Sum Of Maximal Monotone Operators Research Articles

Forward-reflected-backward splitting method without cocoercivity for the sum of maximal monotone operators in Banach spaces

Let E be a real 2-uniformly convex Banach space with topological dual . We prove weak convergence of the forward-reflected-backward splitting method to a solution of inclusion of sum of two monotone operators. Moreover, we give a variant of the method in which the step size does not depend on the Lipschitz constant of one of the operators. Finally, our results extend and complement several existing results in the literature. We also give some numerical illustrations of the proposed method in comparison with other method in the literature to further demonstrate the applicability and efficiency of our method.

Generalized relaxed inertial method with regularization for solving split feasibility problems in real Hilbert spaces

In this paper, we propose a new modified relaxed inertial regularization method for finding a common solution of a generalized split feasibility problem, the zeros of sum of maximal monotone operators, and fixed point problem of two nonlinear mappings in real Hilbert spaces. We prove that the proposed method converges strongly to a minimum-norm solution of the aforementioned problems without using the conventional two cases approach. In addition, we apply our convergence results to the classical variational inequality and equilibrium problems, and present some numerical experiments to show the efficiency and applicability of the proposed method in comparison with other existing methods in the literature. The results obtained in this paper extend, generalize and improve several results in this direction.

Another proof and a generalization of a theorem of H. H. Bauschke on monotone operators

ABSTRACT Using the concept of partial-inverse of monotone operators due to Spingarn, we present a new and simple proof of a result – Theorem 2 in Bauschke [A note on the paper by Eckstein and Svaiter on “General projective splitting methods for sums of maximal monotone operators”. SIAM J Control Optim. 2009;48(4):2513–2515] – of Heinz H. Bauschke. Our proof is based on the maximal monotonicity of the partial-inverse and on the (asymptotic) closedness principle on the graph of maximal monotone operators in the topology. We also present a generalization of Bauschke's theorem to the more general setting of ε–enlargements of monotone maps.

Projective splitting with forward steps

This work is concerned with the classical problem of finding a zero of a sum of maximal monotone operators. For the projective splitting framework recently proposed by Combettes and Eckstein, we show how to replace the fundamental subproblem calculation using a backward step with one based on two forward steps. The resulting algorithms have the same kind of coordination procedure and can be implemented in the same block-iterative and highly flexible manner, but may perform backward steps on some operators and forward steps on others. Prior algorithms in the projective splitting family have used only backward steps. Forward steps can be used for any Lipschitz-continuous operators provided the stepsize is bounded by the inverse of the Lipschitz constant. If the Lipschitz constant is unknown, a simple backtracking linesearch procedure may be used. For affine operators, the stepsize can be chosen adaptively without knowledge of the Lipschitz constant and without any additional forward steps. We close the paper by empirically studying the performance of several kinds of splitting algorithms on a large-scale rare feature selection problem.

Convergence Results of Forward-Backward Algorithms for Sum of Monotone Operators in Banach Spaces

It is well known that many problems in image recovery, signal processing, and machine learning can be modeled as finding zeros of the sum of maximal monotone and Lipschitz continuous monotone operators. Many papers have studied forward-backward splitting methods for finding zeros of the sum of two monotone operators in Hilbert spaces. Most of the proposed splitting methods in the literature have been proposed for the sum of maximal monotone and inverse-strongly monotone operators in Hilbert spaces. In this paper, we consider splitting methods for finding zeros of the sum of maximal monotone operators and Lipschitz continuous monotone operators in Banach spaces. We obtain weak and strong convergence results for the zeros of the sum of maximal monotone and Lipschitz continuous monotone operators in Banach spaces. Many already studied problems in the literature can be considered as special cases of this paper.

