doi: 10.3934/jimo.2018134

An efficient algorithm for non-convex sparse optimization

1. 

School of Mathematics, Tianjin University, Tianjin 300072, China

2. 

Department of Computing, Curtin University, WA, 6102, Australia

3. 

Department of Mathematics and Statistics, Curtin University, WA, 6102, Australia

* Corresponding author: Wanquan Liu

Received  March 2017 Revised  April 2018 Published  September 2018

It is a popular research topic in computer vision community to find a solution for the zero norm minimization problem via solving its non-convex relaxation problem. In fact, there are already many existing algorithms to solve the non-convex relaxation problem. However, most of them are computationally expensive due to the non-Lipschitz property of this problem and thus these existing algorithms are not suitable for many engineering problems with large dimensions.

In this paper, we first develop an efficient algorithm to solve the non-convex relaxation problem via solving a sequence of non-convex sub-problems based on our recent work. To this end, we reformulate the minimization problem into another non-convex one but with non-negative constraint. Then we can transform the non-Lipschitz continuous non-convex problem with the non-negative constraint into a Lipschitz continuous problem, which allows us to use some efficient existing algorithms for its solution. Based on the proposed algorithm, an important relation between the solutions of relaxation problem and the original zero norm minimization problem is established from a different point of view. The results in this paper reveal two important issues: ⅰ) The solution of non-convex relaxation minimization problem converges to the solution of the original problem; ⅱ) The general non-convex relaxation problem can be solved efficiently with another reformulated high dimension problem with nonnegative constraint. Finally, some numerical results are used to demonstrate effectiveness of the proposed algorithm.

Citation: Yong Wang, Wanquan Liu, Guanglu Zhou. An efficient algorithm for non-convex sparse optimization. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018134
References:
[1]

R. H. ByrdP. Lu and J. Nocedal, A limited memory algorithm for bound constrained optimization, SIAM J. Sci. Stat. Comput., 16 (1995), 1190-1208. doi: 10.1137/0916069.

[2]

S. BoydN. ParikhE. ChuB. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learning, 3 (2010), 1-122.

[3]

S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511804441.

[4]

A. CohenW. Dahmen and R. DeVore, Compressed sensing and best $k$-term approximation, J. Amer. Math. Soc., 22 (2009), 211-231. doi: 10.1090/S0894-0347-08-00610-3.

[5]

R. Chartrand, Nonconvex compressed sensing and error correction, IEEE International Conference on Acoustics, Speech and Signal Processing, (2007), 889-892.

[6]

A. Charkrabarti and F. Hirakawa, Efective separation of sparse and non-sparse image features for denoising, in Proc. Int. Conf. Acoust., Speech, Signal Process. (ICASSP), (2008), 857-860.

[7]

X. ChenK. Ng. Michael and C. Zhang, Non-Lipschitz-Regularization and box constrained model for image restoration, IEEE Trans. Image Processing, 21 (2012), 4709-4721. doi: 10.1109/TIP.2012.2214051.

[8]

E. J. Candès and J. Romberg, The code package $l_1$-magic. Available from: http://statweb.stanford.edu/candes/l1magic/.

[9]

E. J. CandèsJ. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Commun. Pure Appl. Math., 59 (2006), 1207-1223. doi: 10.1002/cpa.20124.

[10]

R. Chartrand and V. Staneva, Restricted isometry properties and nonconvex compressive sensing, Inverse Problems, 24 (2008), 035020, 14 pp. doi: 10.1088/0266-5611/24/3/035020.

[11]

E. J. Candès and T. Tao, Decoding by linear programming, IEEE Trans. Inf. Theory, 51 (2005), 4203-4215. doi: 10.1109/TIT.2005.858979.

[12]

X. Chen and S. Xiang, Sparse solutions of linear complementarity problems, Math. Program., 159 (2016), 539-556. doi: 10.1007/s10107-015-0950-x.

[13]

R. Chartrand and W. Yin, Iteratively reweighted algorithms for compressive sensing, IEEE International Conference on Acoustics, Speech and Signal Processing, (2008), 3869-3872.

