Penalty schemes with inertial effects for monotone inclusion problems

We introduce a penalty term-based splitting algorithm with inertial effects designed for solving monotone inclusion problems involving the sum of maximally monotone operators and the convex normal cone to the (nonempty) set of zeros of a monotone and Lipschitz continuous operator. We show weak ergodic convergence of the generated sequence of iterates to a solution of the monotone inclusion problem, provided a condition expressed via the Fitzpatrick function of the operator describing the underlying set of the normal cone is verified. Under strong monotonicity assumptions we can even show strong nonergodic convergence of the iterates. This approach constitutes the starting point for investigating from a similar perspective monotone inclusion problems involving linear compositions of parallel-sum operators and, further, for the minimization of a complexly structured convex objective function subject to the set of minima of another convex and differentiable function.     

Tight global linear convergence rate bounds for Douglas–Rachford splitting

Recently, several authors have shown local and global convergence rate results for Douglas–Rachford splitting under strong monotonicity, Lipschitz continuity, and cocoercivity assumptions. Most of these focus on the convex optimization setting. In the more general monotone inclusion setting, Lions and Mercier showed a linear convergence rate bound under the assumption that one of the two operators is strongly monotone and Lipschitz continuous. We show that this bound is not tight, meaning that no problem from the considered class converges exactly with that rate. In this paper, we present tight global linear convergence rate bounds for that class of problems. We also provide tight linear convergence rate bounds under the assumptions that one of the operators is strongly monotone and cocoercive, and that one of the operators is strongly monotone and the other is cocoercive. All our linear convergence results are obtained by proving the stronger property that the Douglas–Rachford operator is contractive.     

A stochastic inertial forward–backward splitting algorithm for multivariate monotone inclusions

We propose an inertial forward–backward splitting algorithm to compute a zero of a sum of two monotone operators allowing for stochastic errors in the computation of the operators. More precisely, we establish almost sure convergence in real Hilbert spaces of the sequence of iterates to an optimal solution. Then, based on this analysis, we introduce two new classes of stochastic inertial primal–dual splitting methods for solving structured systems of composite monotone inclusions and prove their convergence. Our results extend to the stochastic and inertial setting various types of structured monotone inclusion problems and corresponding algorithmic solutions. Application to minimization problems is discussed.     

A proximal point method for the sum of maximal monotone operators

In this paper, we concentrate on the maximal inclusion problem of locating the zeros of the sum of maximal monotone operators in the framework of proximal point method. Such problems arise widely in several applied mathematical fields such as signal and image processing. We define two new maximal monotone operators and characterize the solutions of the considered problem via the zeros of the new operators. The maximal monotonicity and resolvent of both of the defined operators are proved and calculated, respectively. The traditional proximal point algorithm can be therefore applied to the considered maximal inclusion problem, and the convergence is ensured. Furthermore, by exploring the relationship between the proposed method and the generalized forward‐backward splitting algorithm, we point out that this algorithm is essentially the proximal point algorithm when the operator corresponding to the forward step is the zero operator. Copyright © 2013 John Wiley & Sons, Ltd.     

Primal-Dual Splitting Algorithm for Solving Inclusions with Mixtures of Composite, Lipschitzian, and Parallel-Sum Type Monotone Operators

We propose a primal-dual splitting algorithm for solving monotone inclusions involving a mixture of sums, linear compositions, and parallel sums of set-valued and Lipschitzian operators. An important feature of the algorithm is that the Lipschitzian operators present in the formulation can be processed individually via explicit steps, while the set-valued operators are processed individually via their resolvents. In addition, the algorithm is highly parallel in that most of its steps can be executed simultaneously. This work brings together and notably extends various types of structured monotone inclusion problems and their solution methods. The application to convex minimization problems is given special attention.     

