projection methods

Feasibility-Seeking and Superiorization Algorithms Applied to Inverse Treatment Planning in Radiation Therapy

  • Yair Censor
  • Ran Davidi
  • Sarah Geneser
  • Reinhard W. Schulte
  • Lei Xing
    • Categories Biomedical Applications, Convex Optimization Tags , , , ,

    We apply the recently proposed superiorization methodology (SM) to the inverse planning problem in radiation therapy. The inverse planning problem is represented here as a constrained minimization problem of the total variation (TV) of the intensity vector over a large system of linear two-sided inequalities. The SM can be viewed conceptually as lying between feasibility-seeking … Read more

    A Relaxed-Projection Splitting Algorithm for Variational Inequalities in Hilbert Spaces

  • Yunier Bello-Cruz
  • R. Diaz Millan
    • Categories Complementarity and Variational Inequalities, Nonlinear Optimization, Nonsmooth Optimization Tags , , , , ,

    We introduce a relaxed-projection splitting algorithm for solving variational inequalities in Hilbert spaces for the sum of nonsmooth maximal monotone operators, where the feasible set is defined by a nonlinear and nonsmooth continuous convex function inequality. In our scheme, the orthogonal projections onto the feasible set are replaced by projections onto separating hyperplanes. Furthermore, each … Read more

    About uniform regularity of collections of sets

  • Alexander Y. Kruger
  • Nguyen H. Thao
    • Categories Convex and Nonsmooth Optimization Tags ,

    We further investigate the uniform regularity property of collections of sets via primal and dual characterizing constants. These constants play an important role in determining convergence rates of projection algorithms for solving feasibility problems. Citation Published in Serdica Math. J. 39, 287–312 (2013) http://www.math.bas.bg/serdica/2013/2013-287-312.pdf Article Download View About uniform regularity of collections of sets

    Convergence and Perturbation Resilience of Dynamic String-Averaging Projection Methods

  • Yair Censor
  • Alexander J. Zaslavski
    • Categories Biomedical Applications, Convex Optimization, Parallel Algorithms Tags , , , , , , , , ,

    We consider the convex feasibility problem (CFP) in Hilbert space and concentrate on the study of string-averaging projection (SAP) methods for the CFP, analyzing their convergence and their perturbation resilience. In the past, SAP methods were formulated with a single predetermined set of strings and a single predetermined set of weights. Here we extend the … Read more

    Boosting the Feasibility Pump

  • Natashia Boland
  • Andrew C. Eberhard
  • Faramroze G. Engineer
  • Matteo Fischetti
  • Martin Savelsbergh
  • Angelos Tsoukalas
    • Categories (Mixed) Integer Linear Programming Tags , , , ,

    The Feasibility Pump (FP) has proved to be an effective method for finding feasible solutions to mixed integer programming problems. FP iterates between a rounding procedure and a projection procedure, which together provide a sequence of points alternating between LP feasible but fractional solutions, and integer but LP relaxed infeasible solutions. The process attempts to … Read more

    Algorithms for Bilevel Pseudomonotone Variational Inequality Problems

  • Bui Van Dinh
  • Le Dung Muu
    • Categories Complementarity and Variational Inequalities, Convex and Nonsmooth Optimization Tags , , , ,

    We propose easily implementable algorithms for minimizing the norm with pseudomonotone variational inequality constraints. This bilevel problem arises in the Tikhonov regularization method for pseudomonone variational inequalities. Since the solution set of the lower variational inequality is not given explicitly, the available methods of mathematical programming and variational inequality can not be applied directly. With … Read more

    Total variation superiorization schemes in proton computed tomography image reconstruction

  • Yair Censor
  • Scott Penfold
  • Anatoly Rosenfeld
  • Reinhard W. Schulte
    • Categories Biomedical Applications, Constrained Nonlinear Optimization, Other Topics Tags ,

    Purpose: Iterative projection reconstruction algorithms are currently the preferred reconstruction method in proton computed tomography (pCT). However, due to inconsistencies in the measured data arising from proton energy straggling and multiple Coulomb scattering, noise in the reconstructed image increases with successive iterations. In the current work, we investigated the use of total variation superiorization (TVS) … Read more

    Perturbation resilience and superiorization of iterative algorithms

  • Yair Censor
  • Ran Davidi
  • Gabor T. Herman
    • Categories Biomedical Applications, Constrained Nonlinear Optimization Tags , , , ,

    Iterative algorithms aimed at solving some problems are discussed. For certain problems, such as finding a common point in the intersection of a finite number of convex sets, there often exist iterative algorithms that impose very little demand on computer resources. For other problems, such as finding that point in the intersection at which the … Read more

    A Note on the Behavior of the Randomized Kaczmarz Algorithm of Strohmer and Vershynin

  • Yair Censor
  • Gabor T. Herman
  • Ming Jiang
    • Categories Biomedical Applications, Convex Optimization Tags , ,

    In a recent paper by Strohmer and Vershynin (J. Fourier Anal. Appl. 15:262–278, 2009) a “randomized Kaczmarz algorithm” is proposed for solving consistent systems of linear equations {ai, x = bi }m i=1. In that algorithm the next equation to be used in an iterative Kaczmarz process is selected with a probability proportional to ai2. … Read more

    Block-Iterative and String-Averaging Projection Algorithms in Proton Computed Tomography Image Reconstruction

  • Vladimir Bashkirov
  • Yair Censor
  • Scott McAllister
  • Scott Penfold
  • Anatoly Rosenfeld
  • Keith Schubert
  • Reinhard W. Schulte
    • Categories Applications - Science and Engineering, Biomedical Applications Tags , , ,

    Proton computed tomography (pCT) is an imaging modality that has been suggested as a means for reducing the range uncertainty during proton radiation treatments. By measuring the spatial location of individual protons pre- and post-patient, as well as the energy lost along the proton path, three dimensional maps of patient water equivalent electron densities can … Read more