A Fixed Point Method for Convex Systems

Affiliation(s)

Department of Mathematics, Razi University, Kermanshah, Iran.

Department of Mathematics, Payame Noor University, Tehran, Iran.

Department of Mathematics, Razi University, Kermanshah, Iran.

Department of Mathematics, Payame Noor University, Tehran, Iran.

ABSTRACT

We present a new fixed point technique to solve a system of convex equations in several variables. Our approach is based on two powerful algorithmic ideas: operator-splitting and steepest descent direction. The quadratic convergence of the proposed approach is established under some reasonable conditions. Preliminary numerical results are also reported.

We present a new fixed point technique to solve a system of convex equations in several variables. Our approach is based on two powerful algorithmic ideas: operator-splitting and steepest descent direction. The quadratic convergence of the proposed approach is established under some reasonable conditions. Preliminary numerical results are also reported.

KEYWORDS

Convex Equations; Least Squares; -Regularization Problems; Fixed Point; Quadratically Convergence

Convex Equations; Least Squares; -Regularization Problems; Fixed Point; Quadratically Convergence

Cite this paper

M. Kimiaei and F. Rahpeymaii, "A Fixed Point Method for Convex Systems,"*Applied Mathematics*, Vol. 3 No. 10, 2012, pp. 1327-1333. doi: 10.4236/am.2012.330189.

M. Kimiaei and F. Rahpeymaii, "A Fixed Point Method for Convex Systems,"

References

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[2] E. T. Hale, W. Yin and Y. Zhang, “A Fixed-Point Continuation Method for -Regularized Minimization with Applications to Compressed Sensing,” CAAM Technical Report TR07-07, 7 July 2007.

[3] A. Chambolle, R. A. De Vore, N. Y. Lee and B. J. Lucier, “Nonlinear Wavelet Image Processing: Variational Problems, Compression, and Noise Removal through Wavelet Shrinkage,” IEEE Transactions on Image Processing, Vol. 7, No. 3, 1998, pp. 319-335. doi:10.1109/83.661182

[4] M. A. T. Figueiredo and R. D. Nowa, “An EM Algorithm for Wavelet-Based Image Restoration,” IEEE Transactions on Image Processing, Vol. 12, No. 8, 2003, pp. 906-916. doi:10.1109/TIP.2003.814255

[5] I. Daubechies, M. Defrise and C. D. Mol, “An Iterative Thresholding Algorithm for Linear Inverse Problems with a Sparsity Constraint,” Communications on Pure and Applied Mathematics, Vol. 57, No. 11, 2004, pp. 1413-1457. doi:10.1002/cpa.20042

[6] C. Vonesch and M. Unser, “Fast Iterative Thresholding N. Algorithm for Wavelet-Regularized Deconvolution,” Proceedings of the SPIE Optics and Photonics 2007 Conference on Mathematical Methods: Wavelet XII, Vol. 6701, San Diego, 26-29 August 2007, p. 15.

[7] B. Martinet, “Régularisation d’Inéquations Variation Nelles par Approximations Successives,” Recherche Operationnelle, Vol. 4, No. 3, 1970, pp. 154-158

[8] G. L. Yuan, Z. X. Wei and S. Lu, “Limited Memory BFGS Method with Backtracking for Symmetric Nonlinear Equations,” Mathematical and Computer Modelling, Vol. 54, No. 1-2, 2011, pp. 367-377. doi:10.1016/j.mcm.2011.02.021

[9] M. Moga and C. Smutnicki, “Test Functions for Optimization Need,” 2005. www.zsd.itc.pwr.pc/file/docs/function.pdf.

[10] L. Luk?an and J. Vl?ek, “Sparse and Partially Separable Test Problems for Unconstrained and Equality Constrained Optimization,” Technical Report, Vol. 767, 1999.

[11] E. D. Dolan and J. J. Moré, “Benchmarking Optimization Software with Performance Profiles,” Mathematical Programming, Vol. 91, No. 2, 2002, pp. 201-203.

[1] Z. Wen, W. Yin, D. Goldfarb and Y. Zhang, “A Fast Algorithm for Sparse Reconstruction Based on Sharinkage,” Optimization, and Continuation, CAAM Technical Report TR09-01, 2009.

[2] E. T. Hale, W. Yin and Y. Zhang, “A Fixed-Point Continuation Method for -Regularized Minimization with Applications to Compressed Sensing,” CAAM Technical Report TR07-07, 7 July 2007.

[3] A. Chambolle, R. A. De Vore, N. Y. Lee and B. J. Lucier, “Nonlinear Wavelet Image Processing: Variational Problems, Compression, and Noise Removal through Wavelet Shrinkage,” IEEE Transactions on Image Processing, Vol. 7, No. 3, 1998, pp. 319-335. doi:10.1109/83.661182

[4] M. A. T. Figueiredo and R. D. Nowa, “An EM Algorithm for Wavelet-Based Image Restoration,” IEEE Transactions on Image Processing, Vol. 12, No. 8, 2003, pp. 906-916. doi:10.1109/TIP.2003.814255

[5] I. Daubechies, M. Defrise and C. D. Mol, “An Iterative Thresholding Algorithm for Linear Inverse Problems with a Sparsity Constraint,” Communications on Pure and Applied Mathematics, Vol. 57, No. 11, 2004, pp. 1413-1457. doi:10.1002/cpa.20042

[6] C. Vonesch and M. Unser, “Fast Iterative Thresholding N. Algorithm for Wavelet-Regularized Deconvolution,” Proceedings of the SPIE Optics and Photonics 2007 Conference on Mathematical Methods: Wavelet XII, Vol. 6701, San Diego, 26-29 August 2007, p. 15.

[7] B. Martinet, “Régularisation d’Inéquations Variation Nelles par Approximations Successives,” Recherche Operationnelle, Vol. 4, No. 3, 1970, pp. 154-158

[8] G. L. Yuan, Z. X. Wei and S. Lu, “Limited Memory BFGS Method with Backtracking for Symmetric Nonlinear Equations,” Mathematical and Computer Modelling, Vol. 54, No. 1-2, 2011, pp. 367-377. doi:10.1016/j.mcm.2011.02.021

[9] M. Moga and C. Smutnicki, “Test Functions for Optimization Need,” 2005. www.zsd.itc.pwr.pc/file/docs/function.pdf.

[10] L. Luk?an and J. Vl?ek, “Sparse and Partially Separable Test Problems for Unconstrained and Equality Constrained Optimization,” Technical Report, Vol. 767, 1999.

[11] E. D. Dolan and J. J. Moré, “Benchmarking Optimization Software with Performance Profiles,” Mathematical Programming, Vol. 91, No. 2, 2002, pp. 201-203.