The aim of this article is to present two different primal-dual methods for solving structured monotone inclusions involving parallel sums of compositions of maximally monotone operators with linear bounded operators. By employing some elaborated splitting techniques, all of the operators occurring in the problem formulation are processed individually via forward or backward steps. The treatment of parallel sums of linearly composed maximally monotone operators is motivated by applications in imaging which involve first- and second-order total variation functionals, to which a special attention is given.
The most famous open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar's constraint qualification holds. In this paper, we prove the maximal monotonicity of $A+B$ provided that $A, B$ are maximally monotone and $A$ is a linear relation, as soon as Rockafellar's constraint qualification holds: $\dom A\cap\inte\dom B\neq\varnothing$. Moreover, $A+B$ is of type (FPV).
The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that Rockafellar's constraint qualification holds. In this paper, we prove the maximal monotonicity of $A+\partial f$ provided that $A$ is a maximally monotone linear relation, and $f$ is a proper lower semicontinuous convex function satisfying $\dom A\cap\inte\dom \partial f\neq\varnothing$. Moreover, $A+\partial f$ is of type (FPV). The maximal monotonicity of $A+\partial f$ when $\intdom A\cap\dom \partial f\neq\varnothing$ follows from a resu…
In this paper, we consider the structure of maximally monotone operators in Banach space whose domains have nonempty interior and we present new and explicit structure formulas for such operators. Along the way, we provide new proofs of the norm-to-weak$^{*}$ closedness and of property (Q) for these operators (as recently proven by Voisei). Various applications and limiting examples are given.
The notion of a firmly nonexpansive mapping is central in fixed point theory because of attractive convergence properties for iterates and the correspondence with maximal monotone operators due to Minty. In this paper, we systematically analyze the relationship between properties of firmly nonexpansive mappings and associated maximal monotone operators. Dual and self-dual properties are also identified. The results are illustrated through several examples.
Monotone operator theory and fixed point theory for nonexpansive mappings are central areas in modern nonlinear analysis and optimization. Although these areas are fairly well developed, almost all examples published are based on subdifferential operators, linear relations, or combinations thereof. In this paper, we construct an intriguing maximally monotone operator induced by a certain nonexpansive retraction. We analyze this operator, which does not appear to be assembled from subdifferential operators or linear relations, in some detail. Particular emphasis is placed on duality and strong …
Monotone linear relations play important roles in variational inequality problems and quadratic optimizations. In this paper, we give explicit maximally monotone linear subspace extensions of a monotone linear relation in finite dimensional spaces. Examples are provided to illustrate our extensions. Our results generalize a recent result by Crouzeix and Anaya.
A common theme in mathematics is to define generalized solutions to deal with problems that potentially do not have solutions. A classical example is the introduction of least squares solutions via the normal equations associated with a possibly infeasible system of linear equations. In this paper, we introduce a "normal problem" associated with finding a zero of the sum of two maximally monotone operators. If the original problem admits solutions, then the normal problem returns this same set of solutions. The normal problem may yield solutions when the original problem does not admit any; fu…
We present a counterexample showing that the graphical limit of maximally monotone operators might not be maximally monotone. We also characterize the directional differentiability of the resolvent of an operator $B$ in terms of existence and maximal monotonicity of the proto-derivative of $B$.
In his recent Proceedings of the AMS paper "Gossez's skew linear map and its pathological maximally monotone multifunctions", Stephen Simons proved that the closure of the range of the sum of the Gossez operator and a multiple of the duality map is nonconvex whenever the scalar is between 0 and 4. The problem of the convexity of that range when the scalar is equal to 4 was explicitly stated. In this paper, we answer this question in the negative for any scalar greater than or equal to 4. We derive this result from an abstract framework that allows us to also obtain a corresponding result for t…
In their recent SIAM J. Control Optim. paper from 2009, 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.
We provide a concise analysis about what is known regarding when the closure of the domain of a maximally monotone operator on an arbitrary real Banach space is convex. In doing so, we also provide an affirmative answer to a problem posed by Simons.
We show that every maximally monotone operator of Fitzpatrick-Phelps type defined on a real Banach space must be of dense type. This provides an affirmative answer to a question posed by Stephen Simons in 2001 and implies that various important notions of monotonicity coincide.
Enlargements have proven to be useful tools for studying maximally monotone mappings. It is therefore natural to ask in which cases the enlargement does not change the original mapping. Svaiter has recently characterized non-enlargeable operators in reflexive Banach spaces and has also given some partial results in the nonreflexive case. In the present paper, we provide another characterization of non-enlargeable operators in nonreflexive Banach spaces under a closedness assumption on the graph. Furthermore, and still for general Banach spaces, we present a new proof of the maximality of the s…
The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar's constraint qualification holds. In this paper, we establish the maximal monotonicity of $A+B$ provided that $A$ and $B$ are maximally monotone operators such that $\sta(\dom A)\cap\inte\dom B\neq\varnothing$, and $A$ is of type (FPV). We show that when also $\dom A$ is convex, the sum operator: $A+B$ is also of type (FPV). Our result generalizes and unifies several recent sum theorems.
