Non-smooth equations based method for -norm problems with applications to compressed sensing

Xiao, Yunhai; Wang, Qiuyu; Hu, Quan

doi:10.1016/j.na.2011.02.040

Cited by 128 publications

(93 citation statements)

References 30 publications

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“…The problem has an optimal solution * = (0.3461, 0.0850, 0, 0, 0.3719, 0) in [14,15]. The optimal solution of our method is * 1 = (0.3461, 0.0850, 0, 0.0003, 0.3720, −0.0001).…”

Section: Examplementioning

confidence: 94%

“…The numerical experiments are also considered in [9,14,15]. In computing Examples 1, 2, 3, and 4, we compare Algorithm 4 with the smoothing gradient method in [15].…”

Section: Numerical Testsmentioning

confidence: 99%

“…We consider a typical CS scenario, which is also considered in [1][2][3][4][13][14][15]. The goal is to reconstruct a lengthsparse signal from observations, where < .…”

Section: Examplementioning

confidence: 99%

“…Furthermore, the × matrix is obtained by first filling it with independent samples of a standard Gaussian distribution and then orthonormalizing its rows. In this example, we choose 2 = 10 −4 and = 0.1‖ ‖ ∞ as suggested in [14]. Figure 15 shows the results of the signal reconstruction.…”

Section: Examplementioning

confidence: 99%

“…For instance, gradient projection for sparse reconstruction was proposed by Figueiredo et al in [9], a two-step iterative shrinkage thresholding (IST) method was proposed by Bioucas-Dias and Figueiredo in [10], a fast IST algorithm was presented by Beck and Teboulle in [11], SPGL1 (A solver for large-scale sparse reconstruction) was proposed by Ewout and Friedlander in [12], who consider a least-squares problem with 1 -norm constraint and use a spectral gradient projection method, and ADM method was proposed by Yang and Zhang in [13]. Problem (3) was formulated to a convex quadratic problem in [14]. Among all the references mentioned above, there is no use of the relationship between 2 Mathematical Problems in Engineering the linear complementarity problems and the absolute value equations to solve (3).…”

Section: Introductionmentioning

confidence: 99%

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The Smoothing FR Conjugate Gradient Method for Solving a Kind of Nonsmooth Optimization Problem with l1-Norm

Chen

2018

Mathematical Problems in Engineering

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We study the method for solving a kind of nonsmooth optimization problems with 1 -norm, which is widely used in the problem of compressed sensing, image processing, and some related optimization problems with wide application background in engineering technology. Transformated by the absolute value equations, this kind of nonsmooth optimization problem is rewritten as a general unconstrained optimization problem, and the transformed problem is solved by a smoothing FR conjugate gradient method. Finally, the numerical experiments show the effectiveness of the given smoothing FR conjugate gradient method.

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“…The problem has an optimal solution * = (0.3461, 0.0850, 0, 0, 0.3719, 0) in [14,15]. The optimal solution of our method is * 1 = (0.3461, 0.0850, 0, 0.0003, 0.3720, −0.0001).…”

Section: Examplementioning

confidence: 94%

“…The numerical experiments are also considered in [9,14,15]. In computing Examples 1, 2, 3, and 4, we compare Algorithm 4 with the smoothing gradient method in [15].…”

Section: Numerical Testsmentioning

confidence: 99%

“…We consider a typical CS scenario, which is also considered in [1][2][3][4][13][14][15]. The goal is to reconstruct a lengthsparse signal from observations, where < .…”

Section: Examplementioning

confidence: 99%

Section: Examplementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Smoothing FR Conjugate Gradient Method for Solving a Kind of Nonsmooth Optimization Problem with l1-Norm

Chen

2018

Mathematical Problems in Engineering

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An improved Dai‐Liao‐style hybrid conjugate gradient‐based method for solving unconstrained nonconvex optimization and extension to constrained nonlinear monotone equations

Yuan,

Shao,

Zeng

et al. 2024

Math Methods in App Sciences

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In this work, for unconstrained optimization, we introduce an improved Dai‐Liao‐style hybrid conjugate gradient method based on the hybridization‐based self‐adaptive technique, and the search direction generated fulfills the sufficient descent and trust region properties regardless of any line search. The global convergence is established under standard Wolfe line search and common assumptions. Then, combining the hyperplane projection technique and a new self‐adaptive line search, we extend the proposed conjugate gradient method and obtain an improved Dai‐Liao‐style hybrid conjugate gradient projection method to solve constrained nonlinear monotone equations. Under mild conditions, we obtain its global convergence without Lipschitz continuity. In addition, the convergence rates for the two proposed methods are analyzed, respectively. Finally, numerical experiments are conducted to demonstrate the effectiveness of the proposed methods.

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Another hybrid approach for solving monotone operator equations and application to signal processing

Abubakar

Ibrahim

Kura³

et al. 2022

Math Methods in App Sciences

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This paper presents a hybrid conjugate gradient (CG) approach for solving nonlinear equations and signal reconstruction. The CG parameter of the approach is a convex combination of the Dai‐Yuan (DY)‐like and Hestenes‐Stiefel (HS)‐like parameters. Independent of any line search, the search direction is descent and bounded. Under some reasonable assumptions, the global convergence of the hybrid approach is proved. Numerical experiments on some benchmark test problems show that the proposed approach is efficient compared with some existing algorithms. Finally, the proposed approach is applied in signal reconstruction.

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Non-smooth equations based method for -norm problems with applications to compressed sensing

Cited by 128 publications

References 30 publications

The Smoothing FR Conjugate Gradient Method for Solving a Kind of Nonsmooth Optimization Problem with l1-Norm

The Smoothing FR Conjugate Gradient Method for Solving a Kind of Nonsmooth Optimization Problem with l1-Norm

An improved Dai‐Liao‐style hybrid conjugate gradient‐based method for solving unconstrained nonconvex optimization and extension to constrained nonlinear monotone equations

Another hybrid approach for solving monotone operator equations and application to signal processing

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