Backward–forward algorithms for structured monotone inclusions in Hilbert spaces

In this paper, we study the backward–forward algorithm as a splitting method to solve structured monotone inclusions, and convex minimization problems in Hilbert spaces. It has a natural link with the forward–backward algorithm and has the same computational complexity, since it involves the same basic blocks, but organized differently. Surprisingly enough, this kind of iteration arises when studying the time discretization of the regularized Newton method for maximally monotone operators. First, we show that these two methods enjoy remarkable involutive relations, which go far beyond the evident inversion of the order in which the forward and backward steps are applied. Next, we establish several convergence properties for both methods, some of which were unknown even for the forward–backward algorithm. This brings further insight into this well-known scheme. Finally, we specialize our results to structured convex minimization problems, the gradient-projection algorithms, and give a numerical illustration of theoretical interest.     

A note on Solodov and Tseng’s methods for maximal monotone mappings

This paper considers the problem of finding a zero of the sum of a single-valued Lipschitz continuous mapping A and a maximal monotone mapping B in a closed convex set C. We first give some projection-type methods and extend a modified projection method proposed by Solodov and Tseng for the special case of B=N_C to this problem, then we give a refinement of Tseng’s method that replaces P_C by P_Ck. Finally, convergence of these methods is established.     

Convergence and stability of extended block boundary value methods for Volterra delay integro-differential equations

In this paper, we construct a class of extended block boundary value methods (B₂VMs) for Volterra delay integro-differential equations and analyze the convergence and stability of the methods. It is proven under the classical Lipschitz condition that an extended B₂VM is convergent of order p if the underlying boundary value methods (BVM) has consistent order p. The analysis shows that a B₂VM extended by an A-stable BVM can preserve the delay-independent stability of the underlying linear systems. Moreover, under some suitable conditions, the extended B₂VMs can also keep the delay-dependent stability of the underlying linear systems. In the end, we test the computational effectiveness by applying the introduced methods to the Volterra delay dynamical model of two interacting species, where the theoretical precision of the methods is further verified.     

Maximality of the Sum of a Maximally Monotone Linear Relation and a Maximally Monotone Operator

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: ${\operatorname{dom}}\,A\cap{\operatorname{int}}\,{\operatorname{dom}}\,B\neq\varnothing$ . Moreover, A?+?B is of type (FPV).     

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.     

Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems

The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points F(S) of a nonexpansive mapping S and the set of solutions Ω _A of the variational inequality for a monotone, Lipschitz continuous mapping A. We introduce a hybrid extragradient-like approximation method which is based on the well-known extragradient method and a hybrid (or outer approximation) method. The method produces three sequences which are shown to converge strongly to the same common element of \({F(S)\cap\Omega_{A}}\). As applications, the method provides an algorithm for finding the common fixed point of a nonexpansive mapping and a pseudocontractive mapping, or a common zero of a monotone Lipschitz continuous mapping and a maximal monotone mapping.     

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.     

Lyusternik-Schnirelman theory and eigenvalue problems for monotone potential operators

We treat the eigenvalue problem Ax = λBx, where A and B are odd potential operators, A is strictly monotone, bounded, coercive, and continuously invertible, and B is monotone and compact. A naturally defined iteration operator is employed, together with the Lyusternik-Schnirelman theory, to prove the existence of infinitely many nontrivial eigenfunctions. With the possible exception of the multiplicity assertion the results which we obtain are not new. The method which we use, however, has not been applied before to problems of this type. It exploits both the potential character and the monotonicity of the operators and makes the treatment of the infinite dimensional problem essentially as simple as that of its finite dimensional analog. This simplification results primarily from the compactness properties of the iteration operator.     

A Trotter-Kato type result for a second order difference inclusion in a Hilbert space

A Trotter-Kato type result is proved for a class of second order difference inclusions in a real Hilbert space. The equation contains a nonhomogeneous term f and is governed by a nonlinear operator A, which is supposed to be maximal monotone and strongly monotone. The associated boundary conditions are also of monotone type. One shows that, if An is a sequence of operators which converges to A in the sense of resolvent and fn converges to f in a weighted l²-space, then under additional hypotheses, the sequence of the solutions of the difference inclusion associated to An and fn is uniformly convergent to the solution of the original problem.     

Higher Differentiability of Solutions of Elliptic Systems with Sobolev Coefficients: The Case p = n = 2

We establish higher differentiability results for local solutions of elliptic systems of the type $$\text{div} A(x,Du)=0 $$ in a bounded open set in ?². The operator A(x, ξ) is assumed to be strictly monotone and Lipschitz continuous with respect to variable ξ. The novelty of the paper is that we allow discontinuous dependence with respect to the x-variable, through a suitable Sobolev function.     

The small quantum cohomology of a weighted projective space, a mirror D-module and their classical limits

We first describe a mirror partner (B-model) of the small quantum orbifold cohomology of weighted projective spaces (A-model) in the framework of differential equations: we attach to the A-model (resp. B-model) a quantum differential system (that is a trivial bundle equipped with a suitable flat meromorphic connection and a flat bilinear form) and we give an explicit isomorphism between these two quantum differential systems. On the A-side (resp. on the B-side), the quantum differential system alluded to is naturally produced by the small quantum cohomology (resp. a solution of the Birkhoff problem for the Brieskorn lattice of a Landau–Ginzburg model). Then we study the degenerations of these quantum differential systems and we apply our results to the construction of (classical, limit, logarithmic) Frobenius manifolds.     

