doi: 10.3934/jimo.2019034

The space decomposition method for the sum of nonlinear convex maximum eigenvalues and its applications

1. 

School of Science, Dalian Maritime University, Dalian 116026, China

2. 

School of Control Science and Engineering, Dalian University of Technology, Dalian 116024, China

3. 

School of Economics and Management, Hebei University of Technology, Tianjin 300401, China

4. 

College of Science, Dalian Minzu University, Dalian 116600, China

5. 

School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

* Corresponding author

Published  May 2019

Fund Project: The first author's work was supported in part by the National Natural Science Foundation of China under projects No.11626053, 11701063; the Project funded by China Postdoctoral Science Foundation under No.2016M601296 and the Fundamental Research Funds for the Central Universities under project No.3132017053, 3132018215, 3132018218 and 3132018219; the Scientific Research Foundation Funds of DLMU under project No.02501102. The third author's work was supported in part by the National Natural Science Foundation of China under Grant No.11501080

In this paper, we mainly consider optimization problems involving the sum of largest eigenvalues of nonlinear symmetric matrices. One of the difficulties with numerical analysis of such problems is that the eigenvalues, regarded as functions of a symmetric matrix, are not differentiable at those points where they coalesce. The $\mathcal {U}$-Lagrangian theory is applied to the function of the sum of the largest eigenvalues, with convex matrix-valued mappings, which doesn't need to be affine. Some of the results generalize the corresponding conclusions for linear mapping. In the approach, we reformulate the first- and second-order derivatives of ${\mathcal U}$-Lagrangian in the space of decision variables $R^m$ under some mild conditions in terms of $\mathcal{VU}$-space decomposition. We characterize smooth trajectory, along which the function has a second-order expansion. Moreover, an algorithm framework with superlinear convergence is presented. Finally, an application of $\mathcal{VU}$-decomposition derivatives shows that $\mathcal{U}$-Lagrangian possesses proper execution in matrix variable.

Citation: Ming Huang, Cong Cheng, Yang Li, Zun Quan Xia. The space decomposition method for the sum of nonlinear convex maximum eigenvalues and its applications. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019034
References:
[1]

R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Applied Mathematical Sciences Volume 75, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-1029-0.

[2]

P. ApkarianD. NollJ.-B. Thevenet and H. D. Tuan, A spectral quadratic-SDP method with applications to fixed-order $H_2$ and $H_{\infty}$ synthesis, European Journal of Control, 10 (2004), 527-538. doi: 10.3166/ejc.10.527-538.

[3]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1394-9.

[4]

S. Cox and R. Lipton, Extremal eigenvalue problems for two-phase conductors, Archive for Rational Mechanics and Analysis, 136 (1996), 101-117. doi: 10.1007/BF02316974.

[5]

J. CullumW. E. Donath and P. Wolfe, The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices, Mathematical Programming Study, 3 (1975), 35-55. doi: 10.1007/bfb0120698.

[6]

A. R. Díaz and N. Kikuchi, Solutions to shape and topology eigenvalue optimization problems using a homogenization method, International Journal for Numerical Methods in Engineering, 35 (1992), 1487-1502. doi: 10.1002/nme.1620350707.

[7]

R. Fletcher, Semi-definite matrix constraints in optimization, SIAM Journal on Control and Optimization, 23 (1985), 493-513. doi: 10.1137/0323032.

[8]

C. HelmbergF. Rendl and R. Weismantel, A semidefinite programming approach to the quadratic knapsack problem, Journal of Combinatorial Optimization, 4 (2000), 197-215. doi: 10.1023/A:1009898604624.

[9]

C. HelmbergM. L. Overton and F. Rendl, The spectral bundle method with second-order information, Optimization Methods and Software, 29 (2014), 855-876. doi: 10.1080/10556788.2013.858155.

[10]

M. HuangL. P. Pang and Z. Q. Xia, The space decomposition theory for a class of eigenvalue optimizations, Computational Optimization and Applications, 58 (2014), 423-454. doi: 10.1007/s10589-013-9624-x.

[11]

M. Huang, L. P. Pang, X. J. Liang and Z. Q. Xia, The space decomposition theory for a class of semi-infinite maximum eigenvalue optimizations, Abstract and Applied Analysis, 2014, Article ID 845017, 12 pages. doi: 10.1155/2014/845017.

[12]

M. HuangL. P. PangX. J. Liang and F. Y. Meng, A second-order bundle method based on $\mathcal {VU}$-decomposition strategy for a special class of eigenvalue optimizations, Numerical Functional Analysis and Optimization, 37 (2016), 554-582. doi: 10.1080/01630563.2016.1138969.