[14]

X. Chen and W. Zhou, Convergence of Reweighted $l_1$ Minimization Algorithms and Unique Solution of Truncated $l_p$ Minimization, Tech. rep., Hong Kong Polytechnic University, 2010.

[15]

D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.

[16]

D. L. Donoho and X. Huo, Uncertainty principles and ideal atomic decomposition, IEEE Trans. Inf. Theory, 47 (2001), 2845-2862. doi: 10.1109/18.959265.

[17]

M. Elad and A. M. Bruckstein, A generalized uncertainly priciple and sparse representation in pairs of bases, IEEE Trans. Inf. Theory, 48 (2002), 2558-2567. doi: 10.1109/TIT.2002.801410.

[18]

G. Fung and O. Mangasarian, Equivalence of minimal $l_0-$ and $l_p-$norm solutions of linear equalities, inequalities and linear programs for sufficiently small $p$, J. Optim. Theory Appl., 151 (2011), 1-10. doi: 10.1007/s10957-011-9871-x.

[19]

S. Foucart and M. Lai, Sparsest solutions of underdetermined linear systems via $l_q$ minimization for $0 < q < 1$, Applied and Computational Harmonic Analysis, 26 (2009), 395-407. doi: 10.1016/j.acha.2008.09.001.

[20]

M. A. T. FigueiredoR. D. Nowak and S. J. Wright, Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems, IEEE J. Select. Top. Signal Process., 1 (2007), 585-597.

[21]

G. GassoA. Rakotomamonjy and S. Canu, Recovering sparse signals with a certain family of nonconvex penalties and DC programming, IEEE Trans. Signal Process., 57 (2009), 4686-4698. doi: 10.1109/TSP.2009.2026004.

[22]

M. Hyder and K. Mahata, An approximate $l_0$ norm minimization algorithm for compressed sensing, in IEEE International Conference on Acoustics, Speech and Signal Precessing(ICASSP), (2009), 3365-3368.

[23]

E. T. Hale, W. Yin and Y. Zhang, A fixed-point continuation method for $l_1$-regularized minimization with applications to compressed sensing, CAAM Technical Report TR07-07, Rice University, Houston, TX, 2007.

[24]

D. Krishnan and R. Fergus, Fast Image Deconvolution Using Hyper-Laplacian Priors, Neural Information Processing Systems., Cambridge, MA: MIT Press, 2009.

[25]

K. Koh, S.-J. Kim and S. Boyd, The code package l1_ls. Available from: http://www.standord.edu/boyd/l1_ls.

[26]

Q. LyuZ. LinY. She and C. Zhang, A comparison of typical $l_p$ minimization algorithms, Neurocomputing, 119 (2013), 413-424.

[27]

D. C. Liu and J. Nocedal, On the limited memory method for large scale optimization, Mathematical Programming B, 45 (1989), 503-528. doi: 10.1007/BF01589116.

[28]

M. Lai and J. Wang, An unconstrained $l_q$ minimization with $0 < q < 1$ for sparse solution of under-determined linear systems, SIAM J. Optim., 21 (2011), 82-101. doi: 10.1137/090775397.

[29]

B. K. Natraajan, Sparse approximation to linear systems, SIAM J. Comput., 24 (1995), 227-234. doi: 10.1137/S0097539792240406.

[30]

P. Ochs, A. Dosovitskiy, T. Brox and T. Pock, An iterated $l_1$ algorithm for non-smooth non-convex optimization in computer vision, in Computer Vision and Pattern Recognition (CVPR), IEEE Conference, (2013), 1759-1766.

[31]

J. K. PantW. S. Lu and A. Antoniou, New improved algorithms for compressive sensing based on $l_p$ norm, IEEE Trans. on Circuits and Systems-Ⅱ: Express Briefs, 61 (2014), 198-202.

[32]

J. PengS. Yue and H. Li, NP/CMP equivalence: A phenomenon hidden among sparsity models $l_ {0}$ minimization and $l_ {p}$ minimization for information processing, IEEE Trans. Inf. Theory, 61 (2015), 4028-4033. doi: 10.1109/TIT.2015.2429611.