Improved gradient method for monotone and lipschitz continuous mappings in banach spaces

Let C be a nonempty closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E and {A_n}_(n∈N) be a family of monotone and Lipschitz continuos mappings of C into E~*. In this article, we consider the improved gradient method by the hybrid method in mathematical programming [10] for solving the variational inequality problem for{A_n} and prove strong convergence theorems. And we get several results which improve the well-known results in a real 2-uniformly convex and uniformly smooth Banach space and a real Hilbert space.     

On the efficiency of a class of a-stable methods

The numerical solution of systems of differential equations of the formB dx/dt=σ(t)Ax(t)+f(t),x(0) given, whereB andA (withB and —(A+A ^T) positive definite) are supposed to be large sparse matrices, is considered.A-stable methods like the Implicit Runge-Kutta methods based on Radau quadrature are combined with iterative methods for the solution of the algebraic systems of equations.     

On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators

Nemirovski’s analysis (SIAM J. Optim. 15:229–251, 2005) indicates that the extragradient method has the O(1/t) convergence rate for variational inequalities with Lipschitz continuous monotone operators. For the same problems, in the last decades, a class of Fejér monotone projection and contraction methods is developed. Until now, only convergence results are available to these projection and contraction methods, though the numerical experiments indicate that they always outperform the extragradient method. The reason is that the former benefits from the ‘optimal’ step size in the contraction sense. In this paper, we prove the convergence rate under a unified conceptual framework, which includes the projection and contraction methods as special cases and thus perfects the theory of the existing projection and contraction methods. Preliminary numerical results demonstrate that the projection and contraction methods converge twice faster than the extragradient method.     

On the method of alternating resolvents

The work of Hundal [H. Hundal, An alternating projection that does not converge in norm, Nonlinear Anal. 57 (1) (2004) 35-61] has revealed that the sequence generated by the method of alternating projections converges weakly, but not strongly in general. In this paper, we present several algorithms based on alternating resolvents of two maximal monotone operators, A and B, that can be used to approximate common zeros of A and B. In particular, we prove that the sequences generated by our algorithms converge strongly. A particular case of such algorithms enables one to approximate minimum values of certain convex functionals.     

Multiplicative perturbation of nonlinear m-accretive operators

Criteria are obtained for when an accretive product (i.e., composition) BA of nonlinear m-accretive operators A and B in a Banach space X will be itself m-accretive; and, in particular, when a monotone product of two maximal monotone operators in a Hilbert space will be maximal monotone. This extends the theory of multiplicative perturbation of infinitesimal generators of contraction semigroups to the nonlinear case. Also obtained as a biproduct are existence theorems for certain Hammerstein integral equations.     

Invariant sets and Lyapunov pairs for differential inclusions with maximal monotone operators

We give different conditions for the invariance of closed sets with respect to differential inclusions governed by a maximal monotone operator defined on Hilbert spaces, which is subject to a Lipschitz continuous perturbation depending on the state. These sets are not necessarily weakly closed as in [3], [4], while the invariance criteria are still written by using only the data of the system. So, no need to the explicit knowledge of neither the solution of this differential inclusion, nor the semi-group generated by the maximal monotone operator. These invariant/viability results are next applied to derive explicit criteria for a-Lyapunov pairs of lower semi-continuous (not necessarily weakly-lsc) functions associated to these differential inclusions. The lack of differentiability of the candidate Lyapunov functions and the consideration of general invariant sets (possibly not convex or smooth) are carried out by using techniques from nonsmooth analysis.     

A convergence analysis of an inexact Newton-Landweber iteration method for nonlinear problem

In this paper, we study the convergence and the convergence rates of an inexact Newton–Landweber iteration method for solving nonlinear inverse problems in Banach spaces. Opposed to the traditional methods, we analyze an inexact Newton–Landweber iteration depending on the Hölder continuity of the inverse mapping when the data are not contaminated by noise. With the namely Hölder-type stability and the Lipschitz continuity of DF, we prove convergence and monotonicity of the residuals defined by the sequence induced by the iteration. Finally, we discuss the convergence rates.     

A family of projective splitting methods for the sum of two maximal monotone operators

A splitting method for two monotone operators A and B is an algorithm that attempts to converge to a zero of the sum A + B by solving a sequence of subproblems, each of which involves only the operator A, or only the operator B. Prior algorithms of this type can all in essence be categorized into three main classes, the Douglas/Peaceman-Rachford class, the forward-backward class, and the little-used double-backward class. Through a certain “extended” solution set in a product space, we construct a fundamentally new class of splitting methods for pairs of general maximal monotone operators in Hilbert space. Our algorithms are essentially standard projection methods, using splitting decomposition to construct separators. We prove convergence through Fejér monotonicity techniques, but showing Fejér convergence of a different sequence to a different set than in earlier splitting methods. Our projective algorithms converge under more general conditions than prior splitting methods, allowing the proximal parameter to vary from iteration to iteration, and even from operator to operator, while retaining convergence for essentially arbitrary pairs of operators. The new projective splitting class also contains noteworthy preexisting methods either as conventional special cases or excluded boundary cases. Dedicated to Clovis Gonzaga on the occassion of his 60th birthday.     

Strongly preserver problems in Banach algebras and C1-algebras

Let A and B be Banach algebras. Assume that A is unital. We prove that an additive map $T:A\to B$ strongly preserves Drazin (or equivalently group) invertibility, if and only if T is a Jordan triple homomorphism. When A and B are C¹-algebras, we characterize the linear maps strongly preserving generalized invertibility (in the Jordan systems’ sense), and as consequence we determine the structure of selfadjoint linear maps strongly preserving Moore–Penrose invertibility.