AM  Vol.3 No.10 A , October 2012
A Fixed Point Method for Convex Systems
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.
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.

[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.