superiorization

Approaches to iterative algorithms for solving nonlinear equations with an application in tomographic absorption spectroscopy

  • Francisco J. Aragón Artacho
  • Weiwei Cai
  • Yair Censor
  • Aviv Gibali
  • Chongyuan Shui
  • David Torregrosa-Belén
    • Categories Biomedical Applications, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , , ,

    In this paper we propose an approach for solving systems of nonlinear equations without computing function derivatives. Motivated by the application area of tomographic absorption spectroscopy, which is a highly-nonlinear problem with variables coupling, we consider a situation where straightforward translation to a fixed point problem is not possible because the operators that represent the … Read more

    Superiorization: The asymmetric roles of feasibility-seeking and objective function reduction

  • Yair Censor
    • Categories Biomedical Applications, Constrained Nonlinear Optimization, Convex Optimization Tags , , , , , , , , , ,

    The superiorization methodology can be thought of as lying conceptually between feasibility-seeking and constrained minimization. It is not trying to solve the full-fledged constrained minimization problem composed from the modeling constraints and the chosen objective function. Rather, the task is to find a feasible point which is “superior” (in a well-defined manner) with respect to … Read more

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

  • Francisco J. Aragón Artacho
  • Yair Censor
  • Aviv Gibali
  • David Torregrosa-Belén
    • Categories Biomedical Applications, Constrained Nonlinear Optimization, Convex Optimization Tags , , , , , ,

    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

    Developments in mathematical algorithms and computational tools for proton CT and particle therapy treatment planning

  • Yair Censor
  • Keith E. Schubert
  • Reinhard W. Schulte
    • Categories Basic Sciences Applications, Biomedical Applications, Constrained Nonlinear Optimization Tags , , , , , , , ,

    We summarize recent results and ongoing activities in mathematical algorithms and computer science methods related to proton computed tomography (pCT) and intensitymodulated particle therapy (IMPT) treatment planning. Proton therapy necessitates a high level of delivery accuracy to exploit the selective targeting imparted by the Bragg peak. For this purpose, pCT utilizes the proton beam itself … Read more

    Superiorization vs. Accelerated Convex Optimization: The Superiorized/Regularized Least-Squares Case

  • Yair Censor
  • Stefania Petra
  • Christoph Schnörr
    • Categories Biomedical Applications, Constrained Nonlinear Optimization, Convex Optimization Tags , , , , , ,

    In this paper we conduct a study of both superiorization and optimization approaches for the reconstruction problem of superiorized/regularized solutions to underdetermined systems of linear equations with nonnegativity variable bounds. Specifically, we study a (smoothed) total variation regularized least-squares problem with nonnegativity constraints. We consider two approaches: (a) a superiorization approach that, in contrast to … Read more

    An analysis of the superiorization method via the principle of concentration of measure

  • Yair Censor
  • Eliahu Levy
    • Categories Constrained Nonlinear Optimization, Convex and Nonsmooth Optimization, Nonlinear Optimization Tags , , , , , , , ,

    The superiorization methodology is intended to work with input data of constrained minimization problems, i.e., a target function and a constraints set. However, it is based on an antipodal way of thinking to the thinking that leads constrained minimization methods. Instead of adapting unconstrained minimization algorithms to handling constraints, it adapts feasibility-seeking algorithms to reduce … Read more

    Derivative-Free Superiorization With Component-Wise Perturbations

  • Yair Censor
  • Howard Heaton
  • Reinhard W. Schulte
    • Categories Biomedical Applications, Constrained Nonlinear Optimization, Nonsmooth Optimization Tags , , , , ,

    Superiorization reduces, not necessarily minimizes, the value of a target function while seeking constraints-compatibility. This is done by taking a solely feasibility-seeking algorithm, analyzing its perturbations resilience, and proactively perturbing its iterates accordingly to steer them toward a feasible point with reduced value of the target function. When the perturbation steps are computationally efficient, this … Read more

    Superiorization and perturbation resilience of algorithms: A continuously updated bibliography

  • Yair Censor
    • Categories Biomedical Applications, Constrained Nonlinear Optimization Tags ,

    This document presents a, chronologically ordered, bibliography of scientific publications on the superiorization methodology and perturbation resilience of algorithms which is compiled and continuously updated by us at: http://math.haifa.ac.il/yair/bib-superiorization-censor.html. Since the topic is relatively new it is possible to trace the work that has been published about it since its inception. To the best of … Read more

    An Improved Method of Total Variation Superiorization Applied to Reconstruction in Proton Computed Tomography

  • Yair Censor
  • Paniz Karbasi
  • Keith E. Schubert
  • Reinhard W. Schulte
  • Blake Schultze
    • Categories Biomedical Applications, Constrained Nonlinear Optimization, Convex Optimization Tags , , , , ,

    Previous work showed that total variation superiorization (TVS) improves reconstructed image quality in proton computed tomography (pCT). The structure of the TVS algorithm has evolved since then and this work investigated if this new algorithmic structure provides additional benefits to pCT image quality. Structural and parametric changes introduced to the original TVS algorithm included: (1) … Read more

    Can linear superiorization be useful for linear optimization problems?

  • Yair Censor
    • Categories Constrained Nonlinear Optimization, Linear Programming, Problem Solving Environments Tags , , , , , , , , ,

    Linear superiorization considers linear programming problems but instead of attempting to solve them with linear optimization methods it employs perturbation resilient feasibility-seeking algorithms and steers them toward reduced (not necessarily minimal) target function values. The two questions that we set out to explore experimentally are (i) Does linear superiorization provide a feasible point whose linear … Read more