The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar's constraint qualification holds, which is called the "sum problem". In this paper, we establish the maximal monotonicity of $A+B$ provided that $A$ and $B$ are maximally monotone operators such that $\dom A\cap\inte\dom B\neq\varnothing$, and $A+N_{\overline{\dom A}}$ is of type (FPV). This generalizes various current results and also gives an affirmative answer to a problem posed by Borwein and Yao. Moreover, we p…
We deal with monotone inclusion problems of the form $0\in Ax+Dx+N_C(x)$ 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 H. Attouch, M.-O. Czarnecki and J. Peypouquet in [3]. 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 monoto…
In this note, we give a generalization of Gossez's example of a maximally monotone multifunction such that the closure of its range is not convex, using more elementary techniques than in Gossez's original papers. We also discuss some new properties of Gossez's skew linear operator and its adjoint. While most of this paper uses elementary functional analysis, we correlate our results with those obtained by using the Stone-Cech compactification of the integers.
It is shown that, for maximally monotone linear relations defined on a general Banach space, the monotonicities of dense type, of negative-infimum type, and of Fitzpatrick-Phelps type are the same and equivalent to monotonicity of the adjoint. This result also provides affirmative answers to two problems: one posed by Phelps and Simons, and the other by Simons.
We study nearly equal and nearly convex sets, ranges of maximally monotone operators, and ranges and fixed points of convex combinations of firmly nonexpansive mappings. The main result states that the range of an average of firmly nonexpansive mappings is nearly equal to the average of the ranges of the mappings. A striking application of this result yields that the average of asymptotically regular firmly nonexpansive mappings is also asymptotically regular. Throughout, examples are provided to illustrate the theory. We also obtain detailed information on the domain and range of the resolven…
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 useful range 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 n…
Fenchel subdifferential operators of lower semicontinuous proper convex functions on real Banach spaces are classically characterized as those operators that are maximally cyclically monotone or, equivalently, maximally monotone and cyclically monotone. This paper presents an alternative characterization, which does not involve cyclic monotonicity. In the case of subdifferential operators of sublinear functions, the new characterization substantially simplifies. Dually, the new characterization of normal cone operators is very simple, too.
In this paper, we construct maximally monotone operators that are not of Gossez's dense-type (D) in many nonreflexive spaces. Many of these operators also fail to possess the Brønsted-Rockafellar (BR) property. Using these operators, we show that the partial inf-convolution of two BC--functions will not always be a BC--function. This provides a negative answer to a challenging question posed by Stephen Simons. Among other consequences, we deduce that every Banach space which contains an isomorphic copy of the James space $\mathbf{J}$ or its dual $\mathbf{J}^*$, or $c_0$ or its dual $\ell^1$, a…
In this paper, we survey recent progress on the theory of maximally monotone operators in general Banach space. We also extend various of the results and leave some open questions.
It is shown that if a Banach space X has the weak Banach-Saks property and the weak fixed point property for nonexpansive mappings and Y satisfies property asymptotic (P) (which is weaker than the condition WCS(Y)>1), then the direct sum of X and Y endowed with a strictly monotone norm enjoys the weak fixed point property. The same conclusion is valid if X admits a 1-unconditional basis.
We discuss "Banach SN spaces", which include Hilbert spaces, negative Hilbert spaces, and the product of any real Banach space with its dual. We introduce "L-positive" sets, which generalize monotone multifunctions from a Banach space into its dual. We introduce the concepts of "$r_L$-density" and its specialization "quasidensity": the closed quasidense monotone multifunctions from a Banach space into its dual form a (generally) strict subset of the maximally monotone ones, though all surjective maximally monotone and all maximally monotone multifunctions on a reflexive space are quasidense. W…
Maximally monotone operators are fundamental objects in modern optimization. The main classes of monotone operators are subdifferential operators and matrices with a positive semidefinite symmetric part. In this paper, we study a nice class of monotone operators: displacement mappings of isometries of finite order. We derive explicit formulas for resolvents, Yosida approximations, and (set-valued and MoorePenrose) inverses. We illustrate our results by considering certain rational rotators and circular shift operators.
We introduce and investigate the convergence properties of an inertial forward-backward-forward splitting algorithm for approaching the set of zeros of the sum of a maximally monotone operator and a single-valued monotone and Lipschitzian operator. By making use of the product space approach, we expand it to the solving of inclusion problems involving mixtures of linearly composed and parallel-sum type monotone operators. We obtain in this way an inertial forward-backward-forward primal-dual splitting algorithm having as main characteristic the fact that in the iterative scheme all operators a…
In this paper, we first investigate an abstract subdifferential for which (using Ekeland's variational principle) we can prove an analog of the Brøndsted-Rockafellar property. We introduce the "$r_L$-density" of a subset of the product of a Banach space with its dual. A closed $r_L$-dense monotone set is maximally monotone, but we will also consider the case of nonmonotone closed $r_L$-dense sets. As a special case of our results, we can prove Rockafellar's result that the subdifferential of a proper convex lower semicontinuous function is maximally monotone.
