求解非单调+Lipschitz包含问题的非线性前向-后向算法及其在伴随失配问题中的应用
Nonlinear Forward-Backward Algorithm for Solving Non-monotone+Lipschitz Inclusions with Applications to Adjoint Mismatch Problems
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中文总结 AI 辅助
本文针对因伴随近似破坏单调性而无法用标准分裂方法的问题，提出非线性前向-后向算法，证明其收敛性并推广了部分流行算法的收敛保证，通过信号恢复实验验证了方法的适用性。
中文摘要 AI 辅助
本文提出了一种新颖的算法框架，用于求解涉及非单调且Lipschitz连续算子的一大类非线性包含问题。我们的主要动机源于伴随失配问题的研究，这类问题常出现在逆问题与数据科学领域，此时线性测量算子的伴随会被近似替代。由于该近似可能固有地破坏经典的单调性性质，分裂方法的标准收敛保证可能不再适用。为解决这一问题，我们考虑了非线性前向-后向算法，并在不假设单调性的情况下严格证明了其收敛性。通过利用扭曲预解式公式和半单调性假设，我们推导了确保弱收敛的显式条件，且在更强假设下可得到R-线性收敛。此外，我们的理论分析为若干流行方法（包括Condat-Vũ算法和前向半反射后向算法）提供了新的收敛保证。最后，我们通过信号恢复中的数值实验说明了理论结果并验证了所提方法的适用性。
英文摘要
This article presents a novel algorithmic framework for solving a broad class of nonlinear inclusion problems involving non-monotone and Lipschitz continuous operators. Our primary motivation originates from the study of adjoint mismatch problems, which frequently arise in inverse problems and data science when the adjoint of a linear measurement operator is replaced by an approximation. Because this approximation may inherently destroy classical monotonicity properties, standard convergence guarantees for splitting methods may no longer apply. To address this, we consider a Nonlinear Forward-Backward algorithm and rigorously establish its convergence without assuming monotonicity. By leveraging a warped resolvent formulation and semimonotonicity assumptions, we derive explicit conditions ensuring both weak convergence and, under stronger assumptions, R-linear convergence. Furthermore, our theoretical analysis yields new convergence guarantees for several popular methods, including the Condat-Vũ and Forward-Half-Reflected-Backward algorithms. Finally, we illustrate our theoretical findings and demonstrate the applicability of our approach through numerical experiments in signal recovery.
