A Barzilai–Borwein scaling-type conjugate gradient algorithm for vector optimization

Numerous conjugate gradient (CG) methods have been studied and extended to vector optimization, but many lose the descent property, which is crucial for establishing convergence. The Barzilai–Borwein scaled conjugate gradient (BBSCG) method, despite its success in scalar optimization, has not yet been explored in the context of vector optimization. This paper introduces a novel variant of the BBSCG method, termed the VBBSCG method, specifically designed for vector optimization. The proposed method satisfies the sufficient descent condition, a property often lost when extending CG methods for scalar optimization to vector setting. Moreover, the VBBSCG method achieves global convergence under the Wolfe line search without requiring restarts or assuming convexity of the objective functions, and it avoids the computation and storage of Hessian matrices. Extensive numerical experiments on benchmark problems from the multi-objective optimization literature demonstrate that the VBBSCG method outperforms several existing CG methods as well as the steepest descent method, highlighting its superior numerical performance and practical relevance for vector optimization problems.