Block-iterative Projection Research Articles

A generalized block-iterative projection method for the common fixed point problem induced by cutters

The block-iterative projections (BIP) method of Aharoni and Censor [Block-iterative projection methods for parallel computation of solutions to convex feasibility problems, Linear Algebra and its Applications 120, (1989), 165–175] is an iterative process for finding asymptotically a point in the nonempty intersection of a family of closed convex subsets. It employs orthogonal projections onto the individual subsets in an algorithmic regime that uses “blocks” of operators and has great flexibility in constructing specific algorithms from it. We extend this algorithmic scheme to handle a family of continuous cutter operators and to find a common fixed point of them. Since the family of continuous cutters includes several important specific operators, our generalized scheme, which ensures global convergence and retains the flexibility of BIP, can handle, in particular, metric (orthogonal) projectors and continuous subgradient projections, which are very important in applications. We also allow a certain kind of adaptive perturbations to be included, and along the way we derive a perturbed Fejér monotonicity lemma which is of independent interest.

Perturbation resilience and superiorization methodology of averaged mappings

We first prove the bounded perturbation resilience for the successive fixed point algorithm of averaged mappings, which extends the string-averaging projection and block-iterative projection methods. We then apply the superiorization methodology to a constrained convex minimization problem where the constraint set is the intersection of fixed point sets of a finite family of averaged mappings.

Convergence of string-averaging method for a class of operators

We analyse a fixed-point iterative method with a finite pool of operators which are subfamily of strictly quasi-nonexpansive operators. These operators, which are not necessarily continuous, may be employed in iterative methods used in convex feasibility problems. Furthermore, members of this subfamily are able to handle convex constraints. The current iterate of the fixed-point iterative method is made by averaging of strings' endpoints and each string consists of a composition of operators which lie in the pool. To examine the study, we deal with two important pools of operators. The first one is a class of operators which define the algebraic iterative methods, as block iterative projection methods, for solving linear systems of equations (inequalities). The second class consists of the parallel subgradient projection operators for solving nonlinear convex feasibility problems. In both classes, we use optimal relaxation or optimal weight parameters which may break the continuity of the operators used in the classes. The advantages and disadvantages of using these parameters are illustrated using some numerical examples.

New Douglas--Rachford Algorithmic Structures and Their Convergence Analyses

In this paper we study new algorithmic structures with Douglas--Rachford (DR) operators to solve convex feasibility problems. We propose to embed the basic two-set-DR algorithmic operator into the string-averaging projections and into the block-iterative projection algorithmic structures, thereby creating new DR algorithmic schemes that include the recently proposed cyclic DR algorithm and the averaged DR algorithm as special cases. We further propose and investigate a new multiple-set-DR algorithmic operator. Convergence of all these algorithmic schemes is studied by using properties of strongly quasi-nonexpansive operators and firmly nonexpansive operators.

Perturbation-Resilient Iterative Methods with an Infinite Pool of Mappings

We study a perturbation-resilient iterative method with an infinite pool of operators having a property, which allows using discontinuous operators, weaker than paracontracting. A convergence result is given under an extra condition which is inherently present in some state-of-the-art iterative methods. To evaluate the study, we develop a new algorithmic scheme which generalizes both the string-averaging algorithm and the block-iterative projection methods, and give its convergence analysis for the consistent case.

Inertial projection algorithms for convex feasibility problem

The purpose of this paper is to apply inertial technique to string averaging projection method and block-iterative projection method in order to get two accelerated projection algorithms for solving convex feasibility problem. Compared with the existing accelerated methods for solving the problem, the inertial technique employs a parameter sequence and two previous iterations to get the next iteration and hence improves the flexibility of the algorithm. Theoretical asymptotic convergence results are presented under some suitable conditions. Numerical simulations illustrate that the new methods have better convergence than the general projection methods. The presented algorithms are inspired by the inertial proximal point algorithm for finding zeros of a maximal monotone operator.

Accelerated perturbation-resilient block-iterative projection methods with application to image reconstruction

We study the convergence of a class of accelerated perturbation-resilient block-iterative projection methods for solving systems of linear equations. We prove convergence to a fixed point of an operator even in the presence of summable perturbations of the iterates, irrespective of the consistency of the linear system. For a consistent system, the limit point is a solution of the system. In the inconsistent case, the symmetric version of our method converges to a weighted least-squares solution. Perturbation resilience is utilized to approximate the minimum of a convex functional subject to the equations. A main contribution, as compared to previously published approaches to achieving similar aims, is a more than an order of magnitude speed-up, as demonstrated by applying the methods to problems of image reconstruction from projections. In addition, the accelerated algorithms are illustrated to be better, in a strict sense provided by the method of statistical hypothesis testing, than their unaccelerated versions for the task of detecting small tumors in the brain from x-ray CT projection data.

