• Previous Article
    Investigating a green supply chain with product recycling under retailer's fairness behavior
  • JIMO Home
  • This Issue
  • Next Article
    Pricing decisions for closed-loop supply chains with technology licensing and carbon constraint under reward-penalty mechanism
doi: 10.3934/jimo.2022008
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

A relaxed parameter condition for the primal-dual hybrid gradient method for saddle-point problem

1. 

Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing, 211167, China

2. 

School of Applied Mathematics, Nanjing University of Finance & Economics, Nanjing, 210023, China

3. 

Hefei 168 Middle School, Hefei, 230031, China

*Corresponding author: Yuan Shen

Received  February 2021 Revised  August 2021 Early access January 2022

Fund Project: The first author is supported by the National Natural Science Foundation of China Grant 71571096 and Nanjing Institute of Technology Research Start-up Foundation Grant YKJ201738. The second author is supported by the National Social Science Foundation of China under Grants 19AZD018, 20BGL028, 19BGL205

The primal-dual hybrid gradient method and the primal-dual algorithm proposed by Chambolle and Pock are both efficient methods for solving saddle point problem. However, the convergence of both methods depends on some assumptions which can be too restrictive or impractical in real applications. In this paper, we propose a new parameter condition for the primal-dual hybrid gradient method. This improvement only requires either the primal or the dual objective function to be strongly convex. The relaxed parameter condition leads to convergence acceleration. Although counter-example shows that the PDHG method is not necessarily convergent with constant step size, it becomes convergent with our relaxed parameter condition. Preliminary experimental results show that PDHG method with our relaxed parameter condition is more efficient than several state-of-art methods.

Citation: Xiayang Zhang, Yuqian Kong, Shanshan Liu, Yuan Shen. A relaxed parameter condition for the primal-dual hybrid gradient method for saddle-point problem. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022008
References:
[1]

K. J. Arrow, L. Hurwicz and H. Uzawa, Studies in Linear and Non-Linear Programming, With contributions by H. B. Chenery, S. M. Johnson, S. Karlin, T. Marschak, R. M. Solow. Stanford Mathematical Studies in the Social Science, Vol. II. Stanford University Press, Stanford, Calif., 1958.

[2]

S. Bonettini and V. Ruggiero, On the convergence of primal-dual hybrid gradient algorithms for total variation image restoration, J. Math. Imaging Vision, 44 (2012), 236-253.  doi: 10.1007/s10851-011-0324-9.

[3]

S. BoydN. ParikhE. ChuB. Peleato and J. Eckstein, Distributed optimization and statistical learning via ADMM, Found. Trends Mach. Learn., 3 (2010), 1-122. 

[4]

J. CaiE. J. Candés and Z. Shen, A singular value thresholding algorithm for matrix completion, SIAM J. Opti., 20 (2010), 1956-1982.  doi: 10.1137/080738970.

[5]

J. CaiS. Osher and Z. Shen, Linearized Bregman iterations for compressed sensing, Math. Comput., 78 (2009), 1515-1536.  doi: 10.1090/S0025-5718-08-02189-3.

[6]

J. CaiS. Osher and Z. Shen, Linearized Bregman iteration for frame based image deblurring, SIAM J. Imaging Sci., 2 (2009), 226-252.  doi: 10.1137/080733371.

[7]

X. CaiD. Han and L. Xu, An improved first-order primal-dual algorithm with a new correction step, J. Global Optim., 57 (2013), 1419-1428.  doi: 10.1007/s10898-012-9999-8.

[8]

A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vison, 20 (2004), 89-97. 

[9]

A. Chambolle and T. Pock, A first-order primal-dual algorithms for convex problem with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.

[10]

A. Chambolle and T. Pock, On the ergodic convergence rates of a first order primal dual algorithm, Math. Program., 159 (2016), 253-287.  doi: 10.1007/s10107-015-0957-3.

[11]

R. ChanS. Ma and J. Yang, Inertial primal dual algorithms for structured convex optimization, SIAM J. Imag. Sci., 8 (2015), 2239-2267. 

