Aragón-Artacho Francisco J, Boţ Radu I, Torregrosa-Belén David
Department of Mathematics, University of Alicante, San Vicente del Raspeig, 03690 Alicante Spain.
Faculty of Mathematics, University of Vienna, Vienna, 1090 Austria.
Numer Algorithms. 2023;93(1):103-130. doi: 10.1007/s11075-022-01405-9. Epub 2022 Nov 18.
In this work, we study resolvent splitting algorithms for solving composite monotone inclusion problems. The objective of these general problems is finding a zero in the sum of maximally monotone operators composed with linear operators. Our main contribution is establishing the first primal-dual splitting algorithm for composite monotone inclusions with minimal lifting. Specifically, the proposed scheme reduces the dimension of the product space where the underlying fixed point operator is defined, in comparison to other algorithms, without requiring additional evaluations of the resolvent operators. We prove the convergence of this new algorithm and analyze its performance in a problem arising in image deblurring and denoising. This work also contributes to the theory of resolvent splitting algorithms by extending the minimal lifting theorem recently proved by Malitsky and Tam to schemes with resolvent parameters.
在这项工作中，我们研究用于求解复合单调包含问题的预解式分裂算法。这些一般问题的目标是在与线性算子复合的极大单调算子之和中找到一个零点。我们的主要贡献是建立了首个具有最小提升的复合单调包含问题的原始对偶分裂算法。具体而言，与其他算法相比，所提出的方案降低了定义基础不动点算子的乘积空间的维度，且无需对预解式算子进行额外求值。我们证明了这种新算法的收敛性，并在图像去模糊和去噪中出现的一个问题中分析了其性能。这项工作还通过将Malitsky和Tam最近证明的最小提升定理扩展到具有预解式参数的方案，为预解式分裂算法理论做出了贡献。