[33]

R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.

[34]

Y. She, Thresholding-based iterative selection procedures for model selection and shrinkage, Electron. J. Stat., 3 (2009), 384-415. doi: 10.1214/08-EJS348.

[35]

Y. She, An iterative algorithm for fitting nonconvex penalized generalized linear models with grouped predictors, Comput. Statist. Data Anal., 9 (2012), 2976-2990. doi: 10.1016/j.csda.2011.11.013.

[36]

R. Saab, R. Chartrand and O. Yilmaz, Stable sparse approximations via nonconvex optimization, in IEEE International Confereence on Acoustics, Speech and Signal Processing, (2008), 3885-3888.

[37]

J. WrightA. YangA. GaneshS. Sastry and Y. Ma, Robust face recognition via sparse representation, IEEE Trans. Pattern Recogn. Anal. Mach. Intell., 31 (2009), 210-227.

[38]

Y. J. WangG. L. ZhouL. Caccetta and W. Q. Liu, An alternating direction algorithm for $l_1$ problems in compressive sensing, IEEE Trans. Signal Process., 59 (2011), 1895-1901.

[39]

Y. Wang and Q. Ma, A fast subspace method for image deblurring, Appl. Math. Comput., 215 (2009), 2359-2377. doi: 10.1016/j.amc.2009.08.033.

[40]

Y. WangG. ZhouX. ZhangW. Liu and L. Caccetta, The non-convex sparse problem with nonnegative constraint for signal reconstruction, J. Optim. Theory App., 170 (2016), 1009-1025. doi: 10.1007/s10957-016-0869-2.

[41]

A. Y. YangZ. ZhouA. G. BalasubramanianS. S. Sastry and Y. Ma, Fast-minimization algorithms for robust face recognition, IEEE Trans. Image Processing, 22 (2013), 3234-3246.

[42]

F. ZouH. FengH. LingC. LiuL. YanP. Li and D. Li, Nonnegative sparse coding induced hashing for image copy detection, Neurocomputing, 105 (2013), 81-95.

[43]

J. ZengS. LinY. Wang and Z. Xu, $L_{1/2}$ regularization: Convergence of iterative half thresholding algorithm, IEEE Trans. Signal Process., 62 (2014), 2317-2329. doi: 10.1109/TSP.2014.2309076.

[44]

W. Zuo, D. Meng, L. Zhang, X. Feng and D. Zhang, A generalized iterated shrinkage algorithm for non-convex sparse coding, in IEEE International Conference on Computer Vision (ICCV), 2013.

show all references

References:
[1]

R. H. ByrdP. Lu and J. Nocedal, A limited memory algorithm for bound constrained optimization, SIAM J. Sci. Stat. Comput., 16 (1995), 1190-1208. doi: 10.1137/0916069.

[2]

S. BoydN. ParikhE. ChuB. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learning, 3 (2010), 1-122.

[3]

S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2004. doi: 10.1017/CBO9780511804441.

[4]

A. CohenW. Dahmen and R. DeVore, Compressed sensing and best $k$-term approximation, J. Amer. Math. Soc., 22 (2009), 211-231. doi: 10.1090/S0894-0347-08-00610-3.

[5]

R. Chartrand, Nonconvex compressed sensing and error correction, IEEE International Conference on Acoustics, Speech and Signal Processing, (2007), 889-892.

[6]

A. Charkrabarti and F. Hirakawa, Efective separation of sparse and non-sparse image features for denoising, in Proc. Int. Conf. Acoust., Speech, Signal Process. (ICASSP), (2008), 857-860.

[7]

X. ChenK. Ng. Michael and C. Zhang, Non-Lipschitz-Regularization and box constrained model for image restoration, IEEE Trans. Image Processing, 21 (2012), 4709-4721. doi: 10.1109/TIP.2012.2214051.

[8]

E. J. Candès and J. Romberg, The code package $l_1$-magic. Available from: http://statweb.stanford.edu/candes/l1magic/.

[9]

E. J. CandèsJ. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Commun. Pure Appl. Math., 59 (2006), 1207-1223. doi: 10.1002/cpa.20124.