[12]

T. ChanG. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), 1964-1977.  doi: 10.1137/S1064827596299767.

[13]

Y. ChenG. Lan and Y. Ouyang, Optimal primal-dual methods for a class of saddle point problems, SIAM J. Optim., 24 (2014), 1779-1814.  doi: 10.1137/130919362.

[14]

E. EsserX. Zhang and T. Chan, A general framework for a class of first order primal-dual algorithms for TV minimization, SIAM J. Imaging Sci., 3 (2010), 1015-1046.  doi: 10.1137/09076934X.

[15]

F. Facchinei and J. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research, Springer Verlag, New York, 2003.

[16]

T. Goldstein, M. Li and X. Yuan, Adaptive primal-dual splitting methods for statistical learning and image processing, Adv. Neural Inform. Process. Syst., (2015), 2089–2097.

[17]

T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM J. Imag. Sci., 2 (2009), 323-343.  doi: 10.1137/080725891.

[18]

B. HeY. You and X. Yuan, On the convergence of primal-dual hybrid gradient algorithm, SIAM J. Imag. Sci., 7 (2014), 2526-2537.  doi: 10.1137/140963467.

[19]

B. He and X. Yuan, Convergence analysis of primal-dual algorithms for a saddle-point problem: From contraction perspective, SIAM J. Imag. Sci., 5 (2012), 119-149.  doi: 10.1137/100814494.

[20]

B. He and X. Yuan, On the $O(1/n)$ convergence rate of Douglas-Rachford alternating direction method, SIAM J. Numer. Anal., 50 (2012), 700-709.  doi: 10.1137/110836936.

[21]

H. HeJ. Desai and K. Wang., A primal-dual prediction–correction algorithm for saddle point optimization, J. Global Optim., 66 (2016), 573-583.  doi: 10.1007/s10898-016-0437-1.

[22]

M. Hestenes, Multiplier and gradient methods, J. Optim. Theory Appli., 4 (1969), 303-320.  doi: 10.1007/BF00927673.

[23]

N. Higham, Computing the nearest correlation matrix –-A problem from finance, IMA J. Numer. Anal., 22 (2002), 329-343.  doi: 10.1093/imanum/22.3.329.

[24]

Y. Nesterov, Gradient methods for minimizing composite objective function, Math. Program., 140 (2013), 125-161.  doi: 10.1007/s10107-012-0629-5.

[25]

J. Pesquet and N. Pustelnik, A parallel inertial proximal optimization method, Pac. J. Optim., 8 (2012), 273-306. 

[26]

M. Powell, A method for nonlinear constraints in minimization problems, In Optimization edited by R. Fletcher, Academic Press, (1969), 283–298.

[27]

L. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.

[28]

Y. ShenQ. Li and J. Wu, A variable step-size primal-dual algorithm based on proximal point algorithm (in Chinese), Math. Numer. Sinica., 40 (2018), 85-95. 

[29]

R. Tibshirani, Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), 267-288.  doi: 10.1111/j.2517-6161.1996.tb02080.x.

[30]

T. Valkonen, Inertial, corrected, primal-dual proximal splitting, SIAM J. Optim., 30 (2020), 1391-1420.  doi: 10.1137/18M1182851.

[31]

Y. WangJ. YangW. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Imaging Sci., 1 (2008), 248-272.  doi: 10.1137/080724265.

[32]

P. WeissL. Blanc-Feraud and G. Aubert, Efficient schemes for total variation minimization under constraints in image processing, SIAM J. Sci. Comput., 31 (2009), 2047-2080.  doi: 10.1137/070696143.

[33]

B. ZhangZ. Zhu and S. Wang, A simple primal-dual method for total variation image restoration, J. Vis. Commun. Image R., 38 (2016), 814-823.  doi: 10.1016/j.jvcir.2016.04.025.

[34]

H. ZhangJ. CaiL. Cheng and J. Zhu, Strongly convex programming for exact matrix completion and robust principal component analysis, Inver. Prob. Imaging, 6 (2012), 357-372.  doi: 10.3934/ipi.2012.6.357.

