. In this work, we consider nonmonotone line searches, i.e., we allow the increase of objective functions values in some iterations. Two types of nonmonotone line searches are considered here: the one that takes the maximum of recent functions values, and the one that takes their average. Under reasonable assumptions, we prove that every accumulation point of the sequence produced by the nonmonotone version of the steepest descent and Newton methods is Pareto critical. Moreover, we present some numerical experiments, showing that the nonmonotone technique is also efficient in the multiobjective case.
Kyoto University, September 2018