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, which 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.

我们推导出有效的算法来计算一般希尔伯特空间中平滑、凸和无约束多目标优化问题的弱帕累托最优解。为此，我们在多目标设置中定义了一种新颖的惯性梯度动力系统，其轨迹弱收敛于帕累托最优解。该系统的离散化产生了惯性多目标算法，该算法生成弱收敛于帕累托最优解的序列。我们采用 Nesterov 加速来定义一种算法，与普通的多目标最速下降法（算法 1）相比，该算法具有更高的收敛速度。通过避免求解二次子问题来计算所有目标函数的公共步骤方向，可以进一步提高效率，这在一阶方法中通常需要。使用类惯性梯度动力系统的不同离散化，我们获得了一种加速多目标梯度方法，该方法不需要在每个步骤中求解子问题（算法 2）。虽然该算法通常不会收敛，但它在测试问题上产生了良好的结果，同时比标准最速下降更快。