Andrei Sipos
Bauschke and Moursi have recently obtained results that implicitly contain the fact that the composition of finitely many averaged mappings on a Hilbert space that have approximate fixed points also has approximate fixed points and thus is asymptotically regular. Using techniques of proof mining, we analyze their arguments to obtain effective uniform rates of asymptotic regularity.
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We analyse the behaviour of the newly introduced cyclic Douglas-Rachford algorithm for finding a point in the intersection of a finite number of closed convex sets. This work considers the case in which the target intersection set is possibly empty.
Cyclic Relaxed Douglas-Rachford Splitting for Inconsistent Nonconvex Feasibility
本文提出循环松弛 Douglas–Rachford 算法以处理非凸不相容可行性问题，刻画其不动点及影子与循环投影不动点的关系，并给出局部定量收敛条件。
The convex feasibility problem (CFP) is to find a feasible point in the intersection of finitely many convex and closed sets. If the intersection is empty then the CFP is inconsistent and a feasible point does not exist. However, algorithmic research of inconsistent CFPs exists and is mainly focused on two directions. One is oriented toward defining other solution concepts that will apply, such as proximity function minimization wherein a proximity function measures in some way the total violation of all constraints. The second direction investigates the behavior of algorithms that are designe…
Stochastic Block Bregman Projection with Polyak-like Stepsize for Possibly Inconsistent Convex Feasibility Problems
本文提出了一种统一的随机双层改写方法，结合了接近函数最小化和结构正则化，以解决可能不一致的凸可行性问题，并通过Polyak-like步长实现了期望精确收敛。
On The Behavior of Subgradient Projections Methods for Convex Feasibility Problems in Euclidean Spaces
针对不一致情况下一般凸集的凸可行性问题，研究次梯度投影方法，提出一种自适应控制松弛参数的策略，保证算法行为并展示计算优势。
Motivated by nonconvex, inconsistent feasibility problems in imaging, the relaxed alternating averaged reflections algorithm, or relaxed Douglas-Rachford algorithm (DR$λ$), was first proposed over a decade ago. Convergence results for this algorithm are limited either to convex feasibility or consistent nonconvex feasibility with strong assumptions on the regularity of the underlying sets. Using an analytical framework depending only on metric subregularity and pointwise almost averagedness, we analyze the convergence behavior of DR$λ$ for feasibility problems that are both nonconvex and incon…
Algorithmic approaches to avoiding bad local minima in nonconvex inconsistent feasibility
该研究针对非凸不一致可行性问题，通过合成与实验室数据，验证了乘积空间上的松弛道格拉斯-拉奇福德算法可过滤不良局部极小值，提出先循环投影找不动点再用该算法脱离极小值的优化策略，与相位检索现有实践相反。
We study the nonexpansivity of reflection mappings in geodesic spaces and apply our findings to the averaged alternating reflection algorithm employed in solving the convex feasibility problem for two sets in a nonlinear context. We show that weak convergence results from Hilbert spaces find natural counterparts in spaces of constant curvature. Moreover, in this particular setting, one obtains strong convergence.
A new theorem of alternatives leading to sufficient conditions for the superiorization guarantee question of Dynamic String-Averaging in the inconsistent case
针对不一致情形下的通用动态串平均法，提出新择一定理，给出优化方法实现更优可行点的充分条件，解决了现有文献未完全覆盖的优化保证问题。
In this paper, we establish sublinear and linear convergence of fixed point iterations generated by averaged operators in a Hilbert space. Our results are achieved under a bounded Hölder regularity assumption which generalizes the well-known notion of bounded linear regularity. As an application of our results, we provide a convergence rate analysis for Krasnoselskii-Mann iterations, the cyclic projection algorithm, and the Douglas-Rachford feasibility algorithm along with some variants. In the important case in which the underlying sets are convex sets described by convex polynomials in a fin…
The Douglas--Rachford algorithm is a popular algorithm for solving both convex and nonconvex feasibility problems. While its behaviour is settled in the convex inconsistent case, the general nonconvex inconsistent case is far from being fully understood. In this paper, we focus on the most simple nonconvex inconsistent case: when one set is a hyperplane and the other a doubleton (i.e., a two-point set). We present a characterization of cycling in this case which --- somewhat surprisingly --- depends on whether the ratio of the distance of the points to the hyperplane is rational or not. Furthe…
The computational content of super strongly nonexpansive mappings and uniformly monotone operators
本文用证明挖掘方法定量研究超强非扩张映射与一致单调算子，构造超强非扩张性模量并关联一致单调性模量，在超强制情形下首次得到定性与定量兼新的不相容可行性结果及渐近正则性有效一致速率。
Residual-Controlled Douglas--Rachford Splitting for Differentiable Solver Layers
针对可微分求解器层固定深度展开的权衡问题，提出残差控制道格拉斯-拉赫福德分裂（RCDRS），理论证明其性质，实验显示其提升解质量等性能。
Various strategies are available to construct iteratively a common fixed point of nonexpansive operators by activating only a block of operators at each iteration. In the more challenging class of composite fixed point problems involving operators that do not share common fixed points, current methods require the activation of all the operators at each iteration, and the question of maintaining convergence while updating only blocks of operators is open. We propose a method that achieves this goal and analyze its asymptotic behavior. Weak, strong, and linear convergence results are established…
We study the convergence of random function iterations for finding an invariant measure of the corresponding Markov operator. We call the problem of finding such an invariant measure the stochastic fixed point problem. This generalizes earlier work studying the stochastic feasibility problem}, namely, to find points that are, with probability 1, fixed points of the random functions [Hermer, Luke, Sturm, 2019]. When no such points exist, the stochastic feasibility problem is called inconsistent, but still under certain assumptions, the more general stochastic fixed point problem has a solution …
We consider the problem of recovering a signal from nonlinear transformations, under convex constraints modeling a priori information. Standard feasibility and optimization methods are ill-suited to tackle this problem due to the nonlinearities. We show that, in many common applications, the transformation model can be associated with fixed point equations involving firmly nonexpansive operators. In turn, the recovery problem is reduced to a tractable common fixed point formulation, which is solved efficiently by a provably convergent, block-iterative algorithm. Applications to signal and imag…
Douglas-Rachford Splitting (DRS) methods based on the proximal point algorithms for the Poisson and Gaussian log-likelihood functions are proposed for ptychography and phase retrieval. Fixed point analysis shows that the DRS iterated sequences are always bounded explicitly in terms of the step size and that the fixed points are attracting if and only if the fixed points are regular solutions. This alleviates two major drawbacks of the classical Douglas-Rachford algorithm: slow convergence when the feasibility problem is consistent and divergent behavior when the feasibility problem is inconsis…
Nonexpansive Markov Operators and Random Function Iterations for Stochastic Fixed Point Problems
该研究提出随机不动点问题框架，推广了不一致的随机可行性问题，利用确定性不动点理论证明马尔可夫算子不变测度存在性及随机函数迭代的依分布收敛性，可应用于分布式计算等场景，为带误差的迭代方法收敛性提供理论基础。
Quantitative convergence analysis of iterated expansive, set-valued mappings
本文提出了一种用于分析扩张多值固定点映射皮卡迭代的定量收敛性的框架，通过扩展单值平均映射的概念和扩展度量子正则性概念，证明了扩张固定点迭代的收敛性，并给出了定量估计，应用该理论证明了非凸循环投影在不一致（和一致）可行性问题中的局部线性收敛性，前向-后向算法在非凸结构优化中的局部线性收敛性，以及道格拉斯-拉奇福德算法在非凸最小化问题中的局部线性收敛性。
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