[10]

R. Chartrand and V. Staneva, Restricted isometry properties and nonconvex compressive sensing, Inverse Problems, 24 (2008), 035020, 14 pp. doi: 10.1088/0266-5611/24/3/035020.

[11]

E. J. Candès and T. Tao, Decoding by linear programming, IEEE Trans. Inf. Theory, 51 (2005), 4203-4215. doi: 10.1109/TIT.2005.858979.

[12]

X. Chen and S. Xiang, Sparse solutions of linear complementarity problems, Math. Program., 159 (2016), 539-556. doi: 10.1007/s10107-015-0950-x.

[13]

R. Chartrand and W. Yin, Iteratively reweighted algorithms for compressive sensing, IEEE International Conference on Acoustics, Speech and Signal Processing, (2008), 3869-3872.

[14]

X. Chen and W. Zhou, Convergence of Reweighted $l_1$ Minimization Algorithms and Unique Solution of Truncated $l_p$ Minimization, Tech. rep., Hong Kong Polytechnic University, 2010.

[15]

D. L. Donoho, Compressed sensing, IEEE Trans. Inf. Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.

[16]

D. L. Donoho and X. Huo, Uncertainty principles and ideal atomic decomposition, IEEE Trans. Inf. Theory, 47 (2001), 2845-2862. doi: 10.1109/18.959265.

[17]

M. Elad and A. M. Bruckstein, A generalized uncertainly priciple and sparse representation in pairs of bases, IEEE Trans. Inf. Theory, 48 (2002), 2558-2567. doi: 10.1109/TIT.2002.801410.

[18]

G. Fung and O. Mangasarian, Equivalence of minimal $l_0-$ and $l_p-$norm solutions of linear equalities, inequalities and linear programs for sufficiently small $p$, J. Optim. Theory Appl., 151 (2011), 1-10. doi: 10.1007/s10957-011-9871-x.

[19]

S. Foucart and M. Lai, Sparsest solutions of underdetermined linear systems via $l_q$ minimization for $0 < q < 1$, Applied and Computational Harmonic Analysis, 26 (2009), 395-407. doi: 10.1016/j.acha.2008.09.001.

[20]

M. A. T. FigueiredoR. D. Nowak and S. J. Wright, Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems, IEEE J. Select. Top. Signal Process., 1 (2007), 585-597.

[21]

G. GassoA. Rakotomamonjy and S. Canu, Recovering sparse signals with a certain family of nonconvex penalties and DC programming, IEEE Trans. Signal Process., 57 (2009), 4686-4698. doi: 10.1109/TSP.2009.2026004.

[22]

M. Hyder and K. Mahata, An approximate $l_0$ norm minimization algorithm for compressed sensing, in IEEE International Conference on Acoustics, Speech and Signal Precessing(ICASSP), (2009), 3365-3368.

[23]

E. T. Hale, W. Yin and Y. Zhang, A fixed-point continuation method for $l_1$-regularized minimization with applications to compressed sensing, CAAM Technical Report TR07-07, Rice University, Houston, TX, 2007.

[24]

D. Krishnan and R. Fergus, Fast Image Deconvolution Using Hyper-Laplacian Priors, Neural Information Processing Systems., Cambridge, MA: MIT Press, 2009.

[25]

K. Koh, S.-J. Kim and S. Boyd, The code package l1_ls. Available from: http://www.standord.edu/boyd/l1_ls.

[26]

Q. LyuZ. LinY. She and C. Zhang, A comparison of typical $l_p$ minimization algorithms, Neurocomputing, 119 (2013), 413-424.

[27]

D. C. Liu and J. Nocedal, On the limited memory method for large scale optimization, Mathematical Programming B, 45 (1989), 503-528. doi: 10.1007/BF01589116.

[28]

M. Lai and J. Wang, An unconstrained $l_q$ minimization with $0 < q < 1$ for sparse solution of under-determined linear systems, SIAM J. Optim., 21 (2011), 82-101. doi: 10.1137/090775397.

[29]

B. K. Natraajan, Sparse approximation to linear systems, SIAM J. Comput., 24 (1995), 227-234. doi: 10.1137/S0097539792240406.

