A new smoothing modified three-term conjugate gradient method for l1$l_{1}$-norm minimization problem

@article{Du2018ANS,
  title={A new smoothing modified three-term conjugate gradient method for l1\$l\_\{1\}\$-norm minimization problem},
  author={Shou-qiang Du and Miao Chen},
  journal={Journal of Inequalities and Applications},
  year={2018},
  volume={2018},
  pages={1-14}
}
  • S. Du, Miao Chen
  • Published 2018
  • Mathematics, Computer Science
  • Journal of Inequalities and Applications
We consider a kind of nonsmooth optimization problems with l1$l_{1}$-norm minimization, which has many applications in compressed sensing, signal reconstruction, and the related engineering problems. Using smoothing approximate techniques, this kind of nonsmooth optimization problem can be transformed into a general unconstrained optimization problem, which can be solved by the proposed smoothing modified three-term conjugate gradient method. The smoothing modified three-term conjugate gradient… 

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References

SHOWING 1-10 OF 34 REFERENCES

The Smoothing FR Conjugate Gradient Method for Solving a Kind of Nonsmooth Optimization Problem with -Norm

TLDR
This work studies the method for solving a kind of nonsmooth optimization problems with -norm, which is widely used in the problem of compressed sensing, image processing, and some related optimization problems, and shows the effectiveness of the given smoothing FR conjugate gradient method.

A New Modified Three-Term Conjugate Gradient Method with Sufficient Descent Property and Its Global Convergence

A new modified three-term conjugate gradient (CG) method is shown for solving the large scale optimization problems. The idea relates to the famous Polak-Ribiere-Polyak (PRP) formula. As the

Smoothing methods for nonsmooth, nonconvex minimization

We consider a class of smoothing methods for minimization problems where the feasible set is convex but the objective function is not convex, not differentiable and perhaps not even locally Lipschitz

An Interior-Point Method for Large-Scale $\ell_1$-Regularized Least Squares

TLDR
A specialized interior-point method for solving large-scale -regularized LSPs that uses the preconditioned conjugate gradients algorithm to compute the search direction and can solve large sparse problems, with a million variables and observations, in a few tens of minutes on a PC.

Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems

TLDR
This paper proposes gradient projection algorithms for the bound-constrained quadratic programming (BCQP) formulation of these problems and test variants of this approach that select the line search parameters in different ways, including techniques based on the Barzilai-Borwein method.

Three modified Polak-Ribière-Polyak conjugate gradient methods with sufficient descent property

TLDR
These modified Polak-Ribière-Polyak (PRP) conjugate gradient methods for unconstrained optimization possess the sufficient descent property without any line searches and converge globally with a Wolfe line search.

An Alternative Lagrange-Dual Based Algorithm for Sparse Signal Reconstruction

TLDR
A new Lagrange-dual reformulation associated with an l1 -norm minimization problem for sparse signal reconstruction and the efficiency and performance of the proposed algorithm are validated via theoretical analysis as well as some illustrative numerical examples.

A Nonlinear Conjugate Gradient Method with a Strong Global Convergence Property

TLDR
This paper presents a new version of the conjugate gradient method, which converges globally, provided the line search satisfies the standard Wolfe conditions.