  The imbalances and conditioning of the objective functions influence the
performance of first-order methods for multiobjective optimization problems
(MOPs). The latter is related to the metric selected in the direction-finding
subproblems. Unlike single-objective optimization problems, capturing the
curvature of all objective functions with a single Hessian matrix is
impossible. On the other hand, second-order methods for MOPs use different
metrics for objectives in direction-finding subproblems, leading to a high
per-iteration cost. To balance per-iteration cost and better curvature
exploration, we propose a Barzilai-Borwein descent method with variable metrics
(BBDMO\_VM). In the direction-finding subproblems, we employ a variable metric
to explore the curvature of all objectives. Subsequently, Barzilai-Borwein's
method relative to the variable metric is applied to tune objectives, which
mitigates the effect of imbalances. We investigate the convergence behaviour of
the BBDMO\_VM, confirming fast linear convergence for well-conditioned problems
relative to the variable metric. In particular, we establish linear convergence
for problems that involve some linear objectives. These convergence results
emphasize the importance of metric selection, motivating us to approximate the
trade-off of Hessian matrices to better capture the geometry of the problem.
Comparative numerical results confirm the efficiency of the proposed method,
even when applied to large-scale and ill-conditioned problems.
