  The efficient optimization method for locally Lipschitz continuous
multiobjective optimization problems from [1] is extended from
finite-dimensional problems to general Hilbert spaces. The method iteratively
computes Pareto critical points, where in each iteration, an approximation of
the subdifferential is computed in an efficient manner and then used to compute
a common descent direction for all objective functions. To prove convergence,
we present some new optimality results for nonsmooth multiobjective
optimization problems in Hilbert spaces. Using these, we can show that every
accumulation point of the sequence generated by our algorithm is Pareto
critical under common assumptions. Computational efficiency for finding Pareto
critical points is numerically demonstrated for multiobjective optimal control
of an obstacle problem.
