An important iterative method for solving systems of linear equations or linear inequalities is the Landweber method which constructs orbits of the Landweber operator. We show that many iterative methods for solving these problems employ, actually, underrelaxations of the Landweber operator or its scaled version. We give estimations of the relaxation parameters for the operators applied in particular methods. Moreover, we present an extrapolated version of the Landweber method which is applied for solving consistent systems of linear inequalities. We show that the extrapolated Landweber method is a special case of the surrogate projection method. We also compare the numerical behavior of particular methods.

We study the product of two relaxed cutters having a common fixed point. We assume that one of the relaxation parameters is greater than two so that the corresponding relaxed cutter is no longer quasi-nonexpansive, but rather demicontractive. We show that if both of the operators are (weakly/linearly) regular, then under certain conditions, the resulting product inherits the same type of regularity. We then apply these results to proving convergence in the weak, norm and linear sense of algorithms that employ such products.

We give properties of strict pseudocontractions and demicontractions defined on a Hilbert space, which constitute wide classes of operators that arise in iterative methods for solving fixed point problems. In particular, we give necessary and sufficient conditions under which a convex combination and composition of strict pseudocontractions as well as demicontractions that share a common fixed point is again a strict pseudocontraction or a demicontraction, respectively. Moreover, we introduce a generalized relaxation of composition of demicontraction and give its properties. We“ apply these properties to prove the weak convergence of a class of algorithms that is wider than the Douglas–Rachford algorithm and projected Landweber algorithms. We have also presented two numerical examples, where we compare the behavior of the presented methods with the Douglas–Rachford method.

In this paper, we analyze the convergence properties of projected non-stationary block iterative methods (P-BIM) aiming to find a constrained solution to large linear, usually both noisy and ill-conditioned, systems of equations. We split the error of the kth iterate into noise error and iteration error, and consider each error separately. The iteration error is treated for a more general algorithm, also suited for solving split feasibility problems in Hilbert space. The results for P-BIM come out as a special case. The algorithmic step involves projecting onto closed convex sets. When these sets are polyhedral, and of finite dimension, it is shown that the algorithm converges linearly. We further derive an upper bound for the noise error of P-BIM. Based on this bound, we suggest a new strategy for choosing relaxation parameters, which assist in speeding up the reconstruction process and improving the quality of obtained images. The relaxation parameters may depend on the noise. The performance of the suggested strategy is shown by examples taken from the field of image reconstruction from projections.

The split equality problem (SEP) seeks a pair of points $(x^{\ast },y^{\ast})\in (C,D)$ with the property that $Ax^{\ast }=By^{\ast }$, where $C,D$ arenonempty closed convex subsets of Hilbert spaces $\mathcal{H}_{1}$ and $%\mathcal{H}_{2}$, respectively, and $A:\mathcal{H}_{1}\rightarrow \mathcal{H}%_{3}$ and $B:\mathcal{H}_{2}\rightarrow \mathcal{H}_{3}$ are bounded linearoperators, where $\mathcal{H}_{3}$ is another Hilbert space. The SEP canequivalently be converted to a split feasibility problem in theproduct space $\mathcal{H}_{1}\times \mathcal{H}_{2}$. Using thisequivalence, we are able to provide a Landweber operator approach tostudying the convergence of several iterative methods for finding a solutionto the SEP. We also discuss the linear regularity of the Landweber operatorassociated with the SEP and linear convergence of the iterative methods.

We consider a Hilbert space that is a product of a finite number of Hilbert spaces and operators that are represented by "componental operators" acting on the Hilbert spaces that form the product space. We attribute operatorial properties to the componental operators rather than to the full operators. The operatorial properties that we discuss include nonexpansivity, firm non-expansivity, relaxed firm nonexpansivity, averagedness, being a cutter, quasi-nonexpansivity, strong quasi-nonexpansivity, strict quasi-nonexpansivity and contraction. Some relationships between operators whose componental operators have such properties and operators that have these properties on the product space are studied. This enables also to define componental fixed point sets and to study their properties. For componental contractions we offer a variant of the Banach fixed point theorem. Our motivation comes from the desire to extend a fully-simultaneous method that takes into account sparsity of the linear system in order to accelerate convergence [Censor et al., On diagonally relaxed orthogonal projection methods, SIAM J. Sci. Comput. 30 (2008), 473-504]. This was originally applicable to the linear case only and gives rise to an iterative process that uses dfferent componental operators during iterations.

