  We propose a descent subgradient algorithm for unconstrained nonsmooth
nonconvex multiobjective optimization problems. To find a descent direction, we
present an iterative process that efficiently approximates the Goldstein
subdifferential of each objective function. To this end, we develop a new
variant of Mifflin's line search in which the subgradients are arbitrary and
its finite convergence is proved under a semismooth assumption. To reduce the
number of subgradient evaluations, we employ a backtracking line search that
identifies the objectives requiring an improvement in the current approximation
of the Goldstein subdifferential. Meanwhile, for the remaining objectives, new
subgradients are not computed. Unlike bundle-type methods, the proposed
approach can handle nonconvexity without the need for algorithmic adjustments.
Moreover, the quadratic subproblems have a simple structure, and hence the
method is easy to implement. We analyze the global convergence of the proposed
method and prove that any accumulation point of the generated sequence
satisfies a necessary Pareto optimality condition. Furthermore, our convergence
analysis addresses a theoretical challenge in a recently developed subgradient
method. Through numerical experiments, we observe the practical capability of
the proposed method and evaluate its efficiency when applied to a diverse range
of nonsmooth test problems.
