The superiorization method with restarted perturbations for split minimization problems with an application to radiotherapy treatment planning

In this paper we study the split minimization problem that consists of two constrained minimization problems in two separate spaces that are connected via a linear operator that maps one space into the other. To handle the data of such a problem we develop a superiorization approach that can reach a feasible point with reduced … Read more

Dynamic string-averaging CQ-methods for the split feasibility problem with percentage violation constraints arising in radiation therapy treatment planning

We study a feasibility-seeking problem with percentage violation constraints. These are additional constraints, that are appended to an existing family of constraints, which single out certain subsets of the existing constraints and declare that up to a specified fraction of the number of constraints in each subset is allowed to be violated by up to … Read more

Dynamic string-averaging CQ-methods for the split feasibility problem with percentage violation constraints arising in radiation therapy treatment planning

In this paper we study a feasibility-seeking problem with percentage violation con- straints. These are additional constraints, that are appended to an existing family of constraints, which single out certain subsets of the existing constraints and declare that up to a speci ed fraction of the number of constraints in each subset is allowed to be … Read more

The Cyclic Douglas-Rachford Algorithm with r-sets-Douglas-Rachford Operators

The Douglas-Rachford (DR) algorithm is an iterative procedure that uses sequential reflections onto convex sets and which has become popular for convex feasibility problems. In this paper we propose a structural generalization that allows to use r-sets-DR operators in a cyclic fashion. We prove convergence and present numerical illustrations of the potential advantage of such … Read more

The implicit convex feasibility problem and its application to adaptive image denoising

The implicit convex feasibility problem attempts to find a point in the intersection of a finite family of convex sets, some of which are not explicitly determined but may vary. We develop simultaneous and sequential projection methods capable of handling such problems and demonstrate their applicability to image denoising in a specific medical imaging situation. … Read more

The Split Common Null Point Problem

We introduce and study the Split Common Null Point Problem (SCNPP) for set-valued maximal monotone mappings in Hilbert spaces. This problem generalizes our Split Variational Inequality Problem (SVIP) [Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numerical Algorithms 59 (2012), 301–323]. The SCNPP with only two set-valued mappings entails … Read more

A von Neumann Alternating Method for Finding Common Solutions to Variational Inequalities

Modifying von Neumann’s alternating projections algorithm, we obtain an alternating method for solving the recently introduced Common Solutions to Variational Inequalities Problem (CSVIP). For simplicity, we mainly confine our attention to the two-set CSVIP, which entails finding common solutions to two unrelated variational inequalities in Hilbert space. Citation Nonlinear Analysis Series A: Theory, Methods & … Read more

Weak and Strong Convergence of Algorithms for the Split Common Null Point Problem

We introduce and study the Split Common Null Point Problem (SCNPP) for set-valued maximal monotone mappings in Hilbert space. This problem generalizes our Split Variational Inequality Problem (SVIP) [Y. Censor, A. Gibali and S. Reich, Algorithms for the split variational inequality problem, Numerical Algorithms, accepted for publication, DOI 10.1007/s11075-011-9490-5]. The SCNPP with only two set-valued … Read more

The Split Variational Inequality Problem

We propose a new variational problem which we call the Split Variational Inequality Problem (SVIP). It entails finding a solution of one Variational Inequality Problem (VIP), the image of which under a given bounded linear transformation is a solution of another VIP. We construct iterative algorithms that solve such problems, under reasonable conditions, in Hilbert … Read more

Projections Onto Super-Half-Spaces for Monotone Variational Inequality Problems in Finite-Dimensional Spaces

The variational inequality problem (VIP) is considered here. We present a general algorithmic scheme which employs projections onto hyperplanes that separate balls from the feasible set of the VIP instead of projections onto the feasible set itself. Our algorithmic scheme includes the classical projection method and Fukushima’s subgradient projection method as special cases. Citation Technical … Read more