On the convergence of steepest descent methods for multiobjective optimization

In this paper we consider the classical unconstrained nonlinear multiobjective optimization problem. For such a problem, it is particularly interesting to compute as many points as possible in an effort to approximate the so-called Pareto front. Consequently, to solve the problem we define an "a posteriori" algorithm whose generic iterate is represented by a set of points rather than by a single one. The proposed algorithm takes advantage of a linesearch with extrapolation along steepest descent directions with respect to (possibly not all of) the objective functions. The sequence of sets of points produced by the algorithm defines a set of "linked" sequences of points. We show that each linked sequence admits at least one limit point (not necessarily distinct from those obtained by other sequences) and that every limit point is Pareto-stationary. We also report numerical results on a collection of multiobjective problems that show efficiency of the proposed approach over more classical ones.