July  2020, 16(4): 1885-1905. 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 research is supported by the National Natural Science Foundation of China under Projects No.11701063, 11501080, 11626053, 71831002; the Innovative Research Groups of the National Natural Science Foundation of China No.71621061 and the 111 Project No.B16009; the Project funded by China Postdoctoral Science Foundation under No.2016M601296; the Fundamental Research Funds for the Central Universities under Projects No.3132017053, 3132018215, 3132019179, 3132019501 and 3132019502; the Program for Innovative Research Team in University No.IRT-17R13.

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, 2020, 16 (4) : 1885-1905. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

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 (2014), Article ID 845017, 12 pages. doi: 10.1155/2014/845017.  Google Scholar

[14]

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

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[10]

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.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

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 (2014), Article ID 845017, 12 pages. doi: 10.1155/2014/845017.  Google Scholar

[14]

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

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[37]

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

[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.  Google Scholar

[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.  Google Scholar

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