projection methods

The alternating simultaneous Halpern-Lions-Wittmann-Bauschke algorithm for finding the best approximation pair for two disjoint intersections of convex sets

  • Yair Censor
  • Rafiq Mansour
  • Daniel Reem
    • Categories Approximation Algorithms, Constrained Nonlinear Optimization, Convex Optimization Tags , , , , ,

    Given two nonempty and disjoint intersections of closed and convex subsets, we look for a best approximation pair relative to them, i.e., a pair of points, one in each intersection, attaining the minimum distance between the disjoint intersections. We propose an iterative process based on projections onto the subsets which generate the intersections. The process … Read more

    A successive centralized circumcentered-reflection method for the convex feasibility problem

  • Yunier Bello-Cruz
  • Roger Behling
  • A. N. Iusem
  • Luiz-Rafael Santos
  • Di Liu
    • Categories Convex Optimization Tags , , ,

    In this paper, we present a successive centralization process for the circumcentered-reflection scheme with several control sequences for solving the convex feasibility problem in Euclidean space. Assuming that a standard error bound holds, we prove the linear convergence of the method with the most violated constraint control sequence. Moreover, under additional smoothness assumptions on the … Read more

    Regularized Nonsmooth Newton Algorithms for Best Approximation

  • Yair Censor
  • Walaa M. Moursi
  • Tyler Weames
  • Henry Wolkowicz
    • Categories Convex Optimization, Linear Programming, Nonsmooth Optimization Tags , , , , ,

    We consider the problem of finding the best approximation point from a polyhedral set, and its applications, in particular to solving large-scale linear programs. The classical projection problem has many various and many applications. We study a regularized nonsmooth Newton type solution method where the Jacobian is singular; and we compare the computational performance to … Read more

    An extragradient method for solving variational inequalities without monotonicity

  • Lei Ming
  • He Yiran
    • Categories Complementarity and Variational Inequalities Tags , , , ,

    A new extragradient projection method is devised in this paper, which does not obviously require generalized monotonicity and assumes only that the so-called dual variational inequality has a solution in order to ensure its global convergence. In particular, it applies to quasimonotone variational inequality having a nontrivial solution. Article Download View An extragradient method for … Read more

    Algorithms and Convergence Results of Projection Methods for Inconsistent Feasibility Problems: A Review

  • Yair Censor
  • Maroun Zaknoon
    • Categories Convex Optimization, Nonlinear Optimization Tags , ,

    The convex feasibility problem (CFP) is to find a feasible point in the intersection of finitely many convex and closed sets. If the intersection is empty then the CFP is inconsistent and a feasible point does not exist. However, algorithmic research of inconsistent CFPs exists and is mainly focused on two directions. One is oriented … Read more

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

  • Yair Censor
  • Aviv Gibali
  • Frank Lenzen
  • Christoph Schnörr
    • Categories Biomedical Applications, Convex and Nonsmooth Optimization Tags , , , , ,

    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

    Techniques in Iterative Proton CT Image Reconstruction

  • Yair Censor
  • Scott Penfold
    • Categories Biomedical Applications, Convex Optimization Tags , , , , , ,

    This is a review paper on some of the physics, modeling, and iterative algorithms in proton computed tomography (pCT) image reconstruction. The primary challenge in pCT image reconstruction lies in the degraded spatial resolution resulting from multiple Coulomb scattering within the imaged object. Analytical models such as the most likely path (MLP) have been proposed … Read more

    Projected Reflected Gradient Methods for Monotone Variational Inequalities

  • Yura Malitsky
    • Categories Complementarity and Variational Inequalities Tags , , ,

    This paper is concerned with some new projection methods for solving variational inequality problems with monotone and Lipschitz-continuous mapping in Hilbert space. First, we propose the projected reflected gradient algorithm with a constant stepsize. It is similar to the projected gradient method, namely, the method requires only one projection onto the feasible set and only … Read more

    On the Iteration Complexity of Some Projection Methods for Monotone Linear Variational Inequalities

  • Caihua Chen
  • Xiaoling Fu
  • Bingsheng He
  • Xiao-Ming Yuan
    • Categories Complementarity and Variational Inequalities Tags , , ,

    Projection type methods are among the most important methods for solving monotone linear variational inequalities. In this note, we analyze the iteration complexity for two projection methods and accordingly establish their worst-case O(1/t) convergence rates measured by the iteration complexity in both the ergodic and nonergodic senses, where t is the iteration counter. Our analysis … Read more

    Projection Methods: An Annotated Bibliography of Books and Reviews

  • Andrzej Cegielski
  • Yair Censor
    • Categories Constrained Nonlinear Optimization, Convex and Nonsmooth Optimization Tags , ,

    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 … Read more