COMPUTING INEXACT K-STEEPEST DESCENT DIRECTIONS A NEW LINE SEARCH PROCEDURE FOR VECTOR OPTIMIZATION

Abstract

In this work, we proposes a new linear search and a way for the computation of σ-approximate direction. Yunda Dong, in 2010 and 2012, introduced a new linear search procedure for Conjugated Gradient methods using only rst-order information, i.e., without working with functional values. We extend his works to Vector Optimization. We stud conjugate gradient methods, showing convergence when the following βk's are used: Fletcher-Reeves, conjugate descent, Dai-Yuan, Polak-Ribière-Polyak, and Hestenes-Stiefel. We also use this line search in the gradient method, showing its convergence. In 2004, Iusem and Graña Drummond introduced the concept of σ-approximate K-steepest descent direction. They showed that by replacing the Cauchy direction with these directions, the convergence result of the generated sequence is the same: every accumulation point is critical. We will present an e cient procedure for computing these directions when the cone K is nitely generated.