Preprint A21/2001
A Steepest Descent Method for Vector Optimization

B. F. Svaiter | GraĆ±a Drummond, L. M.

**Keywords: **
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.