[30]

P. Ochs, A. Dosovitskiy, T. Brox and T. Pock, An iterated $l_1$ algorithm for non-smooth non-convex optimization in computer vision, in Computer Vision and Pattern Recognition (CVPR), IEEE Conference, (2013), 1759-1766.

[31]

J. K. PantW. S. Lu and A. Antoniou, New improved algorithms for compressive sensing based on $l_p$ norm, IEEE Trans. on Circuits and Systems-Ⅱ: Express Briefs, 61 (2014), 198-202.

[32]

J. PengS. Yue and H. Li, NP/CMP equivalence: A phenomenon hidden among sparsity models $l_ {0}$ minimization and $l_ {p}$ minimization for information processing, IEEE Trans. Inf. Theory, 61 (2015), 4028-4033. doi: 10.1109/TIT.2015.2429611.

[33]

R. T. Rockafellar, Convex Analysis, Princeton University Press, 1970.

[34]

Y. She, Thresholding-based iterative selection procedures for model selection and shrinkage, Electron. J. Stat., 3 (2009), 384-415. doi: 10.1214/08-EJS348.

[35]

Y. She, An iterative algorithm for fitting nonconvex penalized generalized linear models with grouped predictors, Comput. Statist. Data Anal., 9 (2012), 2976-2990. doi: 10.1016/j.csda.2011.11.013.

[36]

R. Saab, R. Chartrand and O. Yilmaz, Stable sparse approximations via nonconvex optimization, in IEEE International Confereence on Acoustics, Speech and Signal Processing, (2008), 3885-3888.

[37]

J. WrightA. YangA. GaneshS. Sastry and Y. Ma, Robust face recognition via sparse representation, IEEE Trans. Pattern Recogn. Anal. Mach. Intell., 31 (2009), 210-227.

[38]

Y. J. WangG. L. ZhouL. Caccetta and W. Q. Liu, An alternating direction algorithm for $l_1$ problems in compressive sensing, IEEE Trans. Signal Process., 59 (2011), 1895-1901.

[39]

Y. Wang and Q. Ma, A fast subspace method for image deblurring, Appl. Math. Comput., 215 (2009), 2359-2377. doi: 10.1016/j.amc.2009.08.033.

[40]

Y. WangG. ZhouX. ZhangW. Liu and L. Caccetta, The non-convex sparse problem with nonnegative constraint for signal reconstruction, J. Optim. Theory App., 170 (2016), 1009-1025. doi: 10.1007/s10957-016-0869-2.

[41]

A. Y. YangZ. ZhouA. G. BalasubramanianS. S. Sastry and Y. Ma, Fast-minimization algorithms for robust face recognition, IEEE Trans. Image Processing, 22 (2013), 3234-3246.

[42]

F. ZouH. FengH. LingC. LiuL. YanP. Li and D. Li, Nonnegative sparse coding induced hashing for image copy detection, Neurocomputing, 105 (2013), 81-95.

[43]

J. ZengS. LinY. Wang and Z. Xu, $L_{1/2}$ regularization: Convergence of iterative half thresholding algorithm, IEEE Trans. Signal Process., 62 (2014), 2317-2329. doi: 10.1109/TSP.2014.2309076.

[44]

W. Zuo, D. Meng, L. Zhang, X. Feng and D. Zhang, A generalized iterated shrinkage algorithm for non-convex sparse coding, in IEEE International Conference on Computer Vision (ICCV), 2013.

