On the Dai–Liao conjugate gradient method for vector optimization

Abstract

Several conjugate gradient (CG) parameters have led to promising methods for optimization problems. However, some parameters, such as the Dai-Liao (DL+) and modified Polak-Ribière-Polyak (EPRP) methods, have not yet been explored in the vector optimization setting. This paper addresses this gap by proposing the DL+ and EPRP+ methods to vector optimization. We start by developing a self-adjusting DL+ algorithm to find critical points of vector-valued functions within a closed, convex, pointed cone with a nonempty interior. The algorithm is designed to satisfy the sufficient descent condition (SDC) with a Wolfe line search; if this condition is not satisfied, the algorithm adjusts by redefining the DL+ to meet the SDC. We establish the global convergence of this algorithm under the assumption of SDC without requiring regular restarts or convexity assumption on the objective functions. Similarly, we developed the EPRP+ algorithm using the same approach as the DL+ algorithm. Furthermore, under exact line search, the DL+ and EPRP+ methods reduce to Hestenes-Stiefel (HS) and Polak-Ribière-Polyak (PRP) methods, respectively. We present numerical experiments on some selected test problems sourced from the multiobjective optimization literature that demonstrate the effectiveness and practical implementation of the proposed algorithms, highlighting their promising potential.