涉及并凸集的可行性问题投影方法的全局收敛性与加速
Global convergence and acceleration of projection methods for feasibility problems involving union convex sets
AI总结：
本文针对并凸集可行性问题建立经典投影算法的全局收敛理论，并据此推导稀疏仿射可行性与含 $P$-矩阵线性互补问题的收敛条件，同时提出有收敛保证且数值上更高效的加速算法。
摘要：
我们证明了经典投影算法在涉及并凸集的可行性问题上的全局收敛性，其中并凸集是指可表示为有限个闭凸集之并的集合。我们提出了一种统一的全局收敛性分析策略，其方法是研究一个集值算子的不动点迭代，该算子是有限个紧值上半连续映射的并。这一广义框架使得能够分析一类近端算法，用于最小化一个分段光滑函数与“有限个弱凸函数的逐点最小值减去一个分段光滑凸函数”之差的和。当应用于两集合可行性问题时，该算法类别将交替投影和平均投影恢复为特例，因此我们获得了这些投影算法的全局收敛准则。利用这些一般结果，我们推导了充分条件，以保证若干投影算法在求解稀疏仿射可行性问题以及线性互补问题的可行性重构时具有全局收敛性。值得注意的是，对于涉及 $P$-矩阵的线性互补问题，我们证明了交替投影法和平均投影法均全局收敛到解集。通过利用所考虑问题类别的结构，我们还提出了具有全局收敛保证的加速算法。数值结果进一步表明，所提出的加速方案在效率上显著优于其未加速的对应方法。
英文摘要：
We prove global convergence of classical projection algorithms for feasibility problems involving union convex sets, which refer to sets expressible as the union of a finite number of closed convex sets. We present a unified strategy for analyzing global convergence by means of studying fixed-point iterations of a set-valued operator that is the union of a finite number of compact-valued upper semicontinuous maps. Such a generalized framework permits the analysis of a class of proximal algorithms for minimizing the sum of a piecewise smooth function and the difference between pointwise minimum of finitely many weakly convex functions and a piecewise smooth convex function. When realized on two-set feasibility problems, this algorithm class recovers alternating projections and averaged projections as special cases, and thus we obtain global convergence criterion for these projection algorithms. Using these general results, we derive sufficient conditions to guarantee global convergence for several projection algorithms for solving the sparse affine feasibility problem and a feasibility reformulation of the linear complementarity problem. Notably, we obtain global convergence of both the alternating and the averaged projection methods to the solution set for linear complementarity problems involving $P$-matrices. By leveraging the structures of the classes of problems we consider, we also propose acceleration algorithms with guaranteed global convergence. Numerical results further exemplify that the proposed acceleration schemes significantly improve upon their non-accelerated counterparts in efficiency.
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