[13]

M. HuangX. J. LiangL. P. Pang and Y. Lu, The bundle scheme for solving arbitrary eigenvalue optimizations, Journal of Industrial and Management Optimization, 13 (2017), 659-680. doi: 10.3934/jimo.2016039.

[14]

M. HuangL. P. PangY. Lu and Z. Q. Xia, A fast space-decomposition scheme for nonconvex eigenvalue optimization, Set-Valued and Variational Analysis, 25 (2017), 43-67. doi: 10.1007/s11228-016-0365-8.

[15]

M. HuangY. LuL. P. Pang and Z. Q. Xia, A space decomposition scheme for maximum eigenvalue functions and its applications, Mathematical Methods of Operations Research, 85 (2017), 453-490. doi: 10.1007/s00186-017-0579-z.

[16]

J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I-II, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02796-7.

[17]

J.-B. Hiriart-Urruty and D. Ye, Sensitivity analysis of all eigenvalues of a symmetric matrix, Numerische Mathematik, 70 (1995), 45-72. doi: 10.1007/s002110050109.

[18]

C. Kan and W. Song, Second-order conditions for existence of augmented lagrange multipliers for eigenvalue composite optimization problems, Journal of Global Optimization, 63 (2015), 77-97. doi: 10.1007/s10898-015-0273-8.

[19]

C. LemaréchalF. Oustry and C. Sagastizábal, The ${\mathcal U}$-Lagrangian of a convex function, Transactions of the American Mathematical Society, 352 (2000), 711-729. doi: 10.1090/S0002-9947-99-02243-6.

[20]

A. S. Lewis and M. L. Overton, Eigenvalue optimization, Acta Numerica, 5 (1996), 149-190. doi: 10.1017/S0962492900002646.

[21]

X. LiuX. WangZ. Wen and Y. Yuan, On the Convergence of the Self-Consistent Field Iteration in Kohn-Sham Density Functional Theory, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 546-558. doi: 10.1137/130911032.

[22]

X. LiuZ. WenX. WangM. Ulbrich and Y. Yuan, On the analysis of the discretized kohn-sham density functional theory, SIAM Journal on Numerical Analysis, 53 (2015), 1758-1785. doi: 10.1137/140957962.

[23]

R. Mifflin and C. Sagastizábal, A $\mathcal {VU}$-algorithm for convex minimization, Mathematical Programming Ser.B, 104 (2005), 583-608. doi: 10.1007/s10107-005-0630-3.

[24]

D. Noll and P. Apkarian, Spectral bundle method for nonconvex maximum eigenvalue functions: Second-order methods, Mathematical Programming Ser. B, 104 (2005), 729-747. doi: 10.1007/s10107-005-0635-y.

[25]

D. NollM. Torki and P. Apkarian, Partially augmented Lagrangian method for matrix inequality constraints, SIAM Journal on Optimization, 15 (2004), 161-184. doi: 10.1137/S1052623402413963.

[26]

F. Oustry, The ${\mathcal U}$-Lagrangian of the maximum eigenvalue functions, SIAM Journal on Optimization, 9 (1999), 526-549. doi: 10.1137/S1052623496311776.

[27]

M. L. Overton, Large-scale optimization of eigenvalues, SIAM Journal on Optimization, 2 (1992), 88-120. doi: 10.1137/0802007.

[28]

M. L. Overton and R. S. Womersley, Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices, Mathematical Programming, 62 (1993), 321-357. doi: 10.1007/BF01585173.

[29]

M. L. Overton and R. S. Womersley, Second derivatives for optimizing eigenvalues of symmetric matrices, SIAM Journal on Matrix Analysis and Applications, 16 (1995), 697-718. doi: 10.1137/S089547989324598X.

[30]

M. L. Overton and X. Ye, Towards second-order methods for structured nonsmooth optimization, In: S. Gomez, J.-P. Hennart eds., Advances in Optimization and Numerical Analysis, 97–109, Volume 275 of the series Mathematics and Its Applications, Kluwer Academic Publishers, Norwell, MA, 1994. doi: 10.1007/978-94-015-8330-5_7.

[31]

G. Pataki, On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues, Mathematics of Operations Research, 23 (1998), 339-358. doi: 10.1287/moor.23.2.339.

[32] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1997.
[33]

A. Shapiro and M. K. H. Fan, On eigenvalue optimization, SIAM Journal on Optimization, 5 (1995), 552-569. doi: 10.1137/0805028.

