The Dai–Liao-type conjugate gradient methods for solving vector optimization problems

Abstract

This paper attempts to propose Dai–Liao (DL)-type nonlinear conjugate gradient (CG) methods for solving vector optimization problems. Four variants of the DL method are extended and analysed from the scalar case to the vector setting. We first give a direct vector extended version of the modified Dai–Liao (DL+) method. Then the second algorithm is presented by combining a new extension form of the well-known Hager–Zhang (HZ) parameter. This new scheme equips a good descent property and its parameter also reduces to the classical one in the scalar case. The last two algorithms extend two sufficient descent DL-type methods. Particularly, two different vector forms of their parameters are considered and compared. As a result, we reveal some differences between scalar and vector cases to a certain extent. Without any convex assumption, the sufficient descent property and global convergence of all algorithms are established under inexact line searches. Finally, preliminary numerical experiments illustrate the practical behaviour of our algorithms by using Dolan and Moré performance profile.