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A new smoothing modified three-term conjugate gradient method for [Formula: see text]-norm minimization problem.

Shouqiang Du1, Miao Chen1

  • 1School of Mathematics and Stochastic, Qingdao University, Qingdao, China.

Journal of Inequalities and Applications
|May 15, 2018
PubMed
Summary

This study introduces a new smoothing modified three-term conjugate gradient method for solving nonsmooth optimization problems common in compressed sensing and signal reconstruction. The method demonstrates efficiency and global convergence without requiring line searches.

Keywords:
Global convergenceNonsmooth optimization problemSmoothing modified three-term conjugate gradient method

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Area of Science:

  • Optimization Theory
  • Applied Mathematics
  • Signal Processing

Background:

  • Nonsmooth optimization problems involving the [Formula: see text]-norm are prevalent in compressed sensing and signal reconstruction.
  • These problems present significant computational challenges due to their nonsmooth nature.

Purpose of the Study:

  • To develop an efficient and globally convergent method for solving [Formula: see text]-norm minimization problems.
  • To transform nonsmooth problems into a solvable unconstrained optimization framework.

Main Methods:

  • Utilizing smoothing approximation techniques to convert the nonsmooth problem into an unconstrained one.
  • Proposing a novel smoothing modified three-term conjugate gradient method, building upon the Polak-Ribière-Polyak algorithm.
  • Ensuring the method possesses sufficient descent properties without necessitating line searches.

Main Results:

  • The proposed smoothing modified three-term conjugate gradient method is proven to be globally convergent.
  • The method exhibits sufficient descent properties, enhancing its practical applicability.
  • Numerical experiments validate the efficiency of the developed algorithm.

Conclusions:

  • The smoothing modified three-term conjugate gradient method offers an effective approach for [Formula: see text]-norm minimization problems.
  • The method's global convergence and efficiency make it suitable for applications in compressed sensing and signal reconstruction.