[34]

A. Shapiro, First and second order analysis of nonlinear semidefinite programs, Mathematical Programming, 77 (1997), 301-320. doi: 10.1007/BF02614439.

[35]

M. Torki, First- and second-order epi-differentiability in eigenvalue optimization, Journal of Mathematical Analysis and Applications, 234 (1999), 391-416. doi: 10.1006/jmaa.1999.6320.

[36]

M. Torki, Second-order directional derivatives of all eigenvalues of a symmetric matrix, Nonlinear Analysis. Theory, Methods & Applications, 46 (2001), 1133-1150. doi: 10.1016/S0362-546X(00)00165-6.

[37]

L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Review, 38 (1996), 49-95. doi: 10.1137/1038003.

[38]

Z. WenC. YangX. Liu and Y. Zhang, Trace-penalty minimization for large-scale eigenspace computation, J. Sci. Comput., 66 (2016), 1175-1203. doi: 10.1007/s10915-015-0061-0.

[39]

Z. ZhaoB. BraamsM. FukudaM. Overton and J. Percus, The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions, Journal of Chemical Physics, 120 (2004), 2095-2104. doi: 10.1063/1.1636721.

show all references

References:
[1]

R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Applied Mathematical Sciences Volume 75, Springer-Verlag, 1988. doi: 10.1007/978-1-4612-1029-0.

[2]

P. ApkarianD. NollJ.-B. Thevenet and H. D. Tuan, A spectral quadratic-SDP method with applications to fixed-order $H_2$ and $H_{\infty}$ synthesis, European Journal of Control, 10 (2004), 527-538. doi: 10.3166/ejc.10.527-538.

[3]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1394-9.

[4]

S. Cox and R. Lipton, Extremal eigenvalue problems for two-phase conductors, Archive for Rational Mechanics and Analysis, 136 (1996), 101-117. doi: 10.1007/BF02316974.

[5]

J. CullumW. E. Donath and P. Wolfe, The minimization of certain nondifferentiable sums of eigenvalues of symmetric matrices, Mathematical Programming Study, 3 (1975), 35-55. doi: 10.1007/bfb0120698.

[6]

A. R. Díaz and N. Kikuchi, Solutions to shape and topology eigenvalue optimization problems using a homogenization method, International Journal for Numerical Methods in Engineering, 35 (1992), 1487-1502. doi: 10.1002/nme.1620350707.

[7]

R. Fletcher, Semi-definite matrix constraints in optimization, SIAM Journal on Control and Optimization, 23 (1985), 493-513. doi: 10.1137/0323032.

[8]

C. HelmbergF. Rendl and R. Weismantel, A semidefinite programming approach to the quadratic knapsack problem, Journal of Combinatorial Optimization, 4 (2000), 197-215. doi: 10.1023/A:1009898604624.

[9]

C. HelmbergM. L. Overton and F. Rendl, The spectral bundle method with second-order information, Optimization Methods and Software, 29 (2014), 855-876. doi: 10.1080/10556788.2013.858155.

[10]

M. HuangL. P. Pang and Z. Q. Xia, The space decomposition theory for a class of eigenvalue optimizations, Computational Optimization and Applications, 58 (2014), 423-454. doi: 10.1007/s10589-013-9624-x.

[11]

M. Huang, L. P. Pang, X. J. Liang and Z. Q. Xia, The space decomposition theory for a class of semi-infinite maximum eigenvalue optimizations, Abstract and Applied Analysis, 2014, Article ID 845017, 12 pages. doi: 10.1155/2014/845017.

[12]

M. HuangL. P. PangX. J. Liang and F. Y. Meng, A second-order bundle method based on $\mathcal {VU}$-decomposition strategy for a special class of eigenvalue optimizations, Numerical Functional Analysis and Optimization, 37 (2016), 554-582. doi: 10.1080/01630563.2016.1138969.

[13]

M. HuangX. J. LiangL. P. Pang and Y. Lu, The bundle scheme for solving arbitrary eigenvalue optimizations, Journal of Industrial and Management Optimization, 13 (2017), 659-680. doi: 10.3934/jimo.2016039.

[14]

M. HuangL. P. PangY. Lu and Z. Q. Xia, A fast space-decomposition scheme for nonconvex eigenvalue optimization, Set-Valued and Variational Analysis, 25 (2017), 43-67. doi: 10.1007/s11228-016-0365-8.

[15]

M. HuangY. LuL. P. Pang and Z. Q. Xia, A space decomposition scheme for maximum eigenvalue functions and its applications, Mathematical Methods of Operations Research, 85 (2017), 453-490. doi: 10.1007/s00186-017-0579-z.

