[论文解读] Douglas-Rachford splitting for nonconvex feasibility problems

Guoyin Li, Ting Kei Pong|arXiv (Cornell University)|Sep 30, 2014

Optimization and Variational Analysis参考文献 26被引用 6

一句话总结

本文通过分析在Lipschitz连续梯度条件下，对一个适当闭函数与一个光滑函数之和进行最小化问题的收敛性，将Douglas-Rachford分裂法推广至非凸可行性问题。研究证明：当步长低于可计算的阈值且序列存在聚点时，序列收敛于驻点；在半代数假设下，可实现全局收敛与局部线性收敛速率，并将该方法应用于求解凸集与一般闭集交集中的点。

ABSTRACT

We adapt the Douglas-Rachford (DR) splitting method to solve nonconvex feasibility problems by studying this method for a class of nonconvex optimization problem. While the convergence properties of the method for convex problems have been well studied, far less is known in the nonconvex setting. In this paper, for the direct adaptation of the method to minimize the sum of a proper closed function g and a smooth function f with a Lipschitz continuous gradient, we show that if the step-size parameter is smaller than a computable threshold and the sequence generated has a cluster point, then it gives a stationary point of the optimization problem. Convergence of the whole sequence and a local convergence rate are also established under the additional assumption that f and g are semi-algebraic. We also give simple sufficient conditions guaranteeing the boundedness of the sequence generated. We then apply our nonconvex DR splitting method to finding a point in the intersection of a closed convex set C and a general closed set D by minimizing the square distance to C subject to D. We show that if either set is bounded and the step-size parameter is smaller than a computable threshold, then the sequence generated from the DR splitting method is actually bounded. Consequently, the sequence generated will have cluster points that are stationary for an optimization problem, and the whole sequence is convergent under an additional assumption that C and D are semi-algebraic. We achieve these results based on a new merit function constructed particularly for the DR splitting method. Our preliminary numerical results indicate that the DR splitting method usually outperforms the alternating projection method in finding a sparse solution of a linear system, in terms of both the solution quality and the number of iterations taken.

研究动机与目标

• 将经典的Douglas-Rachford分裂法扩展至非凸优化问题，其中收敛性尚不明确。
• 在可计算的步长条件与有界性假设下，建立收敛至驻点的理论保证。
• 当目标函数为半代数函数时，提供全局收敛性与局部线性收敛速率。
• 将该方法应用于涉及凸集与一般闭集的可行性问题，特别关注稀疏解的恢复。
• 通过实验表明，该方法在解的质量与迭代次数方面优于交替投影法。

提出的方法

• 通过最小化一个适当闭函数g与一个具有Lipschitz连续梯度的光滑函数f之和，将经典Douglas-Rachford分裂法推广至非凸问题。
• 提出一种专为非凸DR分裂框架中的收敛性分析而设计的新颖增广函数。
• 当步长低于可计算的阈值且序列存在聚点时，证明序列收敛于优化问题的驻点。
• 在f与g均为半代数函数的附加假设下，证明全局收敛性与局部线性收敛速率。
• 通过最小化到闭凸集C的平方距离并满足属于一般闭集D的约束，将该方法应用于可行性问题。
• 利用有界性条件——特别是当C或D有界时——确保序列有界，从而保证存在聚点。

实验结果

研究问题

• RQ1 在何种条件下，直接的Douglas-Rachford分裂法可收敛于非凸问题的驻点？
• RQ2 能否推导出一个可计算的步长阈值，以确保在存在聚点时收敛于驻点？
• RQ3 在非凸设置下，需要哪些额外假设才能保证全局收敛与局部线性收敛速率？
• RQ4 如何将DR分裂法应用于涉及凸集与一般闭集的可行性问题？
• RQ5 在非凸设置下，何种条件可确保DR分裂法生成序列的有界性？

主要发现

• 若步长小于可计算的阈值且序列存在聚点，则序列收敛于优化问题的驻点。
• 当f与g均为半代数函数时，可实现全局收敛与局部线性收敛速率。
• 当凸集C或一般集D有界且步长低于阈值时，DR分裂法生成的序列有界。
• 在求解线性系统稀疏解时，该方法在解的质量与迭代次数方面优于交替投影法。
• 所提出的增广函数使得在传统李雅普诺夫方法失效的非凸设置下，仍可实现收敛性分析。