Split common fixed point and common null point problem

In this paper, we present a new algorithm for solving the split common null point and common fixed point problem, to find a point that belongs to the common element of common zero points of an infinite family of maximal monotone operators and common fixed points of an infinite family of demicontractive mappings such that its image under a linear transformation belongs to the common zero points of another infinite family of maximal monotone operators and its image under another linear transformation belongs to the common fixed point of another infinite family of demicontractive mappings in the image space. We establish strong convergence for the algorithm to find a unique solution of the variational inequality, which is the optimality condition for the minimization problem. As special cases, we shall use our results to study the split equilibrium problems and the split optimization problems.     

Outer Approximation Method for Constrained Composite Fixed Point Problems Involving Lipschitz Pseudo Contractive Operators

We propose a method for solving constrained fixed point problems involving compositions of Lipschitz pseudo contractive and firmly nonexpansive operators in Hilbert spaces. Each iteration of the method uses separate evaluations of these operators and an outer approximation given by the projection onto a closed half-space containing the constraint set. Its convergence is established and applications to monotone inclusion splitting and constrained equilibrium problems are demonstrated.     

Representable Monotone Operators and Limits of Sequences of Maximal Monotone Operators

We show that the lower limit of a sequence of maximal monotone operators on a reflexive Banach space is a representable monotone operator. As a consequence, we obtain that the variational sum of maximal monotone operators and the variational composition of a maximal monotone operator with a linear continuous operator are both representable monotone operators.     

Strong convergence of the forward–backward splitting method with multiple parameters in Hilbert spaces

Many problems arising from machine learning, signal & image recovery, and compressed sensing can be casted into a monotone inclusion problem for finding a zero of the sum of two monotone operators. The forward–backward splitting algorithm is one of the most powerful and successful methods for solving such a problem. However, this algorithm has only weak convergence in the infinite dimensional settings. In this paper, we propose a new modification of the FBA so that it possesses a norm convergent property. Moreover, we establish two strong convergence theorems of the proposed algorithms under more general conditions.     

关于α-凸凹混合单调算子的不动点定理

引入了α-凸凹混合单调算子的概念.借助于集值分析的方法,利用锥理论讨论了这类混合单调算子,得到了若干α-凸凹混合单调算子不动点的存在性和唯一性定理.     

Examples of discontinuous maximal monotone linear operators and the solution to a recent problem posed by B.F. Svaiter

In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator S on ?² is skew. We show its domain is a proper subset of the domain of its adjoint S^∗, and −S^∗ is not maximal monotone. This gives a negative answer to a recent question posed by Svaiter. The second unbounded linear maximal monotone operator is the inverse Volterra operator T on L²[0,1]. We compare the domain of T with the domain of its adjoint T^∗ and show that the skew part of T admits two distinct linear maximal monotone skew extensions. These unbounded linear maximal monotone operators show that the constraint qualification for the maximality of the sum of maximal monotone operators cannot be significantly weakened, and they are simpler than the example given by Phelps-Simons. Interesting consequences on Fitzpatrick functions for sums of two maximal monotone operators are also given.     

On the generalized parallel sum of two maximal monotone operators of Gossez type (D)

The generalized parallel sum of two monotone operators via a linear continuous mapping is defined as the inverse of the sum of the inverse of one of the operators and with inverse of the composition of the second one with the linear continuous mapping. In this article, by assuming that the operators are maximal monotone of Gossez type (D), we provide sufficient conditions of both interiority- and closedness-type for guaranteeing that their generalized sum via a linear continuous mapping is maximal monotone of Gossez type (D), too. This result will follow as a particular instance of a more general one concerning the maximal monotonicity of Gossez type (D) of an extended parallel sum defined for the maximal monotone extensions of the two operators to the corresponding biduals.     

Banach空间中极大单调算子零点的投影算法

本文设计了一种极大单调算子零点的带误差项的新投影迭代算法,并在Banach空间中,利用Lyapunov泛函与广义投影映射等技巧,证明了迭代序列强收敛于极大单调算子零点的结论.     

