Box-constrained vector optimization: a steepest descent method without “a priori” scalarization

In this paper a notion of descent direction for a vector function defined on a box is introduced. This concept is based on an appropriate convex combination of the “projected” gradients of the components of the objective functions. The proposed approach does not involve an “apriori” scalarization since the coefficients of the convex combination of the projected gradients are the solutions of a suitable minimization problem depending on the feasible point considered. Subsequently, the descent directions are considered in the formulation of a first order optimality condition for Pareto optimality in a box-constrained multiobjective optimization problem. Moreover, a computational method is proposed to solve box-constrained multiobjective optimization problems. This method determines the critical points of the box constrained multiobjective optimization problem following the trajectories defined through the descent directions mentioned above. The convergence of the method to the critical points is proved. The numerical experience shows that the computational method efficiently determines the whole local Pareto front.