Revisiting jointly firmly nonexpansive families of mappings

渐近正则强非扩张映象的复合和凸组合也是渐近正则的

Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regular

由于Minty关于强非扩张映射与极大单调算子之间的经典对应关系，强非扩张映射的概念在不动点理论、单调算子理论和凸优化中被证明是非常重要的。本文证明了：如果给定实希尔伯特空间上的n个强非扩张映射，且每个映射都是渐近正则的，等价于说它们有或“几乎有”不动点，那么它们的合成也是如此.这显著地推广了Bauschke在2003年关于投影机（最近点映射）的结果。证明是在Hilbert积空间中进行的，它依赖于Brezis-Haraux值域逼近的结果.通过在适当尺度的Hilbert积空间中工作，我们还建立了凸组合的渐近正则性。

On the minimal displacement vector of compositions and convex combinations of nonexpansive mappings

Monotone operators and (firmly) nonexpansive mappings are fundamental objects in modern analysis and computational optimization. Five years ago, it was shown that if finitely many firmly nonexpansive mappings have or "almost have" fixed points, then the same is true for compositions and convex combinations. More recently, sharp information about the minimal displacement vector of compositions and of convex combinations of firmly nonexpansive mappings was obtained in terms of the displacement vectors of the underlying operators. Using a new proof technique based on the Brezis-Haraux theorem and…

严格非扩张迭代的Cesàro均值不一定强收敛

Cesàro means of firmly nonexpansive iterates need not converge strongly

本文通过构造无穷维Hilbert空间中的反例，证明严格非扩张算子的迭代Cesàro均值可能不收敛到不动点，揭示了平均技术在无穷维优化中的固有限制。

A circumcentered-reflection method for finding common fixed points of firmly nonexpansive operators

The circumcentered-reflection method (CRM) has been recently proposed as a methodology for accelerating several algorithms for solving the Convex Feasibility Problem (CFP), equivalent to finding a common fixed-point of the orthogonal projections onto a finite number of closed and convex sets. In this paper, we apply CRM to the more general Fixed Point Problem (denoted as FPP), consisting of finding a common fixed-point of operators belonging to a larger family of operators, namely firmly nonexpansive operators. We prove than in this setting, CRM is globally convergent to a common fixed-point (…

学习牢固非扩张算子

Learning Firmly Nonexpansive Operators

本文提出一种数据驱动方法学习牢固非扩张算子，用于即插即用图像去噪，并证明经验风险最小化收敛于期望风险最小化，且算子收敛到最优经验解。

$α$-Firmly Nonexpansive Operators on Metric Spaces

We extend to $p$-uniformly convex spaces tools from the analysis of fixed point iterations in linear spaces. This study is restricted to an appropriate generalization of single-valued, pointwise $α$-averaged mappings. Our main contribution is establishing a calculus for these mappings in p-uniformly convex spaces, showing in particular how the property is preserved under compositions and convex combinations. This is of central importance to splitting algorithms that are built by such convex combinations and compositions, and reduces the convergence analysis to simply verifying $α$-firm nonexpa…

Demiclosedness principles for generalized nonexpansive mappings

Demiclosedness principles are powerful tools in the study of convergence of iterative methods. For instance, a multi-operator demiclosedness principle for firmly nonexpansive mappings is useful in obtaining simple and transparent arguments for the weak convergence of the shadow sequence generated by the Douglas-Rachford algorithm. We provide extensions of this principle which are compatible with the framework of more general families of mappings such as cocoercive and conically averaged mappings. As an application, we derive the weak convergence of the shadow sequence generated by the adaptive…

How averaged is the composition of two linear projections?

Projection operators are fundamental algorithmic operators in Analysis and Optimization. It is well known that these operators are firmly nonexpansive; however, their composition is generally only averaged and no longer firmly nonexpansive. In this note, we introduce the modulus of averagedness and provide an exact result for the composition of two linear projection operators. As a consequence, we deduce that the Ogura-Yamada bound for the modulus of the composition is sharp.

Generalized monotone operators and their averaged resolvents

The correspondence between the monotonicity of a (possibly) set-valued operator and the firm nonexpansiveness of its resolvent is a key ingredient in the convergence analysis of many optimization algorithms. Firmly nonexpansive operators form a proper subclass of the more general - but still pleasant from an algorithmic perspective - class of averaged operators. In this paper, we introduce the new notion of conically nonexpansive operators which generalize nonexpansive mappings. We characterize averaged operators as being resolvents of comonotone operators under appropriate scaling. As a conse…

Weak, Strong and Linear Convergence of the CQ-Method Via the Regularity of Landweber Operators

We consider the split convex feasibility problem in a fixed point setting. Motivated by the well-known CQ-method of Byrne (2002), we define an abstract andweber transform which applies to more general operators than the metric projection. We call the result of this transform a Landweber operator. It turns out that the Landweber transform preserves many interesting properties. For example, the Landweber transform of a (quasi/firmly) nonexpansive mapping is again (quasi/firmly) nonexpansive. Moreover, the Landweber transform of a (weakly/linearly) regular mapping is again (weakly/linearly) regul…

A Variational Inequality Model for the Construction of Signals from Inconsistent Nonlinear Equations

Building up on classical linear formulations, we posit that a broad class of problems in signal synthesis and in signal recovery are reducible to the basic task of finding a point in a closed convex subset of a Hilbert space that satisfies a number of nonlinear equations involving firmly nonexpansive operators. We investigate this formalism in the case when, due to inaccurate modeling or perturbations, the nonlinear equations are inconsistent. A relaxed formulation of the original problem is proposed in the form of a variational inequality. The properties of the relaxed problem are investigate…

Multi-marginal maximal monotonicity and convex analysis

Monotonicity and convex analysis arise naturally in the framework of multi-marginal optimal transport theory. However, a comprehensive multi-marginal monotonicity and convex analysis theory is still missing. To this end we study extensions of classical monotone operator theory and convex analysis into the multi-marginal setting. We characterize multi-marginal c-monotonicity in terms of classical monotonicity and firmly nonexpansive mappings. We provide Minty type, continuity and conjugacy criteria for multi-marginal maximal monotonicity. We extend the partition of the identity into a sum of fi…

An abstract proximal point algorithm

The proximal point algorithm is a widely used tool for solving a variety of convex optimization problems such as finding zeros of maximally monotone operators, fixed points of nonexpansive mappings, as well as minimizing convex functions. The algorithm works by applying successively so-called "resolvent" mappings associated to the original object that one aims to optimize. In this paper we abstract from the corresponding resolvents employed in these problems the natural notion of jointly firmly nonexpansive families of mappings. This leads to a streamlined method of proving weak convergence of…