[35]

X. ZhangM. Burger and S. Osher, A unified primal-dual algorithm framework based on Bregman iteration, J. Sci. Comput., 46 (2011), 20-46.  doi: 10.1007/s10915-010-9408-8.

[36]

M. Zhu and T. Chan, An Efficient Primal-dual Hybrid Gradient Algorithm for Total Variation Image Restoration, CAM Report 08-34, UCLA, USA, 2008.

show all references

References:
[1]

K. J. Arrow, L. Hurwicz and H. Uzawa, Studies in Linear and Non-Linear Programming, With contributions by H. B. Chenery, S. M. Johnson, S. Karlin, T. Marschak, R. M. Solow. Stanford Mathematical Studies in the Social Science, Vol. II. Stanford University Press, Stanford, Calif., 1958.

[2]

S. Bonettini and V. Ruggiero, On the convergence of primal-dual hybrid gradient algorithms for total variation image restoration, J. Math. Imaging Vision, 44 (2012), 236-253.  doi: 10.1007/s10851-011-0324-9.

[3]

S. BoydN. ParikhE. ChuB. Peleato and J. Eckstein, Distributed optimization and statistical learning via ADMM, Found. Trends Mach. Learn., 3 (2010), 1-122. 

[4]

J. CaiE. J. Candés and Z. Shen, A singular value thresholding algorithm for matrix completion, SIAM J. Opti., 20 (2010), 1956-1982.  doi: 10.1137/080738970.

[5]

J. CaiS. Osher and Z. Shen, Linearized Bregman iterations for compressed sensing, Math. Comput., 78 (2009), 1515-1536.  doi: 10.1090/S0025-5718-08-02189-3.

[6]

J. CaiS. Osher and Z. Shen, Linearized Bregman iteration for frame based image deblurring, SIAM J. Imaging Sci., 2 (2009), 226-252.  doi: 10.1137/080733371.

[7]

X. CaiD. Han and L. Xu, An improved first-order primal-dual algorithm with a new correction step, J. Global Optim., 57 (2013), 1419-1428.  doi: 10.1007/s10898-012-9999-8.

[8]

A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vison, 20 (2004), 89-97. 

[9]

A. Chambolle and T. Pock, A first-order primal-dual algorithms for convex problem with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.

[10]

A. Chambolle and T. Pock, On the ergodic convergence rates of a first order primal dual algorithm, Math. Program., 159 (2016), 253-287.  doi: 10.1007/s10107-015-0957-3.

[11]

R. ChanS. Ma and J. Yang, Inertial primal dual algorithms for structured convex optimization, SIAM J. Imag. Sci., 8 (2015), 2239-2267. 

[12]

T. ChanG. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), 1964-1977.  doi: 10.1137/S1064827596299767.

[13]

Y. ChenG. Lan and Y. Ouyang, Optimal primal-dual methods for a class of saddle point problems, SIAM J. Optim., 24 (2014), 1779-1814.  doi: 10.1137/130919362.

[14]

E. EsserX. Zhang and T. Chan, A general framework for a class of first order primal-dual algorithms for TV minimization, SIAM J. Imaging Sci., 3 (2010), 1015-1046.  doi: 10.1137/09076934X.

[15]

F. Facchinei and J. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research, Springer Verlag, New York, 2003.

[16]

T. Goldstein, M. Li and X. Yuan, Adaptive primal-dual splitting methods for statistical learning and image processing, Adv. Neural Inform. Process. Syst., (2015), 2089–2097.

[17]

T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM J. Imag. Sci., 2 (2009), 323-343.  doi: 10.1137/080725891.

[18]

B. HeY. You and X. Yuan, On the convergence of primal-dual hybrid gradient algorithm, SIAM J. Imag. Sci., 7 (2014), 2526-2537.  doi: 10.1137/140963467.

[19]

B. He and X. Yuan, Convergence analysis of primal-dual algorithms for a saddle-point problem: From contraction perspective, SIAM J. Imag. Sci., 5 (2012), 119-149.  doi: 10.1137/100814494.

[20]

B. He and X. Yuan, On the $O(1/n)$ convergence rate of Douglas-Rachford alternating direction method, SIAM J. Numer. Anal., 50 (2012), 700-709.  doi: 10.1137/110836936.