Perturbation‐resilient block‐iterative projection methods with application to image reconstruction from projections

A block-iterative projection algorithm for solving the consistent convex feasibility problem in a finite-dimensional Euclidean space that is resilient to bounded and summable perturbations (in the sense that convergence to a feasible point is retained even if such perturbations are introduced in each iterative step of the algorithm) is proposed. This resilience can be used to steer the iterative process towards a feasible point that is superior in the sense of some functional on the points in the Euclidean space having a small value. The potential usefulness of this is illustrated in image reconstruction from projections, using both total variation and negative entropy as the functional.

On Diagonally Relaxed Orthogonal Projection Methods

We propose and study a block-iterative projection method for solving linear equations and/or inequalities. The method allows diagonal componentwise relaxation in conjunction with orthogonal projections onto the individual hyperplanes of the system, and is thus called diagonally relaxed orthogonal projections (DROP). Diagonal relaxation has proven useful in accelerating the initial convergence of simultaneous and block-iterative projection algorithms, but until now it was available only in conjunction with generalized oblique projections in which there is a special relation between the weighting and the oblique projections. DROP has been used by practitioners, and in this paper a contribution to its convergence theory is provided. The mathematical analysis is complemented by some experiments in image reconstruction from projections which illustrate the performance of DROP.

A projection method for semidefinite linear systems and its applications

We study the solution of consistent, semidefinite and symmetric linear systems by iterative techniques. Given a finite sequence of subspaces a block-iterative projection type algorithm is considered. For two specific choices of iteration parameters we show convergence. We apply our results to over and under determined linear equations. These methods are based on decomposing the system matrix into blocks of rows or blocks of columns. Thereby several algorithms, many used in image reconstruction, are presented in a unified way.

Block-Iterative Algorithms with Underrelaxed Bregman Projections

The notion of relaxation is well understood for orthogonal projections onto convex sets. For general Bregman projections it was considered only for hyperplanes, and the question of how to relax Bregman projections onto convex sets that are not linear (i.e., not hyperplanes or half-spaces) has remained open. A definition of the underrelaxation of Bregman projections onto general convex sets is given here, which includes as special cases the underrelaxed orthogonal projections and the underrelaxed Bregman projections onto linear sets as given by De Pierro and Iusem [J. Optim. Theory Appl., 51 (1986), pp. 421--440]. With this new definition, we construct a block-iterative projection algorithmic scheme and prove its convergence to a solution of the convex feasibility problem. The practical importance of relaxation parameters in the application of such projection algorithms to real-world problems is demonstrated on a problem of image reconstruction from projections.

Strong convergence of almost simultaneous block-iterative projection methods in Hilbert spaces

We prove strong convergence of a class of block-iterative projection methods for finding a common point of a finite family of closed convex subsets in a Hilbert space.

Convergence criteria for generalized gradient methods of solving locally Lipschitz feasibility problems

We prove the convergence of a class of iterative algorithms for solving locally Lipschitz feasibility problems, that is, finite systems of inequalities f i (x)≤0, (i ∈ I), where each f i is a locally Lipschitz functional on ℝ n . We also obtain a new convergence criterion for the so-called block-iterative projection methods of finding common points of finite families of convex closed subsets of ℝ n as defined by Aharoni and Censor ([3]).

On the behavior of a block-iterative projection method for solving convex feasibility problems

The behavior of a class of block-iterative projection algorithms for solving convex feasibility problems is studied. A limit characterization theorem and a convergence criterion are proven. Ways of accelerating the computational procedures are pointed out.

Block-iterative projection methods for parallel computation of solutions to convex feasibility problems

An iterative method is proposed for solving convex feasibility problems. Each iteration is a convex combination of projections onto the given convex sets where the weights of the combination may vary from step to step. It is shown that any sequence of iterations generated by the algorithm converges if the intersection of the given family of convex sets is nonempty and that the limit point of the sequence belongs to this intersection under mild conditions on the sequence of weight functions. Special cases are block-iterative processes where in each iterative step a certain subfamily of the given family of convex sets is used. In particular, a block-iterative version of the Agmon-Motzkin-Schoenberg relaxation method for solving systems of linear inequalities is derived. Such processes lend themselves to parallel implementation and will be useful in various areas of applications, including image reconstruction from projections, image restoration, and other fully discretized inversion problems.