[16]

J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I-II, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02796-7.

[17]

J.-B. Hiriart-Urruty and D. Ye, Sensitivity analysis of all eigenvalues of a symmetric matrix, Numerische Mathematik, 70 (1995), 45-72. doi: 10.1007/s002110050109.

[18]

C. Kan and W. Song, Second-order conditions for existence of augmented lagrange multipliers for eigenvalue composite optimization problems, Journal of Global Optimization, 63 (2015), 77-97. doi: 10.1007/s10898-015-0273-8.

[19]

C. LemaréchalF. Oustry and C. Sagastizábal, The ${\mathcal U}$-Lagrangian of a convex function, Transactions of the American Mathematical Society, 352 (2000), 711-729. doi: 10.1090/S0002-9947-99-02243-6.

[20]

A. S. Lewis and M. L. Overton, Eigenvalue optimization, Acta Numerica, 5 (1996), 149-190. doi: 10.1017/S0962492900002646.

[21]

X. LiuX. WangZ. Wen and Y. Yuan, On the Convergence of the Self-Consistent Field Iteration in Kohn-Sham Density Functional Theory, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 546-558. doi: 10.1137/130911032.

[22]

X. LiuZ. WenX. WangM. Ulbrich and Y. Yuan, On the analysis of the discretized kohn-sham density functional theory, SIAM Journal on Numerical Analysis, 53 (2015), 1758-1785. doi: 10.1137/140957962.

[23]

R. Mifflin and C. Sagastizábal, A $\mathcal {VU}$-algorithm for convex minimization, Mathematical Programming Ser.B, 104 (2005), 583-608. doi: 10.1007/s10107-005-0630-3.

[24]

D. Noll and P. Apkarian, Spectral bundle method for nonconvex maximum eigenvalue functions: Second-order methods, Mathematical Programming Ser. B, 104 (2005), 729-747. doi: 10.1007/s10107-005-0635-y.

[25]

D. NollM. Torki and P. Apkarian, Partially augmented Lagrangian method for matrix inequality constraints, SIAM Journal on Optimization, 15 (2004), 161-184. doi: 10.1137/S1052623402413963.

[26]

F. Oustry, The ${\mathcal U}$-Lagrangian of the maximum eigenvalue functions, SIAM Journal on Optimization, 9 (1999), 526-549. doi: 10.1137/S1052623496311776.

[27]

M. L. Overton, Large-scale optimization of eigenvalues, SIAM Journal on Optimization, 2 (1992), 88-120. doi: 10.1137/0802007.

[28]

M. L. Overton and R. S. Womersley, Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices, Mathematical Programming, 62 (1993), 321-357. doi: 10.1007/BF01585173.

[29]

M. L. Overton and R. S. Womersley, Second derivatives for optimizing eigenvalues of symmetric matrices, SIAM Journal on Matrix Analysis and Applications, 16 (1995), 697-718. doi: 10.1137/S089547989324598X.

[30]

M. L. Overton and X. Ye, Towards second-order methods for structured nonsmooth optimization, In: S. Gomez, J.-P. Hennart eds., Advances in Optimization and Numerical Analysis, 97–109, Volume 275 of the series Mathematics and Its Applications, Kluwer Academic Publishers, Norwell, MA, 1994. doi: 10.1007/978-94-015-8330-5_7.

[31]

G. Pataki, On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues, Mathematics of Operations Research, 23 (1998), 339-358. doi: 10.1287/moor.23.2.339.

[32] R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1997.
[33]

A. Shapiro and M. K. H. Fan, On eigenvalue optimization, SIAM Journal on Optimization, 5 (1995), 552-569. doi: 10.1137/0805028.

[34]

A. Shapiro, First and second order analysis of nonlinear semidefinite programs, Mathematical Programming, 77 (1997), 301-320. doi: 10.1007/BF02614439.

[35]

M. Torki, First- and second-order epi-differentiability in eigenvalue optimization, Journal of Mathematical Analysis and Applications, 234 (1999), 391-416. doi: 10.1006/jmaa.1999.6320.

[36]

M. Torki, Second-order directional derivatives of all eigenvalues of a symmetric matrix, Nonlinear Analysis. Theory, Methods & Applications, 46 (2001), 1133-1150. doi: 10.1016/S0362-546X(00)00165-6.