In this work we investigate dynamical systems designed to approach the solution sets of inclusion problems involving the sum of two maximally monotone operators. Our aim is to design methods which guarantee strong convergence of trajectories towards the minimum norm solution of the underlying monotone inclusion problem. To that end, we investigate in detail the asymptotic behavior of dynamical systems perturbed by a Tikhonov regularization where either the maximally monotone operators themselves, or the vector field of the dynamical system is regularized. In both cases we prove strong converge…
In this paper, we give two explicit examples of unbounded linear maximal monotone operators. The first unbounded linear maximal monotone operator $S$ on $\ell^{2}$ 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 monot…
We show that the direct sum of Banach spaces $X_{1},..., X_{r}$ with a strictly monotone norm has the weak fixed point property for nonexpansive mappings whenever $M(X_{i})>1$ for each $i=1,...,r$. In particular, $(X_{1} \oplus ... \oplus X_{r})_ψ$ enjoys the fixed point property if Banach spaces $X_{i}$ are uniformly nonsquare. This combined with the earlier results gives a definitive answer for r=2: the direct sum of uniformly nonsquare spaces $X_{1}, X_{2}$ with any monotone norm has FPP. Our results are extended for asymptotically nonexpansive mappings in the intermediate sense.
We provide quantitative information in the form of a rate of metastability in the sense of T. Tao and (under a metric regularity assumption) a rate of convergence for an algorithm approximating zeros of differences of maximally monotone operators due to A. Moudafi by using techniques from `proof mining', a subdiscipline of mathematical logic. For the rate of convergence, we provide an abstract and general result on the construction of rates of convergence for quasi-Fejér monotone sequences with metric regularity assumptions, generalizing previous results for Fejér monotone sequences due to U. …
In this paper, we study convex analysis and its theoretical applications. We first apply important tools of convex analysis to Optimization and to Analysis. We then show various deep applications of convex analysis and especially infimal convolution in Monotone Operator Theory. Among other things, we recapture the Minty surjectivity theorem in Hilbert space, and present a new proof of the sum theorem in reflexive spaces. More technically, we also discuss autoconjugate representers for maximally monotone operators. Finally, we consider various other applications in mathematical analysis.
We investigate the convergence rates of the trajectories generated by implicit first and second order dynamical systems associated to the determination of the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space. We show that these trajectories strongly converge with exponential rate to a zero of the sum, provided the latter is strongly monotone. We derive from here convergence rates for the trajectories generated by dynamical systems associated to the minimization of the sum of a proper, convex and lower semicontinuous function …
We begin by considering second order dynamical systems of the from $\ddot x(t) + γ(t)\dot x(t) + λ(t)B(x(t))=0$, where $B: {\cal H}\rightarrow{\cal H}$ is a cocoercive operator defined on a real Hilbert space ${\cal H}$, $λ:[0,+\infty)\rightarrow [0,+\infty)$ is a relaxation function and $γ:[0,+\infty)\rightarrow [0,+\infty)$ a damping function, both depending on time. For the generated trajectories, we show existence and uniqueness of the generated trajectories as well as their weak asymptotic convergence to a zero of the operator $B$. The framework allows to address from similar perspectives…
Because of Minty's classical correspondence between firmly nonexpansive mappings and maximally monotone operators, the notion of a firmly nonexpansive mapping has proven to be of basic importance in fixed point theory, monotone operator theory, and convex optimization. In this note, we show that if finitely many firmly nonexpansive mappings defined on a real Hilbert space are given and each of these mappings is asymptotically regular, which is equivalent to saying that they have or "almost have" fixed points, then the same is true for their composition. This significantly generalizes the resul…
The correspondence between the class of nonexpansive mappings and the class of maximally monotone operators via the reflected resolvents of the latter has played an instrumental role in the convergence analysis of the splitting methods. Indeed, the performance of some of these methods, e.g., Douglas-Rachford and Peaceman-Rachford methods hinges on iterating the so-called splitting operator associated with the individual operators. This splitting operator is a function of the composition of the reflected resolvents of the underlying operators. In this paper, we provide a comprehensive study of …
In this paper, we develop some of the theory of SSD spaces and SSDB spaces, and deduce some results on maximally monotone multifunctions on a reflexive Banach space.
In this paper, we show how the Brezis-Browder theorem for maximally monotone multifunctions with a linear graph on a reflexive Banach space, and a consequence of it due to Yao, can be generalized to SSDB spaces.
In this paper, we give three different new proofs of the validity of the geometry conjecture about cycles of projections onto nonempty closed, convex subsets of a Hilbert space. The first uses a simple minimax theorem, which depends on the finite dimensional Hahn-Banach theorem. The second uses Fan's inequality, which has found many applications in optimization and mathematical economics. The third uses three results on maximally monotone operators on a Hilbert space.