[21]

H. HeJ. Desai and K. Wang., A primal-dual prediction–correction algorithm for saddle point optimization, J. Global Optim., 66 (2016), 573-583.  doi: 10.1007/s10898-016-0437-1.

[22]

M. Hestenes, Multiplier and gradient methods, J. Optim. Theory Appli., 4 (1969), 303-320.  doi: 10.1007/BF00927673.

[23]

N. Higham, Computing the nearest correlation matrix –-A problem from finance, IMA J. Numer. Anal., 22 (2002), 329-343.  doi: 10.1093/imanum/22.3.329.

[24]

Y. Nesterov, Gradient methods for minimizing composite objective function, Math. Program., 140 (2013), 125-161.  doi: 10.1007/s10107-012-0629-5.

[25]

J. Pesquet and N. Pustelnik, A parallel inertial proximal optimization method, Pac. J. Optim., 8 (2012), 273-306. 

[26]

M. Powell, A method for nonlinear constraints in minimization problems, In Optimization edited by R. Fletcher, Academic Press, (1969), 283–298.

[27]

L. RudinS. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F.

[28]

Y. ShenQ. Li and J. Wu, A variable step-size primal-dual algorithm based on proximal point algorithm (in Chinese), Math. Numer. Sinica., 40 (2018), 85-95. 

[29]

R. Tibshirani, Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B, 58 (1996), 267-288.  doi: 10.1111/j.2517-6161.1996.tb02080.x.

[30]

T. Valkonen, Inertial, corrected, primal-dual proximal splitting, SIAM J. Optim., 30 (2020), 1391-1420.  doi: 10.1137/18M1182851.

[31]

Y. WangJ. YangW. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Imaging Sci., 1 (2008), 248-272.  doi: 10.1137/080724265.

[32]

P. WeissL. Blanc-Feraud and G. Aubert, Efficient schemes for total variation minimization under constraints in image processing, SIAM J. Sci. Comput., 31 (2009), 2047-2080.  doi: 10.1137/070696143.

[33]

B. ZhangZ. Zhu and S. Wang, A simple primal-dual method for total variation image restoration, J. Vis. Commun. Image R., 38 (2016), 814-823.  doi: 10.1016/j.jvcir.2016.04.025.

[34]

H. ZhangJ. CaiL. Cheng and J. Zhu, Strongly convex programming for exact matrix completion and robust principal component analysis, Inver. Prob. Imaging, 6 (2012), 357-372.  doi: 10.3934/ipi.2012.6.357.

[35]

X. ZhangM. Burger and S. Osher, A unified primal-dual algorithm framework based on Bregman iteration, J. Sci. Comput., 46 (2011), 20-46.  doi: 10.1007/s10915-010-9408-8.

[36]

M. Zhu and T. Chan, An Efficient Primal-dual Hybrid Gradient Algorithm for Total Variation Image Restoration, CAM Report 08-34, UCLA, USA, 2008.