Menger PN-空间中两类混合单调算子新的公共不动点定理

在Menger PN-空间,引入(Co)类压缩型算子半群的有关概念.研究了两类混合单调算子新的公共不动点的存在与唯一性,不要求算子具有任何紧性、凹凸性和连续性,从而获得一些新的结论,改进和推广Banach空间中的有关研究结论.     

Metric subregularity and the proximal point method

We examine the linear convergence rates of variants of the proximal point method for finding zeros of maximal monotone operators. We begin by showing how metric subregularity is sufficient for local linear convergence to a zero of a maximal monotone operator. This result is then generalized to obtain convergence rates for the problem of finding a common zero of multiple monotone operators by considering randomized and averaged proximal methods.     

Rectangularity and paramonotonicity of maximally monotone operators

Maximally monotone operators play a key role in modern optimization and variational analysis. Two useful subclasses are rectangular (also known as star monotone) and paramonotone operators, which were introduced by Brezis and Haraux, and by Censor, Iusem and Zenios, respectively. The former class has a useful range of properties while the latter class is of importance for interior point methods and duality theory. Both notions are automatic for subdifferential operators and known to coincide for certain matrices; however, more precise relationships between rectangularity and paramonotonicity were not known. Our aim is to provide new results and examples concerning these notions. It is shown that rectangularity and paramonotonicity are actually independent. Moreover, for linear relations, rectangularity implies paramonotonicity but the converse implication requires additional assumptions. We also consider continuous linear monotone operators, and we point out that in the Hilbert space both notions are automatic for certain displacement mappings.     

Nonlinear Carleman operators on Banach lattices

An operator, not necessarily linear, will be called a Carleman operator if the image of the positive elements in the unit ball are bounded in the universal completion of the range space. For certain Banach lattices, a class of (not necessarily linear) Carleman operators is characterized in terms of an integral representation and in a more general setting as operators satisfying a pointwise finiteness condition. These operators though not linear are orthogonally additive and monotone.

     

Fixed points for mixed monotone multivalued operators in Banach spaces with applications

In this paper, the existence and iterative approximation of fixed points for a class of systems of mixed monotone multivalued operator are discussed. We present some new fixed point theorems of mixed monotone operators and increasing operators which need not be continuous or satisfy a compactness condition. We also give some applications to differential inclusions with discontinuous right hand side in Banach spaces and to Hammerstein integral inclusions on RN.     

Forward-Backward and Tseng’s Type Penalty Schemes for Monotone Inclusion Problems

We deal with monotone inclusion problems of the form 0 ∈ A x + D x + N _C (x) in real Hilbert spaces, where A is a maximally monotone operator, D a cocoercive operator and C the nonempty set of zeros of another cocoercive operator. We propose a forward-backward penalty algorithm for solving this problem which extends the one proposed by Attouch et al. (SIAM J. Optim. 21(4): 1251-1274, 2011). The condition which guarantees the weak ergodic convergence of the sequence of iterates generated by the proposed scheme is formulated by means of the Fitzpatrick function associated to the maximally monotone operator that describes the set C. In the second part we introduce a forward-backward-forward algorithm for monotone inclusion problems having the same structure, but this time by replacing the cocoercivity hypotheses with Lipschitz continuity conditions. The latter penalty type algorithm opens the gate to handle monotone inclusion problems with more complicated structures, for instance, involving compositions of maximally monotone operators with linear continuous ones.     

A Douglas–Rachford splitting method for solving equilibrium problems

We propose a splitting method for solving equilibrium problems involving the sum of two bifunctions satisfying standard conditions. We prove that this problem is equivalent to find a zero of the sum of two appropriate maximally monotone operators under a suitable qualification condition. Our algorithm is a consequence of the Douglas–Rachford splitting applied to this auxiliary monotone inclusion. Connections between monotone inclusions and equilibrium problems are studied.