  We derive efficient algorithms to compute weakly Pareto optimal solutions for
smooth, convex and unconstrained multiobjective optimization problems in
general Hilbert spaces. To this end, we define a novel inertial gradient-like
dynamical system in the multiobjective setting, whose trajectories converge
weakly to Pareto optimal solutions. Discretization of this system yields an
inertial multiobjective algorithm which generates sequences that converge
weakly to Pareto optimal solutions. We employ Nesterov acceleration to define
an algorithm with an improved convergence rate compared to the plain
multiobjective steepest descent method (Algorithm 1).
  A further improvement in terms of efficiency is achieved by avoiding the
solution of a quadratic subproblem to compute a common step direction for all
objective functions, which is usually required in first order methods. Using a
different discretization of our inertial gradient-like dynamical system, we
obtain an accelerated multiobjective gradient method that does not require the
solution of a subproblem in each step (Algorithm 2). While this algorithm does
not converge in general, it yields good results on test problems while being
faster than standard steepest descent by two to three orders of magnitude.

Optimization and Control