Figure 1.  Comparison of parameter domain with different setting of $ \alpha $
Figure 2.  Results on LASSO problem with different sample size $ m $ and feature No. $ n $
Figure 3.  Results on LASSO problem with sparsity and balancing parameter
Figure 4.  From left to right, up to bottom: the original figure 'sunflower.png($ 900\times824 $)', noisy figure with additive Gaussian noise(Standard deviation = 0.05), figure recovered by PDHG method($ \lambda = 8 $) and figure recovered by PDHG-R($ \lambda = 8 $)
Figure 5.  From left to right, up to bottom: the original figure 'boat.png($ 512\times512 $)', noisy figure with additive Gaussian noise(Standard deviation = 0.05), figure recovered by PDHG method($ \lambda = 8 $) and figure recovered by PDHG-R($ \lambda = 8 $)
Figure 6.  Results of image denoising with different images and $ \lambda $
Figure 7.  Results of image denoising with different stepsize $ \tau $ and $ \sigma $
Table 1.  Results on LASSO problem with different sample size $ m $ and feature No. $ n $
100$ \times $1000 100$ \times $10000
Tol=$ 10^{-4} $ Tol=$ 10^{-6} $ Tol=$ 10^{-4} $ Tol=$ 10^{-6} $
PDHG 24 48 172 284
CP 24 39 140 254
CP-A 30 51 908 7993
FISTA 35 78 217 553
PDHG-R 23 36 124 202
100$ \times $100000 10000$ \times $100000
Tol=$ 10^{-4} $ Tol=$ 10^{-6} $ Tol=$ 10^{-4} $ Tol=$ 10^{-6} $
PDHG 1279 2264 12 44
CP 1181 1990 13 31
CP-A 3977 >10000 33 48
FISTA 639 1593 15 53
PDHG-R 990 1723 13 30
100$ \times $1000 100$ \times $10000
Tol=$ 10^{-4} $ Tol=$ 10^{-6} $ Tol=$ 10^{-4} $ Tol=$ 10^{-6} $
PDHG 24 48 172 284
CP 24 39 140 254
CP-A 30 51 908 7993
FISTA 35 78 217 553
PDHG-R 23 36 124 202
100$ \times $100000 10000$ \times $100000
Tol=$ 10^{-4} $ Tol=$ 10^{-6} $ Tol=$ 10^{-4} $ Tol=$ 10^{-6} $
PDHG 1279 2264 12 44
CP 1181 1990 13 31
CP-A 3977 >10000 33 48
FISTA 639 1593 15 53
PDHG-R 990 1723 13 30
Table 2.  Results on LASSO problem with sparsity and balancing parameter
s=5 s=20
Tol=$ 10^{-4} $ Tol=$ 10^{-6} $ Tol=$ 10^{-4} $ Tol=$ 10^{-6} $
PDHG 118 312 161 279
CP 157 249 149 236
CP-A 86 131 533 3450
FISTA 149 298 142 447
PDHG-R 142 225 114 207
$ \beta $=0.1 $ \beta $=0.5
Tol=$ 10^{-4} $ Tol=$ 10^{-6} $ Tol=$ 10^{-4} $ Tol=$ 10^{-6} $
PDHG 193 307 180 290
CP 185 278 166 218
CP-A >10000 >10000 87 138
FISTA 313 755 121 212
PDHG-R 150 249 137 197
s=5 s=20
Tol=$ 10^{-4} $ Tol=$ 10^{-6} $ Tol=$ 10^{-4} $ Tol=$ 10^{-6} $
PDHG 118 312 161 279
CP 157 249 149 236
CP-A 86 131 533 3450
FISTA 149 298 142 447
PDHG-R 142 225 114 207
$ \beta $=0.1 $ \beta $=0.5
Tol=$ 10^{-4} $ Tol=$ 10^{-6} $ Tol=$ 10^{-4} $ Tol=$ 10^{-6} $
PDHG 193 307 180 290
CP 185 278 166 218
CP-A >10000 >10000 87 138
FISTA 313 755 121 212
PDHG-R 150 249 137 197
[1]

Yu-Hong Dai, Xin-Wei Liu, Jie Sun. A primal-dual interior-point method capable of rapidly detecting infeasibility for nonlinear programs. Journal of Industrial and Management Optimization, 2020, 16 (2) : 1009-1035. doi: 10.3934/jimo.2018190

[2]

Kai Wang, Deren Han. On the linear convergence of the general first order primal-dual algorithm. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021134

[3]

Xiaojing Ye, Haomin Zhou. Fast total variation wavelet inpainting via approximated primal-dual hybrid gradient algorithm. Inverse Problems and Imaging, 2013, 7 (3) : 1031-1050. doi: 10.3934/ipi.2013.7.1031

[4]

Nadia Hazzam, Zakia Kebbiche. A primal-dual interior point method for $ P_{\ast }\left( \kappa \right) $-HLCP based on a class of parametric kernel functions. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 513-531. doi: 10.3934/naco.2020053

[5]

