Steepest descent methods for multicriteria optimization

We propose a steepest descent method for unconstrained multicriteria optimi zation and a "feasible descent direction" method for the constrained case. In the unconstrained case, the objective functions are assumed to be contin uously differentiable. In the constrained case, objective and constraint fu nctions are assumed to be Lipshitz-continuously differentiable and a constr aint qualification is assumed. Under these conditions, it is shown that the se methods converge to a point satisfying certain first-order necessary con ditions for Pareto optimality. Both methods do not scalarize the original v ector optimization problem. Neither ordering information nor weighting fact ors for the different objective functions are assumed to be known. In the s ingle objective case, we retrieve the Steepest descent method and Zoutendij k's method of feasible directions, respectively.