On the convergence of maximal monotone operators

We study the convergence of maximal monotone operators with the help of representations by convex functions. In particular, we prove the convergence of a sequence of sums of maximal monotone operators under a general qualification condition of the Attouch-Brezis type.

     

On difference of two monotone operators

In this paper, we introduce and consider the problem of finding zeroes of difference of two monotone operators in a Hilbert space. Using the resolvent operator technique, we show that this problem is equivalent to the fixed point problem. This equivalence is used to suggest and analyze an iterative method for finding a zero of difference of two monotone operators. We also discuss the convergence of the iterative method under suitable conditions. Our method of proof is very simple as compared with other techniques.     

On preimages of a class of generalized monotone operators

In this paper we first provide a geometric interpretation of the Minty-Browder monotonicity which allows us to extend this concept to the so called h-monotonicity, still formulated in an analytic way. A topological concept of monotonicity is also known in the literature: it requires the connectedness of all preimages of the operator involved. This fact is important since combined with the local injectivity, it ensures global injectivity. When a linear structure is present on the source space, one can ask for the preimages to even be convex. In an earlier paper, the authors have shown that Minty-Browder monotone operators defined on convex open sets do have convex preimages, obtaining as a by-product global injectivity theorems. In this paper we study the preimages of h-monotone operators, by showing that they are not divisible by closed connected hypersurfaces, and investigate them from the dimensional point of view. As a consequence we deduce that h-monotone local homeomorphisms are actually global homeomorphisms, as the proved properties of their preimages combined with local injectivity still produce global injectivity.     

On classifying monotone complete algebras of operators

We give a classification of “small” monotone complete C ^*-algebras by order properties. We construct a corresponding semigroup. This classification filters out von Neumann algebras; they are mapped to the zero of the classifying semigroup. We show that there are 2 ^c distinct equivalence classes (where c is the cardinality of the continuum). This remains true when the classification is restricted to special classes of monotone complete C ^*-algebras e.g. factors, injective factors, injective operator systems and commutative algebras which are subalgebras of ℓ^∞. Some examples and applications are given.      

On a generalization of the operators of fractional integration and differentiation

Some general fractional integral operators are studied including those of RIEMANN-LIOUVILLE, HADAMARD and others. They are used to solve a generalized ABEL equation.     

Accelerated proximal point method for maximally monotone operators

Mathematical Programming - This paper proposes an accelerated proximal point method for maximally monotone operators. The proof is computer-assisted via the performance estimation problem approach....     

On the zeros of monotone operators of retarded type in a Banach space

The existence of zeros of operators which map into a Banach space E but whose domain is the space of continuous functions on some real interval into E are considered.     

On maximal monotone operators with relatively compact range

It is shown that every maximal monotone operator on a real Banach space with relatively compact range is of type NI. Moreover, if the space has a separable dual space then every maximally monotone operator T can be approximated by a sequence of maximal monotone operators of type NI, which converge to T in a reasonable sense (in the sense of Kuratowski-Painleve convergence).     

Pictures of monotone operators

Let E be a real Banach space with dual E ^*. We associate with any nonempty subset H of E×E ^* a certain compact convex subset of the first quadrant in ², which we call the picture of H, (H). In general, (H) may be empty, but (M) is nonempty if M is a nonempty monotone subset of E×E ^*. If E is reflexive and M is maximal monotone then (M) is a single point on the diagonal of the first quadrant of ². On the other hand, we give an example (for E the nonreflexive space L ₁[0,1]) of a maximal monotone subset M of E×E ^* such that (0,1)(M) and (1,1)(M) but (1,0)(M). We show that the results for reflexive spaces can be recovered for general Banach spaces by using monotone operator of type (NI) — a class of multifunctions from E into E ^* which includes the subdifferentials of all proper, convex, lower semicontinuous functions on E, all surjective operators and, if E is reflexive, all maximal monotone operators. Our results lead to a simple proof of Rockafellar's result that if E is reflexive and S is maximal monotone on E then S+J is surjective. Our main tool is a classical minimax theorem.     

Pairs of monotone operators

This note is an addendum to Sum theorems for monotone operators and convex functions. In it, we prove some new results on convex functions and monotone operators, and use them to show that several of the constraint qualifications considered in the preceding paper are, in fact, equivalent.

     

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.     

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.     

On generalized cyclically monotone operators and proper quasimonotonictty

In this paper we consider an abstract subdifferential that fulfills a prioria weak type of a mean value property. We survey and extend some recent results connecting the gener-alized convexity of nonsmooth functions with the generalized cyclic monotonidty of their subdifferentials. It is shown that, for a large class of subdifferentials, a Isc function is quasiconvex if and only if its subdifferential is a cyclically quasimonotone operator. An analogous property holds for pseudoconvexity. It is also shown that the subdiffer-ential of a quasiconvex function is properly quasimonotone. This property is slightly stronger than quasimonotonicity, and is more useful in applications connected with variational inequalities     

On the Douglas—Rachford splitting method and the proximal point algorithm for maximal monotone operators

This paper shows, by means of an operator called asplitting operator, that the Douglas—Rachford splitting method for finding a zero of the sum of two monotone operators is a special case of the proximal point algorithm. Therefore, applications of Douglas—Rachford splitting, such as the alternating direction method of multipliers for convex programming decomposition, are also special cases of the proximal point algorithm. This observation allows the unification and generalization of a variety of convex programming algorithms. By introducing a modified version of the proximal point algorithm, we derive a new,generalized alternating direction method of multipliers for convex programming. Advances of this sort illustrate the power and generality gained by adopting monotone operator theory as a conceptual framework.This paper is drawn largely from the dissertation research of the first author. The dissertation was performed at M.I.T. under the supervision of the second author, and was supported in part by the Army Research Office under grant number DAAL03-86-K-0171, and by the National Science Foundation under grant number ECS-8519058.     

A hybrid of the extragradient method and proximal point algorithm for inverse strongly monotone operators and maximal monotone operators in Banach spaces

In the paper, we introduce two iterative sequences for finding a point in the intersection of the zero set of a inverse strongly monotone or inverse-monotone operator and the zero set of a maximal monotone operator in a uniformly smooth and uniformly convex Banach space. We prove weak convergence theorems under appropriate conditions, respectively.     

On random fixed point theorems of random monotone operators

In this paper, we investigate the existence of random fixed point for random mixed monotone operators and random increasing (decreasing) operators and obtain some new random fixed point theorems.