A global Barzilai and Borwein's normalized gradient descent method for unconstrained multiobjective optimization problems

The Armijo line search may yield excessively small stepsizes along the steepest descent direction in multiobjective optimization, leading to slow convergence. To address this issue, we propose a global Barzilai and Borwein's normalized gradient descent method for unconstrained multiobjective optimization (GBBNMO). The proposed method enhances both the descent direction and the stepsize strategy: a novel normalization technique is incorporated into the direction-finding subproblem to generate an improved descent direction, and a global Barzilai and Borwein's line search for multiobjective optimization is introduced to obtain larger stepsizes. We show that every accumulation point of the sequence generated by GBBNMO is a Pareto critical point. The convergence rates of GBBNMO are established as O(1k) in the non-convex case, O(1k) for convex cases, and O(rk) with 0<r<1 for strongly convex cases. Numerical experiments demonstrate the efficiency of the proposed method.