A New Hybrid Conjugate Gradient Method Based on a Convex Combination for Multiobjective Optimization

Conjugate gradient methods are a crucial class of techniques for solving unconstrained optimization problems. While the Hestenes-Stiefel and other conjugate gradient methods have recently been extended to multiobjective optimization setting-notably the Hestene-Stiefel failed to ensure descent direction. In contrast, the Conjugate Descent method consistently ensures a descent direction in this context. In this paper, we introduce a hybrid conjugate gradient method that combines a modified Hestenes-Stiefel method with the Conjugate Descent method through a convex combination. The convex coefficient parameter is derived using the Dai-Liao conjugacy condition and the condition for Newton’s direction. Crucially, this hybrid approach guarantees sufficient descent property under the Wolfe line search conditions while establishing global convergence under mild assumptions-without necessarily requiring an algorithmic restarts or assuming convexity on the objective functions. Preliminary numerical results demonstrate the clear advantages of this hybrid method over some existing conjugate gradient techniques.