Spectral conjugate gradient methods for vector optimization problems

In this work, we present an extension of the spectral conjugate gradient (SCG) methods for solving unconstrained vector optimization problems, with respect to the partial order induced by a pointed, closed and convex cone with a nonempty interior. We first study the direct extension version of the SCG methods and its global convergence without imposing an explicit restriction on parameters. It shows that the methods may lose their good scalar properties, like yielding descent directions, in the vector setting. By using a truncation technique, we then propose a modified self-adjusting SCG algorithm which is more suitable for various parameters. Global convergence of the new scheme covers the vector extensions of three different spectral parameters and the corresponding Perry, Andrei, and Dai–Kou conjugate parameters (SP, N, and JC schemes, respectively) without regular restarts and any convex assumption. Under inexact line searches, we prove that the sequences generated by the proposed methods find points that satisfy the first-order necessary condition for Pareto-optimality. Finally, numerical experiments illustrating the practical behavior of the methods are presented.