Yixuan Yang, Yuchao Tang, Meng Wen, Tieyong Zeng. Preconditioned Douglas-Rachford type primal-dual method for solving composite monotone inclusion problems with applications. Inverse Problems and Imaging, 2021, 15 (4) : 787-825. doi: 10.3934/ipi.2021014

[6]

Giuseppe Floridia, Hiroshi Takase, Masahiro Yamamoto. A Carleman estimate and an energy method for a first-order symmetric hyperbolic system. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022016

[7]

Yanqin Bai, Xuerui Gao, Guoqiang Wang. Primal-dual interior-point algorithms for convex quadratic circular cone optimization. Numerical Algebra, Control and Optimization, 2015, 5 (2) : 211-231. doi: 10.3934/naco.2015.5.211

[8]

Siqi Li, Weiyi Qian. Analysis of complexity of primal-dual interior-point algorithms based on a new kernel function for linear optimization. Numerical Algebra, Control and Optimization, 2015, 5 (1) : 37-46. doi: 10.3934/naco.2015.5.37

[9]

Guoqiang Wang, Zhongchen Wu, Zhongtuan Zheng, Xinzhong Cai. Complexity analysis of primal-dual interior-point methods for semidefinite optimization based on a parametric kernel function with a trigonometric barrier term. Numerical Algebra, Control and Optimization, 2015, 5 (2) : 101-113. doi: 10.3934/naco.2015.5.101

[10]

Jen-Yen Lin, Hui-Ju Chen, Ruey-Lin Sheu. Augmented Lagrange primal-dual approach for generalized fractional programming problems. Journal of Industrial and Management Optimization, 2013, 9 (4) : 723-741. doi: 10.3934/jimo.2013.9.723

[11]

Gianni Di Pillo, Giampaolo Liuzzi, Stefano Lucidi. A primal-dual algorithm for nonlinear programming exploiting negative curvature directions. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 509-528. doi: 10.3934/naco.2011.1.509

[12]

Jin-Zan Liu, Xin-Wei Liu. A dual Bregman proximal gradient method for relatively-strongly convex optimization. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021028

[13]

Valery Y. Glizer, Oleg Kelis. Singular infinite horizon zero-sum linear-quadratic differential game: Saddle-point equilibrium sequence. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 1-20. doi: 10.3934/naco.2017001

[14]

Monica Lazzo, Paul G. Schmidt. Monotone local semiflows with saddle-point dynamics and applications to semilinear diffusion equations. Conference Publications, 2005, 2005 (Special) : 566-575. doi: 10.3934/proc.2005.2005.566

[15]

Mohammed Al Horani, Angelo Favini. First-order inverse evolution equations. Evolution Equations and Control Theory, 2014, 3 (3) : 355-361. doi: 10.3934/eect.2014.3.355

[16]

Ciro D'Apice, Olha P. Kupenko, Rosanna Manzo. On boundary optimal control problem for an arterial system: First-order optimality conditions. Networks and Heterogeneous Media, 2018, 13 (4) : 585-607. doi: 10.3934/nhm.2018027

[17]

Diogo A. Gomes, Hiroyoshi Mitake, Kengo Terai. The selection problem for some first-order stationary Mean-field games. Networks and Heterogeneous Media, 2020, 15 (4) : 681-710. doi: 10.3934/nhm.2020019

[18]

Fengmin Wang, Dachuan Xu, Donglei Du, Chenchen Wu. Primal-dual approximation algorithms for submodular cost set cover problems with linear/submodular penalties. Numerical Algebra, Control and Optimization, 2015, 5 (2) : 91-100. doi: 10.3934/naco.2015.5.91

[19]

Yu-Hong Dai, Zhouhong Wang, Fengmin Xu. A Primal-dual algorithm for unfolding neutron energy spectrum from multiple activation foils. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2367-2387. doi: 10.3934/jimo.2020073

[20]

Zhongyi Huang. Tailored finite point method for the interface problem. Networks and Heterogeneous Media, 2009, 4 (1) : 91-106. doi: 10.3934/nhm.2009.4.91

2021 Impact Factor: 1.411

Metrics

  • PDF downloads (218)
  • HTML views (214)
  • Cited by (0)

Other articles
by authors

[Back to Top]