We consider inertial iteration methods for Fermat–Weber location problem and primal–dual three-operator splitting in real Hilbert spaces. To do these, we first obtain weak convergence analysis and nonasymptotic O(1/n) convergence rate of the inertial Krasnoselskii–Mann iteration for fixed point of nonexpansive operators in infinite dimensional real Hilbert spaces under some seemingly easy to implement conditions on the iterative parameters. One of our contributions is that the convergence analysis and rate of convergence results are obtained using conditions which appear not complicated and restrictive as assumed in other previous related results in the literature. We then show that Fermat–Weber location problem and primal–dual three-operator splitting are special cases of fixed point problem of nonexpansive mapping and consequently obtain the convergence analysis of inertial iteration methods for Fermat–Weber location problem and primal–dual three-operator splitting in real Hilbert spaces. Some numerical implementations are drawn from primal–dual three-operator splitting to support the theoretical analysis.

我们考虑惯性迭代方法来求解实际希尔伯特空间中的Fermat–Weber位置问题和原对偶三算子分裂。为此，我们首先获得弱收敛分析和非渐近O（1 / n）在看似容易实现的迭代参数条件下，在无穷维实Hilbert空间中非膨胀算子的固定点的惯性Krasnoselskii–Mann迭代的收敛速度。我们的贡献之一是，收敛条件的分析和收敛速度是通过使用条件而获得的，这些条件看起来并不像文献中其他先前的相关结果所假定的那样复杂且具有限制性。然后，我们证明Fermat–Weber位置问题和原双对三算子分裂是非扩张映射不动点问题的特例，因此获得了Fermat–Weber位置问题和原双对偶三算子惯性迭代方法的收敛性分析。在真实的希尔伯特空间中分裂。