Steepest descent methods for multicriteria optimization

We propose a steepest descent method for unconstrained multicriteria op-timization 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 func-tions are assumed to be Lipshitz-continuously differentiable and a constraint qualification is assumed. Under these conditions, it is shown that these meth-ods converge to a point satisfying certain first-order necessary conditions for Pareto optimality. Both methods do not scalarize the original vector opti-mization problem. Neither ordering information nor weighting factors for the different objective functions are assumed to be known. In the single objective case, we retrieve the Steepest descent method and Zoutendijk’s method of feasible directions, respectively.