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A New Spectral Conjugate Subgradient Method with Application in Computed Tomography Image Reconstruction

Milagros Loreto, Thomas Humphries, Chella Raghavan, Kenneth Wu, Sam Kwak

TL;DR

The paper introduces the Spectral Conjugate Subgradient (SCS) method for nonsmooth unconstrained optimization by uniting spectral conjugate gradient ideas with subgradient updates. It analyzes two line-search strategies (nonmonotone and Wolfe) and three formulas for generating conjugate directions, with a focus on the Polak–Ribiere-like $\beta_2$ choice. The algorithm is validated on standard nonsmooth benchmarks and TV-regularized CT reconstruction, showing that the nonmonotone line search paired with $\beta_2$ achieves strong robustness and competitive efficiency relative to the original spectral subgradient method. These findings highlight the practical value of SCS for large-scale, nonsmooth problems, especially in imaging applications where TV regularization induces nondifferentiability.

Abstract

A new spectral conjugate subgradient method is presented to solve nonsmooth unconstrained optimization problems. The method combines the spectral conjugate gradient method for smooth problems with the spectral subgradient method for nonsmooth problems. We study the effect of two different choices of line search, as well as three formulas for determining the conjugate directions. In addition to numerical experiments with standard nonsmooth test problems, we also apply the method to several image reconstruction problems in computed tomography, using total variation regularization. Performance profiles are used to compare the performance of the algorithm using different line search strategies and conjugate directions to that of the original spectral subgradient method. Our results show that the spectral conjugate subgradient algorithm outperforms the original spectral subgradient method, and that the use of the Polak-Ribiere formula for conjugate directions provides the best and most robust performance.

A New Spectral Conjugate Subgradient Method with Application in Computed Tomography Image Reconstruction

TL;DR

The paper introduces the Spectral Conjugate Subgradient (SCS) method for nonsmooth unconstrained optimization by uniting spectral conjugate gradient ideas with subgradient updates. It analyzes two line-search strategies (nonmonotone and Wolfe) and three formulas for generating conjugate directions, with a focus on the Polak–Ribiere-like choice. The algorithm is validated on standard nonsmooth benchmarks and TV-regularized CT reconstruction, showing that the nonmonotone line search paired with achieves strong robustness and competitive efficiency relative to the original spectral subgradient method. These findings highlight the practical value of SCS for large-scale, nonsmooth problems, especially in imaging applications where TV regularization induces nondifferentiability.

Abstract

A new spectral conjugate subgradient method is presented to solve nonsmooth unconstrained optimization problems. The method combines the spectral conjugate gradient method for smooth problems with the spectral subgradient method for nonsmooth problems. We study the effect of two different choices of line search, as well as three formulas for determining the conjugate directions. In addition to numerical experiments with standard nonsmooth test problems, we also apply the method to several image reconstruction problems in computed tomography, using total variation regularization. Performance profiles are used to compare the performance of the algorithm using different line search strategies and conjugate directions to that of the original spectral subgradient method. Our results show that the spectral conjugate subgradient algorithm outperforms the original spectral subgradient method, and that the use of the Polak-Ribiere formula for conjugate directions provides the best and most robust performance.
Paper Structure (15 sections, 5 theorems, 25 equations, 7 figures, 3 tables)

This paper contains 15 sections, 5 theorems, 25 equations, 7 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Spectral Conjugate Gradient Method
  4. Line Search Strategies
  5. Nonmonotone Line Search
  6. Wolfe Line Search
  7. Spectral Conjugate Subgradient Algorithm
  8. CT problem with Total Variation
  9. Convergence Analysis
  10. Numerical Results
  11. Nonsmooth test problems
  12. CT problem with total variation
  13. Final Remarks
  14. References
  15. General proof of convergence for nonmonotone line search with subdifferentiable functions

Key Result

Lemma A.1

Let $f: \mathbb{R}^n \to \mathbb{R}$ be subdifferentiable, and let $x, p \in \mathbb{R}^n$. Define $F:\mathbb{R} \to \mathbb{R}$ as $F(t) = f(x+tp)$. Then $g^Tp \in \partial{F}(t)$ for any $g \in \partial f (x+tp)$

Figures (7)

  • Figure 1: CT imaging example. Left: true $400 \times 400$ pixel digital Shepp-Logan phantom. Center-left: parallel-beam sinogram corresponding to 360 views taken over 180$^\circ$; the affine parameter is the $y$-axis and the angular parameter the $x$-axis. Center-right: low-dose image reconstructed using unregularized least-squares with 20% Gaussian noise added to $b$. Right: Sparse-view mage reconstructed using unregularized least squares with only 60 views taken over 180$^\circ$
  • Figure 2: Performance profile based on error between $f_{\min}$ and $f_{*}$ using the nonmonotone line search with M=7, and the Wolfe line search per each parameter $\beta$.
  • Figure 3: Performance profile based on the number of function evaluations using the nonmonotone line search with M=7, and the Wolfe line search per each parameter $\beta$
  • Figure 4: Performance profile based on CPU using the nonmonotone line search with M=7, and the Wolfe line search per each parameter $\beta$
  • Figure 5: Performance profile for CT reconstruction problems, based on lowest objective function value $f_{\min}$ found by Algorithm SCS using nonmonotone line search with each choice of $\beta$.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Lemma A.1
  • Lemma A.2
  • Lemma A.3
  • Lemma A.4
  • Theorem A.5