Convergence Rates for Projective Splitting

Projective splitting is a family of methods for solving inclusions involving sums of maximal monotone operators. First introduced by Eckstein and Svaiter in 2008, these methods have enjoyed significant innovation in recent years, becoming one of the most flexible operator splitting frameworks available. While weak convergence of the iterates to a solution has been established, there have been few attempts to study convergence rates of projective splitting. The purpose of this paper is to do so under various assumptions. To this end, there are three main contributions. First, in the context of convex optimization, we establish an $O(1/k)$ ergodic function convergence rate. Second, for strongly monotone inclusions, strong convergence is established as well as an ergodic $O(1/\sqrt{k})$ convergence rate for the distance of the iterates to the solution. Finally, for inclusions featuring strong monotonicity and cocoercivity, linear convergence is established.

Strong convergence theorems for quasi-nonexpansive mappings and maximal monotone operators in Hilbert spaces

We present the strong convergence theorem for the iterative scheme for finding a common element of the fixed-point set of a quasi-nonexpansive mapping and the zero set of the sums of maximal monotone operators in Hilbert spaces. Our results extend and improve the recent results of Takahashi et al. (J. Optim. Theory Appl. 147:27-41, 2010) and Takahashi and Takahashi (Nonlinear Anal. 69:1025-1033, 2008).MSC:47H05, 47H09, 47J25.

A proximal point method for the sum of maximal monotone operators

AbstractIn 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.

Strong Convergence Theorems for Maximal Monotone Operators, Fixed-Point Problems, and Equilibrium Problems

We present a new iterative method for finding a common element of the set of fixed points of a nonexpansive mapping, the set of solutions to an equilibrium problem, and the set of zeros of the sum of maximal monotone operators and prove the strong convergence theorems in the Hilbert spaces. We also apply our results to variational inequality and optimization problems.

Strong convergence of a splitting projection method for the sum of maximal monotone operators

In this paper, a projective-splitting method is proposed for finding a zero of the sum of \(n\) maximal monotone operators over a real Hilbert space \(\mathcal{H }\). Without the condition that either \(\mathcal{H }\) is finite dimensional or the sum of \(n\) operators is maximal monotone, we prove that the sequence generated by the proposed method is strongly convergent to an extended solution for the problem, which is closest to the initial point. The main results presented in this paper generalize and improve some recent results in this topic.

Projective splitting methods for sums of maximal monotone operators with applications

Splitting methods for sums of maximal monotone operators are studied in this paper. By formulating the classical splitting methods, including the Douglas/Peaceman–Rachford splitting and the forward–backward splitting, into fixed point iterations, general splitting methods are considered via parameterizing fixed point formulas. Weak convergence results for these general splitting methods are derived with the help of a demiclosedness principle. Moreover, by employing the Haugazeau-like projective method, the weak convergence splitting algorithms are forced to be strongly convergent. Finally, applications to the convex feasibility and best approximation problems are made from the viewpoint of convex optimization; new and improved convergence results for solving these two problems are obtained.

A Primal-Dual Splitting Algorithm for Finding Zeros of Sums of Maximal Monotone Operators

We consider the primal problem of finding the zeros of the sum of a maximal monotone operator and the composition of another maximal monotone operator with a linear continuous operator. By formulating its Attouch--Théra-type dual inclusion problem, a primal-dual splitting algorithm which simultaneously solves the two problems in finite-dimensional spaces is presented. The proposed scheme uses at each iteration the resolvents of the maximal monotone operators involved in separate steps and aims to overcome the shortcoming of classical splitting algorithms when dealing with compositions of maximal monotone and linear continuous operators. The iterative algorithm is used for solving nondifferentiable convex optimization problems arising in image processing and in location theory.

Iterative Methods for Equilibrium Problems and Monotone Inclusion Problems in Hilbert Spaces

We introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of zeros of the sum of maximal monotone operators, and we obtain strong convergence theorems in Hilbert spaces. We also apply our results to the variational inequality and convex minimization problems. Our results extend and improve the recent result of Takahashi et al. (2012).

