A Steepest Descent Method for Vector Optimization
B. F. Svaiter | Graña Drummond, L. M.
Pareto optimality | vector optimization | steepest descent | K-convexity | quasi-Fejer convergence
In this work we propose a Cauchy-like method for solving smooth unconstrained vector optimization problems. When the partial order under consideration is the one induced by the nonnegative orthant, we regain the steepest descent method for multicriteria optimization recently proposed by J. Fliege and B.F. Svaiter. We prove that every accumulation point of the generated sequence satisfies a certain first order necessary condition for optimality, which extends to the vector case the well known ``gradient equal zero'' condition for real-valued minimization. Finally, under some reasonable additional hypotheses, we prove (global) convergence to a weak unconstrained minimizer. As a by-product, we show that the problem of finding a weak constrained minimizer can be viewed as a particular case of the Abstract Equilibrium Problem.