Proximal splitting algorithms for monotone inclusions (and convexoptimization problems) in Hilbert spaces share the common feature to guaranteefor the generated sequences in general weak convergence to a solution. In orderto achieve strong convergence, one usually needs to impose more restrictiveproperties for the involved operators, like strong monotonicity (respectively,strong convexity for optimization problems). In this paper, we propose amodified Krasnosel'ski\u{\i}--Mann algorithm in connection with thedetermination of a fixed point of a nonexpansive mapping and show strongconvergence of the iteratively generated sequence to the minimal norm solutionof the problem. Relying on this, we derive a forward-backward and aDouglas-Rachford algorithm, both endowed with Tikhonov regularization terms,which generate iterates that strongly converge to the minimal norm solution ofthe set of zeros of the sum of two maximally monotone operators. Furthermore,we formulate strong convergent primal-dual algorithms of forward-backward andDouglas-Rachford-type for highly structured monotone inclusion problemsinvolving parallel-sums and compositions with linear operators. The resultingiterative schemes are particularized to the solving of convex minimizationproblems.
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机译：Hilbert空间中单调包含（以及凸优化问题）的近邻分割算法具有共同的特征，即可以保证生成的序列在一般弱收敛到解中。为了实现强收敛，通常需要为所涉及的运算符施加更多的限制性属性，例如强单调性（分别对于优化问题具有强凸性）。本文提出了一种改进的Krasnosel'ski \ u {\ i} -Mann算法，用于确定非膨胀映射的不动点，并证明了迭代生成的序列与该问题的最小范数解的强收敛性。以此为基础，我们推导出了前向-后向和aDouglas-Rachford算法，它们都具有Tikhonov正则化项，它们生成的迭代强烈收敛于两个最大单调算子之和的零集的最小范数解。此外，我们针对涉及并行求和和线性算子的高度结构化单调包含问题，制定了向前-向后和Douglas-Rachford类型的强收敛原对偶算法。所得的迭代方案专门用于解决凸极小化问题。
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