ABSTRACT

This paper considers the problem of finding the resolvent of the sum of two maximal monotone operators. Such a problem arises frequently in practice, but it seems that computation of the solution of the problem is not necessarily easy. It is assumed that both the resolvents of two maximal monotone operators can be easily computed. This enables us to consider the case in which a solution to the problem cannot be computed easily. This paper introduces a new mapping, which satisfies the nonexpansivity property, from the individual resolvents of two maximal monotone operators and investigates some of its properties. In particular, we show that the mapping has a fixed point if and only if the problem has a solution. Then, using this mapping, we propose a splitting method for solving the problem in a real Hilbert space. In particular, we show that the sequences generated by the method converge strongly to the solution to the problem under certain assumptions. Convergence rate analysis of the methods is also provided to illustrate the method's efficiency. Finally, we apply the results to a class of optimization problems.

摘要

本文考虑求解两个最大单调算子之和的问题。这样的问题在实践中经常出现，但问题的解的计算似乎并不一定容易。假设两个最大单调算子的解算子都可以很容易地计算出来。这使我们能够考虑无法轻松计算问题的解决方案的情况。本文从两个最大单调算子的单个分解子中引入了一个满足非扩张性的新映射，并研究了它的一些性质。特别是，我们证明了当且仅当问题有解决方案时，映射才有固定点。然后，使用这种映射，我们提出了一种在真实希尔伯特空间中解决问题的分裂方法。尤其是，我们表明，在某些假设下，该方法生成的序列强烈地收敛于问题的解决方案。还提供了方法的收敛率分析以说明该方法的效率。最后，我们将结果应用于一类优化问题。