A Generalized Forward-Backward Splitting

This paper introduces a generalized forward-backward splitting algorithm for finding a zero of a sum of maximal monotone operators $B + \sum_{i=1}^n A_i$, where $B$ is cocoercive. It involves the computation of $B$ in an explicit (forward) step and the parallel computation of the resolvents of the $A_i$'s in a subsequent implicit (backward) step. We prove the algorithm's convergence in infinite dimension and its robustness to summable errors on the computed operators in the explicit and implicit steps. In particular, this allows efficient minimization of the sum of convex functions $f + \sum_{i=1}^n g_i$, where $f$ has a Lipschitz-continuous gradient and each $g_i$ is simple in the sense that its proximity operator is easy to compute. The resulting method makes use of the regularity of $f$ in the forward step, and the proximity operators of the $g_i$'s are applied in parallel in the backward step. While the forward-backward algorithm cannot deal with more than $n = 1$ nonsmooth function, we generalize it to the case of arbitrary $n$. Examples on inverse problems in imaging demonstrate the advantage of the proposed methods in comparison to other splitting algorithms.

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.

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 ℓ 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 L 2 [ 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.

Maximal monotonicity criteria for the composition and the sum under weak interiority conditions

The main goal of this article is to present several new results on the maximality of the composition and of the sum of maximal monotone operators in Banach spaces under weak interiority conditions involving their domains. Direct applications of our results to the structure of the range and domain of a maximal monotone operator are discussed. The last section of this note studies continuity properties of the duality product between a Banach space X and its dual X* with respect to topologies compatible with the natural duality (X × X*, X* × X).

General Projective Splitting Methods for Sums of Maximal Monotone Operators

We describe a general projective framework for finding a zero of the sum of n maximal monotone operators over a real Hilbert space. Unlike prior methods for this problem, we neither assume $n=2$ nor first reduce the problem to the case $n=2$. Our analysis defines a closed convex extended solution set for which we can construct a separating hyperplane by individually evaluating the resolvent of each operator. At the cost of a single, computationally simple projection step, this framework gives rise to a family of splitting methods of unprecedented flexibility: numerous parameters, including the proximal stepsize, may vary by iteration and by operator. The order of operator evaluation may vary by iteration and may be either serial or parallel. The analysis essentially generalizes our prior results for the case $n=2$. We also include a relative error criterion for approximately evaluating resolvents, which was not present in our earlier work.

A Note on the Paper by Eckstein and Svaiter on “General Projective Splitting Methods for Sums of Maximal Monotone Operators”

In their recent paper from 2009 [SIAM J. Control Optim., 48 (2009), pp. 787–811], J. Eckstein and B. F. Svaiter proposed a very general and flexible splitting framework for finding a zero of the sum of finitely many maximal monotone operators. In this short note, we provide a technical result that allows for the removal of Eckstein and Svaiter's assumption that the sum of the operators be maximal monotone or that the underlying Hilbert space be finite-dimensional.

An inexact method of partial inverses and a parallel bundle method

For a maximal monotone operator T on a Hilbert space H and a closed subspace A of H, we consider the problem of finding (x, y∈T(x)) satisfying x∈A and y∈A ⊥. An equivalent formulation of this problem makes use of the partial inverse operator of Spingarn. The resulting generalized equation can be solved by using the proximal point algorithm. We consider instead the use of hybrid proximal methods. Hybrid methods use enlargements of operators, close in spirit to the concept of ϵ-subdifferentials. We characterize the enlargement of the partial inverse operator in terms of the enlargement of T itself. We present a new algorithm of resolution that combines Spingarn and hybrid methods, we prove for this method global convergence only assuming existence of solutions and maximal monotonicity of T. We also show that, under standard assumptions, the method has a linear rate of convergence. For the important problem of finding a zero of a sum of maximal monotone operators T 1, …, T m , we present a highly parallelizable scheme. Finally, we derive a parallel bundle method for minimizing the sum of polyhedral functions.