American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Algorithms for single-machine scheduling problem with deterioration depending on a novel model
  • JIMO Home
  • This Issue
  • Next Article
    A class of descent four–term extension of the Dai–Liao conjugate gradient method based on the scaled memoryless BFGS update
April  2017, 13(2): 659-680. doi: 10.3934/jimo.2016039

The bundle scheme for solving arbitrary eigenvalue optimizations

Ming Huang 1,2, , Xi-Jun Liang 3, , Yuan Lu 4, and  Li-Ping Pang 5,,

1. 

Department of Mathematics, Dalian Maritime University, Dalian 116026, China
2. 

School of Control Science and Engineering, Dalian University of Technology, Dalian 116024, China
3. 

College of Science, China University of Petroleum, Qingdao 266580, China
4. 

School of Sciences, Shenyang University, Shenyang 110044, China
5. 

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

* Corresponding author

Received  April 2015 Revised  January 2016 Published  May 2016

Fund Project: The first and forth authors' work was supported in part by the National Natural Science Foundation of China under projects No.11171049. The first author was supported in part by National Natural Science Foundation of China under projects No.11626053, the Project funded by China Postdoctoral Science Fundation under No. 2016M601296, Fundamental Research Funds for the Central Universities under projects No.3132016108 and Scientific Research Foundation Funds of DLMU under projects No.02501102. The second author' work was supported in part by the National Natural Science Foundation of China under projects No. 61503412 and Natural Science Foundation of Shandong Province under Grant No. ZR2014AP004. The third author' work was supported in part by the National Natural Science Foundation of China under projects No.11301347, and Program for Liaoning Excellent Talents in University under projects No. LJQ2015075.

Optimization involving eigenvalues arises in a large spectrum of applications in various domains, such as physics, engineering, statistics and finance. In this paper, we consider the arbitrary eigenvalue minimization problems over an affine family of symmetric matrices, which is a special class of eigenvalue function--D.C. function $λ^_{l}$. An explicit proximal bundle approach for solving this class of nonsmooth, nonconvex (D.C.) optimization problem is developed. We prove the global convergence of our method, in the sense that every accumulation point of the sequence of iterates generated by the proposed algorithm is stationary. As an application, we use the proposed bundle method to solve some particular eigenvalue problems. Some encouraging preliminary numerical results indicate that our method can solve the test problems efficiently.

Keywords: Eigenvalue optimization, nonsmooth optimization, D.C. function, nonconvex optimization, first-order proximal bundle method.
Mathematics Subject Classification: Primary: 49J52, 15A18; Secondary: 90C26, 65K10.
Citation: Ming Huang, Xi-Jun Liang, Yuan Lu, Li-Ping Pang. The bundle scheme for solving arbitrary eigenvalue optimizations. Journal of Industrial & Management Optimization, 2017, 13 (2) : 659-680. doi: 10.3934/jimo.2016039
References:
[1]

F. Alizadeh, J.-P. A. Haeberly and M. L. Overton, Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results, SIAM Journal on Optimization, 8 (1998), 746-768.  doi: 10.1137/S1052623496304700.  Google Scholar

[2]

F. Alizadeh, Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM Journal on Optimization, 5 (1995), 13-51.  doi: 10.1137/0805002.  Google Scholar

[3]

R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics, Vol. 169, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0653-8.  Google Scholar

[4]

F. H. Clarke, Optimization and Nonsmooth Analysis, Classics in Applied Mathematics, Vol. 5, Society for Industry and Applied Mathematics (SIAM), Philadelphia, 1990. doi: 10.1137/1.9781611971309.  Google Scholar

[5]

F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, Vol. 178, Springer-Verlag, New York, 1998. doi: 10.1007/b97650.  Google Scholar

[6]

J. Cullum, W. 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

[7]

K. Fan, On a theorem of Weyl concerning eigenvalues of linear transformations. Ⅰ., Proc. Nat. Acad. Sci. U.S.A., 35 (1949), 652-655.  doi: 10.1073/pnas.35.11.652.  Google Scholar

[8]

S. Friedland, J. Nocedal and M. L. Overton, The formulation and analysis of numerical methods for inverse eigenvalue problems, SIAM Journal on Numerical Analysis, 24 (1987), 634-667.  doi: 10.1137/0724043.  Google Scholar

[9]

M. Huang, L. 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

[10]

C. Helmberg and F. Rendl, A spectral bundle method for semidefinite programming, SIAM Journal on Optimization, 10 (2000), 673-696.  doi: 10.1137/S1052623497328987.  Google Scholar

[11]