We consider the split convex feasibility problem in a fixed point setting. Motivated by the well-known CQ-method of Byrne (2002), we define an abstract andweber transform which applies to more general operators than the metric projection. We call the result of this transform a Landweber operator. It turns out that the Landweber transform preserves many interesting properties. For example, the Landweber transform of a (quasi/firmly) nonexpansive mapping is again (quasi/firmly) nonexpansive. Moreover, the Landweber transform of a (weakly/linearly) regular mapping is again (weakly/linearly) regular. The preservation of regularity is important because it leads to (weak/linear) convergence of many CQ-type methods.

We study variational inequalities which are governed by a strongly monotone and Lipschitz continuous operator $F$ over a closed and convex set $S$. We assume that $S=C\cap A^{-1}(Q)$ is the nonempty solution set of a (multiple-set) split convex feasibility problem, where $C$ and $Q$ are both closed and convex subsets of two real Hilbert spaces $\mathcal H_1$ and $\mathcal H_2$, respectively, and the operator $A$ acting between them is linear. We consider a modification of the gradient projection method the main idea of which is to replace at each step the metric projection onto $S$ by another metric projection onto a half-space which contains $S$. We propose three variants of a method for constructing the above-mentioned half-spaces by employing the multiple-set and the split structure of the set $S$. For the split part we make use of the Landweber transform.

We present two versions of the extrapolated cyclic subgradient projections method for solving the convex feasibility problem. Moreover, we present the results of numerical tests, where we compare the methods with the classical cyclic subgradient projections method.

In this paper we present a systematic study of regular sequences of quasi-nonexpansive operators in Hilbert space. We are interested, in particular, in weakly, boundedly and linearly regular sequences of operators. We show that the type of the regularity is preserved under relaxations, convex combinations and products of operators. Moreover, in this connection, we show that weak, bounded and linear regularity lead to weak, strong and linear convergence, respectively, of various iterative methods. This applies, in particular, to block iterative and string averaging projection methods, which, in principle, are based on the above-mentioned algebraic operations applied to projections. Finally, we show an application of regular sequences of operators to variational inequality problems.

The convex feasibility problem is to find a common point of a finite family of closed convex subsets. In many applications one requires something more, namely finding a common point of closed convex subsets which minimizes a continuous convex function. The latter requirement leads to an application of the superiorization methodology which is actually settled between methods for convex feasibility problem and the convex constrained minimization. Inspired by the superiorization idea we introduce a method which sequentially applies a long-step algorithm for a sequence of convex feasibility problems; the method employs quasi-nonexpansive operators as well as subgradient projections with level control and does not require evaluation of the metric projection. We replace a perturbation of the iterations (applied in the superiorization methodology) by a perturbation of the current level in minimizing the objective function. We consider the method in the Euclidean space in order to guarantee the strong convergence, although the method is well defined in a Hilbert space.

We study the convergence properties of an iterative method for a variational inequality defined on a solution set of the split common fixed point problem. The method involves Landweber-type operators related to the problem as well as their extrapolations in an almost cyclic way. The evaluation of these extrapolations does not require prior knowledge of the matrix norm. We prove the strong convergence under the assumption that the operators employed in the method are approximately shrinking.

Iteration-discretization concepts combine approximations of problems in infinite dimensional Hilbert spaces with iteration steps in finite dimensions and a simultaneous, but appropriate refinement of the discretization. In the case of simple iteration maps like gradient projection type steps this results in a problem of nonlinear analysis to find parameter adjustments despite the simultaneous changes of discretization still results in a convergent process. In the papers [1, 7] the analysis of the related parameter changes and simple projected gradient steps applied to variational inequalities with Lipschitz continuous and strongly monotone operators have been provided. Unlike in the majority of publications (see e.g. [8-10,18,20,23,27,29,30,32] the iteration-discretization concept is considered which applies at each discretization level only a finite number of steps. In the present paper the convergence of such processes is analyzed for the case if the strong monotonicity is relaxed to monotonicity only. To extend the previous approach a Tikhonov regularization is applied. Via an adapted parameter control the convergence as one-shot approach can be established. This way the iteration-discretization can be applied to variational inequality problem with only monotone operators.

Our aim is to present several properties of a Landweber operator and of a Landweber-type operator. These operators are widely used in methods for solving the split feasibility problem and the split common fixed point problem. The presented properties can be used in proofs of convergence of related algorithms.

Projections onto sets are used in a wide variety of methods in optimization theory but not every method that uses projections really belongs to the class of projection methods as we mean it here. Here, projection methods are iterative algorithms that use projections onto sets while relying on the general principle that when a family of (usually closed and convex) sets is present, then projections (or approximate projections) onto the given individual sets are easier to perform than projections onto other sets (intersections, image sets under some transformation, etc.) that are derived from the given family of individual sets. Projection methods employ projections (or approximate projections) onto convex sets in various ways. They may use different kinds of projections and, sometimes, even use different projections within the same algorithm. They serve to solve a variety of problems which are either of the feasibility or the optimization types. They have different algorithmic structures, of which some are particularly suitable for parallel computing, and they demonstrate nice convergence properties and/or good initial behavioural patterns. This class of algorithms has witnessed great progress in recent years and its member algorithms have been applied with success to many scientific, technological and mathematical problems. This annotated bibliography includes books and review papers on, or related to, projection methods that we know about, use and like. If you know of books or review papers that should be added to this list please contact us.