[37]

L. Vandenberghe and S. Boyd, Semidefinite programming, SIAM Review, 38 (1996), 49-95. doi: 10.1137/1038003.

[38]

Z. WenC. YangX. Liu and Y. Zhang, Trace-penalty minimization for large-scale eigenspace computation, J. Sci. Comput., 66 (2016), 1175-1203. doi: 10.1007/s10915-015-0061-0.

[39]

Z. ZhaoB. BraamsM. FukudaM. Overton and J. Percus, The reduced density matrix method for electronic structure calculations and the role of three-index representability conditions, Journal of Chemical Physics, 120 (2004), 2095-2104. doi: 10.1063/1.1636721.

[1]

Ekta Mittal, Sunil Joshi. Note on a $ k $-generalised fractional derivative. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 797-804. doi: 10.3934/dcdss.2020045

[2]

Yong Xia, Ruey-Lin Sheu, Shu-Cherng Fang, Wenxun Xing. Double well potential function and its optimization in the $N$ -dimensional real space-part Ⅱ. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1307-1328. doi: 10.3934/jimo.2016074

[3]

Shu-Cherng Fang, David Y. Gao, Gang-Xuan Lin, Ruey-Lin Sheu, Wenxun Xing. Double well potential function and its optimization in the $N$ -dimensional real space-part Ⅰ. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1291-1305. doi: 10.3934/jimo.2016073

[4]

Gang Wang, Yuan Zhang. $ Z $-eigenvalue exclusion theorems for tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2019039

[5]

Caili Sang, Zhen Chen. $ E $-eigenvalue localization sets for tensors. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-19. doi: 10.3934/jimo.2019042

[6]

Lin Du, Yun Zhang. $\mathcal{H}_∞$ filtering for switched nonlinear systems: A state projection method. Journal of Industrial & Management Optimization, 2018, 14 (1) : 19-33. doi: 10.3934/jimo.2017035

[7]

Monica Motta, Caterina Sartori. On ${\mathcal L}^1$ limit solutions in impulsive control. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1201-1218. doi: 10.3934/dcdss.2018068

[8]

Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116

[9]

Alexander Alekseenko, Jeffrey Limbacher. Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $ \mathcal{O}(N^2) $ operations using the discrete fourier transform. Kinetic & Related Models, 2019, 12 (4) : 703-726. doi: 10.3934/krm.2019027

[10]

VicenŢiu D. RǍdulescu, Somayeh Saiedinezhad. A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition. Communications on Pure & Applied Analysis, 2018, 17 (1) : 39-52. doi: 10.3934/cpaa.2018003

[11]

Peter Benner, Ryan Lowe, Matthias Voigt. $\mathcal{L}_{∞}$-norm computation for large-scale descriptor systems using structured iterative eigensolvers. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 119-133. doi: 10.3934/naco.2018007

[12]

Minjia Shi, Yaqi Lu. Cyclic DNA codes over $ \mathbb{F}_2[u,v]/\langle u^3, v^2-v, vu-uv\rangle$. Advances in Mathematics of Communications, 2019, 13 (1) : 157-164. doi: 10.3934/amc.2019009

[13]

Peng Mei, Zhan Zhou, Genghong Lin. Periodic and subharmonic solutions for a 2$n$th-order $\phi_c$-Laplacian difference equation containing both advances and retardations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 2085-2095. doi: 10.3934/dcdss.2019134

[14]

Abdelwahab Bensouilah, Sahbi Keraani. Smoothing property for the $ L^2 $-critical high-order NLS Ⅱ. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2961-2976. doi: 10.3934/dcds.2019123

[15]

Ali Hyder, Juncheng Wei. Higher order conformally invariant equations in $ {\mathbb R}^3 $ with prescribed volume. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2757-2764. doi: 10.3934/cpaa.2019123

[16]

Sugata Gangopadhyay, Goutam Paul, Nishant Sinha, Pantelimon Stǎnicǎ. Generalized nonlinearity of $ S$-boxes. Advances in Mathematics of Communications, 2018, 12 (1) : 115-122. doi: 10.3934/amc.2018007

[17]

Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129

[18]

Haisheng Tan, Liuyan Liu, Hongyu Liang. Total $\{k\}$-domination in special graphs. Mathematical Foundations of Computing, 2018, 1 (3) : 255-263. doi: 10.3934/mfc.2018011

[19]

Zalman Balanov, Yakov Krasnov. On good deformations of $ A_m $-singularities. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1851-1866. doi: 10.3934/dcdss.2019122

[20]

Pak Tung Ho. Prescribing the $ Q' $-curvature in three dimension. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2285-2294. doi: 10.3934/dcds.2019096

2017 Impact Factor: 0.994

Article outline

[Back to Top]