J. -B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms Ⅰ-Ⅱ, Grundlehren der mathematischen Wissenschaften, Vols 305-306, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02796-7.  Google Scholar

[12]

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

[13]

F. Jarre, An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices, SIAM Journal on Control and Optimization, 31 (1993), 1360-1377.  doi: 10.1137/0331064.  Google Scholar

[14]

M. Kojima, M. Shida and S. Shindoh, Local convergence of predictor-corrector infeasible-interior-point algorithms for SDPs and SDLCPs, Mathematical Programming, 80 (1998), 129-160.  doi: 10.1007/BF01581723.  Google Scholar

[15]

K. C. Kiwiel, An aggregate subgradient method for nonsmooth convex minimization, Mathematical Programming, 27 (1983), 320-341.  doi: 10.1007/BF02591907.  Google Scholar

[16]

K. C. Kiwiel, A linearization algorithm for nonsmooth minimization, Mathematics of Operations Research, 10 (1985), 185-194.  doi: 10.1287/moor.10.2.185.  Google Scholar

[17]

K. C. Kiwiel, Proximity control in bundle methods for convex nondifferentiable minimization, Mathematical Programming, 46 (1990), 105-122.  doi: 10.1007/BF01585731.  Google Scholar

[18]

C. Lemaréchal, An extension of davidon methods to nondifferentiable problems, Mathematical Programming Study, 3 (2009), 95-109.  doi: 10.1007/BFb0120700.  Google Scholar

[19]

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

[20]

R. Mifflin, An algorithm for constrained optimization with semismooth functions, Mathematics of Operations Research, 2 (1977), 191-207.  doi: 10.1287/moor.2.2.191.  Google Scholar

[21]

Y. Nesterov, Interior-point methods: An old and new approach to nonlinear programming, Mathematical Programming, 79 (1997), 285-297.  doi: 10.1007/BF02614321.  Google Scholar

[22]

Y. Nesterov and A. Nemirovskii, A General Approach to Polynomial-Time Algorithms Design for Convex Programming, Technical report, Centr. Econ. & Math. Inst., USSR Academy of Sciences, Moscow, USSR, 1988. Google Scholar

[23]

Y. Nesterov and A. Nemirovskii, Interior-Point Polynomial Algorithms in Convex Programming, SIAM Studies in Applied and Numerical Mathematics, Vol. 13, Philadelphia, 1994. doi: 10.1137/1.9781611970791.  Google Scholar

[24]

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

[25]

M. L. Overton, On minimizing the maximum eigenvalue of a symmetric matrix, SIAM Journal on Matrix Analysis and Applications, 9 (1988), 256-268.  doi: 10.1137/0609021.  Google Scholar

[26]

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

[27]

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

[28]

E. Polak, On the mathematical foundations of nondifferentiable optimization in engineering design, SIAM Review, 29 (1987), 21-89.  doi: 10.1137/1029002.  Google Scholar

[29]

E. Polak and Y. Wardi, Nondifferentiable optimization algorithm for designing control systems having singular value inequalities, Automatica, 18 (1982), 267-283.  doi: 10.1016/0005-1098(82)90087-5.  Google Scholar

[30]

S. B. Robinson, On the second eigenvalue for nonhomogeneous quasi-linear operators, SIAM Journal on Mathematical Analysis, 35 (2004), 1241-1249.  doi: 10.1137/S0036141003426008.  Google Scholar

[31]

R. T. Rockafellar, Convex Analysis, Princeton University Press, NJ, 1997.  Google Scholar

[32]

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

[33]

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

[34]

J. Sun, S. Boyd, L. Xiao and P. Diaconis, The fastest mixing markov process on a graph and a connection to a maximum variance unfolding problem, SIAM Review, 48 (2006), 681-699.  doi: 10.1137/S0036144504443821.  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]

J. Vlček and L. Lukšan, Globally convergent variable metric method for nonconvex nondifferentiable unconstrained minimization, Journal of Optimization Theory and Applications, 111 (2001), 407-430.  doi: 10.1023/A:1011990503369.  Google Scholar

[39]

P. Wolfe, A method of conjugate subgradients for minimizing nondifferentiable functions, Mathematical Programming Study, 3 (1975), 145-173.  doi: 10.1007/BFb0120703.  Google Scholar

show all references

References:
[1]

F. Alizadeh, J.-P. A. Haeberly and M. L. Overton, Primal-dual interior-point methods for semidefinite programming: Convergence rates, stability and numerical results, SIAM Journal on Optimization, 8 (1998), 746-768.  doi: 10.1137/S1052623496304700.  Google Scholar

[2]

F. Alizadeh, Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM Journal on Optimization, 5 (1995), 13-51.  doi: 10.1137/0805002.  Google Scholar