Figure 1.  The original signal and the reconstructed signals with $p = 0.8,0.6,0.4,0.2$, respectively
Figure 2.  The original and recovered images with different values of $p$, where SNR denotes the signal-to-noise ratio
Figure 3.  Comparison of the performance for ITM and SMM with large-scale problems. The first line: $p = 0.9$; the second line: $p = 0.7$; the final line: $p = 0.5$. The first column: sparsity; the second column: relative error; the final column: cpu time
[1]

Qilin Wang, Liu He, Shengjie Li. Higher-order weak radial epiderivatives and non-convex set-valued optimization problems. Journal of Industrial & Management Optimization, 2018, 13 (5) : 1-16. doi: 10.3934/jimo.2018051

[2]

Meixia Li, Changyu Wang, Biao Qu. Non-convex semi-infinite min-max optimization with noncompact sets. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1859-1881. doi: 10.3934/jimo.2017022

[3]

Mikhail Krastanov, Michael Malisoff, Peter Wolenski. On the strong invariance property for non-Lipschitz dynamics. Communications on Pure & Applied Analysis, 2006, 5 (1) : 107-124. doi: 10.3934/cpaa.2006.5.107

[4]

Boris Hasselblatt and Amie Wilkinson. Prevalence of non-Lipschitz Anosov foliations. Electronic Research Announcements, 1997, 3: 93-98.

[5]

Yavdat Il'yasov. On critical exponent for an elliptic equation with non-Lipschitz nonlinearity. Conference Publications, 2011, 2011 (Special) : 698-706. doi: 10.3934/proc.2011.2011.698

[6]

Humberto Ramos Quoirin, Kenichiro Umezu. A loop type component in the non-negative solutions set of an indefinite elliptic problem. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1255-1269. doi: 10.3934/cpaa.2018060

[7]

Emmanuele DiBenedetto, Ugo Gianazza, Naian Liao. On the local behavior of non-negative solutions to a logarithmically singular equation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1841-1858. doi: 10.3934/dcdsb.2012.17.1841

[8]

Kewei Zhang. On non-negative quasiconvex functions with quasimonotone gradients and prescribed zero sets. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 353-366. doi: 10.3934/dcds.2008.21.353

[9]

Jingang Xiong, Jiguang Bao. The obstacle problem for Monge-Ampère type equations in non-convex domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 59-68. doi: 10.3934/cpaa.2011.10.59

[10]

Simona Fornaro, Ugo Gianazza. Local properties of non-negative solutions to some doubly non-linear degenerate parabolic equations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 481-492. doi: 10.3934/dcds.2010.26.481

[11]

Genni Fragnelli, Paolo Nistri, Duccio Papini. Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 35-64. doi: 10.3934/dcds.2011.31.35

[12]

Yoon Mo Jung, Taeuk Jeong, Sangwoon Yun. Non-convex TV denoising corrupted by impulse noise. Inverse Problems & Imaging, 2017, 11 (4) : 689-702. doi: 10.3934/ipi.2017032

[13]

Tong Li, Jeungeun Park. Stability of traveling waves of models for image processing with non-convex nonlinearity. Communications on Pure & Applied Analysis, 2018, 17 (3) : 959-985. doi: 10.3934/cpaa.2018047

[14]

Jun Chen, Wenyu Sun, Zhenghao Yang. A non-monotone retrospective trust-region method for unconstrained optimization. Journal of Industrial & Management Optimization, 2013, 9 (4) : 919-944. doi: 10.3934/jimo.2013.9.919

[15]

Genni Fragnelli, Paolo Nistri, Duccio Papini. Corrigendum: Nnon-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3831-3834. doi: 10.3934/dcds.2013.33.3831

[16]

Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539

[17]

C. M. Elliott, B. Gawron, S. Maier-Paape, E. S. Van Vleck. Discrete dynamics for convex and non-convex smoothing functionals in PDE based image restoration. Communications on Pure & Applied Analysis, 2006, 5 (1) : 181-200. doi: 10.3934/cpaa.2006.5.181

[18]

Tong Li, Hui Yin. Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux. Communications on Pure & Applied Analysis, 2014, 13 (2) : 835-858. doi: 10.3934/cpaa.2014.13.835

[19]

Asadollah Aghajani. Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3521-3530. doi: 10.3934/dcds.2017150

[20]

. Adimurthi, Siddhartha Mishra, G.D. Veerappa Gowda. Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes. Networks & Heterogeneous Media, 2007, 2 (1) : 127-157. doi: 10.3934/nhm.2007.2.127

2017 Impact Factor: 0.994

Metrics

  • PDF downloads (10)
  • HTML views (50)
  • Cited by (0)

Other articles
by authors

[Back to Top]