In this paper, we prove the convergence in a norm of sequences generated by an iterative process for solving a variational inequality over the subset of fixed points of a quasi-nonexpansive operator T defined on a Hilbert space. The process employs a sequence of quasi-nonexpansive operators for which the subset of common fixed points contains FixT. We prove the convergence under a demi-closedness type condition for the sequence of operators as well as under the assumption that the process is approximately shrinking. We also give examples of methods satisfying these assumptions.

In this paper we present an application of a class of quasi-nonexpansive operators to iterative methods for solving the following variational inequality problem VIP(F,C): Find 71,. E C such that (F (u) over bar z-(u) over bar) >= 0 for all z is an element of C, where C is a closed and convex subset of a Hilbert space H and F : H -> H is strongly monotone and Lipschitz continuous. A classical method for VIP (F, C) is the gradient projection (GP) method Xk+1 = PC(X-k - mu Fx(k)) which generates sequences converging to the unique solution of VIP(F,C) if mu > 0 is sufficiently small. Unfortunately, in many optimization problems the GP method cannot be applied, because it requires an explicit computation of Pouk in each iteration, where u(k) = x(k)- mu Fx(k). To overcome this disadvantage of the GP method, one can replace the operator Pc employed in the k-th iteration of the method by a quasi-nonexpansive operator T-k and a constant u by lambda(k) >= 0, k >= 0, satisfying boolean AND(infinity)(k=0) FixT(k) superset of FixT and limb(k) lambda(k) = 0. The new method can be presented equivalently as a so called general hybrid steepest descent (GHSD) method in the form Uk+1 = T(k)u(k) - lambda(k)FT(k)u(k). One should, however, suppose something more on the operators Tk in order to guarantee the convergence of uk to the solution of VIP(F,C). In this paper we introduce a class of approximately shrinking operators, prove the closedness of this class with respect to compositions and convex combinations and apply the operators from this class to a general hybrid steepest descent method for solving VIP(F,C). We give sufficient conditions for the convergence of the GHSD method as well as present several examples of methods which satisfy these conditions. In particular, we apply the results in the case, when C = boolean AND(m)(i=1) Fix U-i and U-i ; H -> H are quasi-nonexpansive operators having a common fixed point, i = 1, 2,...,m.

The split common fixed point problem (also called the multiple-sets split feasibility problem) is to find a common fixed point of a finite family of operators in one real Hilbert space, whose image under a bounded linear transformation is a common fixed point of another family of operators in the image space. In the literature one can find many methods for solving this problem as well as for its special case, called the split feasibility problem. We propose a general method for solving both problems. The method is based on a block-iterative procedure, in which we apply quasi-nonexpansive operators satisfying the demi-closedness principle and having a common fixed point. We prove the weak convergence of sequences generated by this method and show that the convergence for methods known from the literature follows from our general result.

There is a wide range of iterative methods in infinite dimensional spaces to treat variational equations or variational inequalities. As a rule, computational handling of problems in infinite dimensional spaces requires some discretization. Any useful discretization of the original problem leads to families of problems over finite dimensional spaces. Thus, two infinite techniques, namely discretization and iteration are embedded into each other. In the present paper, the behaviour of truncated iterative methods is studied, where at each discretization level only a finite number of steps is performed. In our study no accuracy dependent a posteriori stopping criterion is used. From an algorithmic point of view, the considered methods are of iteration–discretization type. The major aim here is to provide the convergence analysis for the introduced abstract iteration–discretization methods. A special emphasis is given on algorithms for the treatment of variational inequalities with strongly monotone operators over fixed point sets of quasi-nonexpansive mappings.

Many convex optimization problems in the Euclidean space can be formulated as a variational inequality over a subset of points satisfying a system of convex inequalities. In this article we propose a method for solving this problem. In the method we combine a hybrid descent idea presented in I. Yamada and N. Ogura, Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mapping, Numer. Funct. Anal. and Optimiz. 25 (2004) 619-655 and an extrapolated simultaneous subgradient projections introduced in L. T. Dos Santos, A parallel subgradient projections method for the convex feasibility problem, J. Comp. and Applied Math. 18 (1987) 307-320. The method does not require computation of the metric projection and can be simply performed. The method provides long steps which seem to be advantageous for the behavior of the method.