[3]

R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics, Vol. 169, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0653-8.  Google Scholar

[4]

F. H. Clarke, Optimization and Nonsmooth Analysis, Classics in Applied Mathematics, Vol. 5, Society for Industry and Applied Mathematics (SIAM), Philadelphia, 1990. doi: 10.1137/1.9781611971309.  Google Scholar

[5]

F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Graduate Texts in Mathematics, Vol. 178, Springer-Verlag, New York, 1998. doi: 10.1007/b97650.  Google Scholar

[6]

J. Cullum, W. 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

[7]

K. Fan, On a theorem of Weyl concerning eigenvalues of linear transformations. Ⅰ., Proc. Nat. Acad. Sci. U.S.A., 35 (1949), 652-655.  doi: 10.1073/pnas.35.11.652.  Google Scholar

[8]

S. Friedland, J. Nocedal and M. L. Overton, The formulation and analysis of numerical methods for inverse eigenvalue problems, SIAM Journal on Numerical Analysis, 24 (1987), 634-667.  doi: 10.1137/0724043.  Google Scholar

[9]

M. Huang, L. 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

[10]

C. Helmberg and F. Rendl, A spectral bundle method for semidefinite programming, SIAM Journal on Optimization, 10 (2000), 673-696.  doi: 10.1137/S1052623497328987.  Google Scholar

[11]

J. -B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms Ⅰ-Ⅱ, Grundlehren der mathematischen Wissenschaften, Vols 305-306, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02796-7.  Google Scholar

[12]

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

[13]

F. Jarre, An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices, SIAM Journal on Control and Optimization, 31 (1993), 1360-1377.  doi: 10.1137/0331064.  Google Scholar

[14]

M. Kojima, M. Shida and S. Shindoh, Local convergence of predictor-corrector infeasible-interior-point algorithms for SDPs and SDLCPs, Mathematical Programming, 80 (1998), 129-160.  doi: 10.1007/BF01581723.  Google Scholar

[15]

K. C. Kiwiel, An aggregate subgradient method for nonsmooth convex minimization, Mathematical Programming, 27 (1983), 320-341.  doi: 10.1007/BF02591907.  Google Scholar

[16]

K. C. Kiwiel, A linearization algorithm for nonsmooth minimization, Mathematics of Operations Research, 10 (1985), 185-194.  doi: 10.1287/moor.10.2.185.  Google Scholar

[17]

K. C. Kiwiel, Proximity control in bundle methods for convex nondifferentiable minimization, Mathematical Programming, 46 (1990), 105-122.  doi: 10.1007/BF01585731.  Google Scholar

[18]

C. Lemaréchal, An extension of davidon methods to nondifferentiable problems, Mathematical Programming Study, 3 (2009), 95-109.  doi: 10.1007/BFb0120700.  Google Scholar

[19]

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

[20]

R. Mifflin, An algorithm for constrained optimization with semismooth functions, Mathematics of Operations Research, 2 (1977), 191-207.  doi: 10.1287/moor.2.2.191.  Google Scholar

[21]

Y. Nesterov, Interior-point methods: An old and new approach to nonlinear programming, Mathematical Programming, 79 (1997), 285-297.  doi: 10.1007/BF02614321.  Google Scholar

[22]

Y. Nesterov and A. Nemirovskii, A General Approach to Polynomial-Time Algorithms Design for Convex Programming, Technical report, Centr. Econ. & Math. Inst., USSR Academy of Sciences, Moscow, USSR, 1988. Google Scholar

[23]

Y. Nesterov and A. Nemirovskii, Interior-Point Polynomial Algorithms in Convex Programming, SIAM Studies in Applied and Numerical Mathematics, Vol. 13, Philadelphia, 1994. doi: 10.1137/1.9781611970791.  Google Scholar

[24]

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

[25]

M. L. Overton, On minimizing the maximum eigenvalue of a symmetric matrix, SIAM Journal on Matrix Analysis and Applications, 9 (1988), 256-268.  doi: 10.1137/0609021.  Google Scholar

[26]

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

[27]

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

[28]

E. Polak, On the mathematical foundations of nondifferentiable optimization in engineering design, SIAM Review, 29 (1987), 21-89.  doi: 10.1137/1029002.  Google Scholar

[29]

E. Polak and Y. Wardi, Nondifferentiable optimization algorithm for designing control systems having singular value inequalities, Automatica, 18 (1982), 267-283.  doi: 10.1016/0005-1098(82)90087-5.  Google Scholar

[30]

S. B. Robinson, On the second eigenvalue for nonhomogeneous quasi-linear operators, SIAM Journal on Mathematical Analysis, 35 (2004), 1241-1249.  doi: 10.1137/S0036141003426008.  Google Scholar

[31]

R. T. Rockafellar, Convex Analysis, Princeton University Press, NJ, 1997.  Google Scholar

[32]

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

[33]

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

[34]

J. Sun, S. Boyd, L. Xiao and P. Diaconis, The fastest mixing markov process on a graph and a connection to a maximum variance unfolding problem, SIAM Review, 48 (2006), 681-699.  doi: 10.1137/S0036144504443821.  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]

J. Vlček and L. Lukšan, Globally convergent variable metric method for nonconvex nondifferentiable unconstrained minimization, Journal of Optimization Theory and Applications, 111 (2001), 407-430.  doi: 10.1023/A:1011990503369.  Google Scholar

[39]

P. Wolfe, A method of conjugate subgradients for minimizing nondifferentiable functions, Mathematical Programming Study, 3 (1975), 145-173.  doi: 10.1007/BFb0120703.  Google Scholar

Table 1.  Numerical results for the second largest eigenvalue
n $n_n$ $n_d$ $\#f/g$ $x^k$ Time $fx^k$ Res
2 11 22 252 $1.0e-06 *(0.4508,0.0685)$ 0.6093 2.8348e-07 4.3320e-07
3 15 25 308 $1.0e-06 *(0.2321,0.3796)$ 0.8073 1.9516e-07 2.9823e-07
4 11 24 270 $1.0e-06 *(0.3036,0.4796)$ 0.7225 2.4712e-07 3.7764e-07
5 17 21 288 $1.0e-06 *(0.4569,0.7587)$ 0.8452 3.8994e-07 5.9589e-07
6 12 23 276 $1.0e-06 *(0.3141,0.2245)$ 0.6242 2.0441e-07 3.1237e-07
7 27 24 266 $1.0e-06 *(0.3944,0.0098)$ 0.8933 2.2090e-07 3.3757e-07
8 15 23 260 $1.0e-06 *(0.3177,0.5355)$ 0.6810 2.7525e-07 4.2062e-07
9 13 24 284 $1.0e-06 *(0.4139,0.0045)$ 0.7899 2.2820e-07 3.4873e-07
10 15 22 302 $1.0e-06 *(0.0770,0.2213)$ 0.7927 1.2711e-07 1.9425e-07
11 16 26 256 $1.0e-06 *(0.1415,0.1247)$ 0.9039 9.1817e-08 1.4031e-07
12 15 24 274 $1.0e-06 *(0.3637,0.0449)$ 0.6943 2.2398e-07 3.4228e-07
13 14 23 270 $1.0e-06 *(0.3033,0.4745)$ 0.7462 2.4475e-07 3.7402e-07
14 18 25 292 $1.0e-06 *(0.1657,0.1121)$ 0.9307 1.0812e-07 1.6522e-07
15 15 22 268 $1.0e-06 *(0.4380,0.0246)$ 0.7305 2.5366e-07 3.8764e-07
16 13 24 284 $1.0e-06 *(0.0836,0.2209)$ 0.7983 1.2407e-07 1.8960e-07
17 14 23 270 $1.0e-06 *(0.3016,0.4746)$ 0.7458 2.4466e-07 3.7388e-07
18 22 16 242 $1.0e-06 *(0.5804,0.7083)$ 0.5951 3.9904e-07 6.0980e-07
19 28 23 354 $1.0e-06 *(0.5475,0.7792)$ 1.1864 4.0963e-07 6.2599e-07
20 17 24 288 $1.0e-06 *(0.3219,0.5632)$ 0.7114 2.8989e-07 4.4300e-07
Table Options

Download as excel

n $n_n$ $n_d$ $\#f/g$ $x^k$ Time $fx^k$ Res
2 11 22 252 $1.0e-06 *(0.4508,0.0685)$ 0.6093 2.8348e-07 4.3320e-07
3 15 25 308 $1.0e-06 *(0.2321,0.3796)$ 0.8073 1.9516e-07 2.9823e-07
4 11 24 270 $1.0e-06 *(0.3036,0.4796)$ 0.7225 2.4712e-07 3.7764e-07
5 17 21 288 $1.0e-06 *(0.4569,0.7587)$ 0.8452 3.8994e-07 5.9589e-07
6 12 23 276 $1.0e-06 *(0.3141,0.2245)$ 0.6242 2.0441e-07 3.1237e-07
7 27 24 266 $1.0e-06 *(0.3944,0.0098)$ 0.8933 2.2090e-07 3.3757e-07
8 15 23 260 $1.0e-06 *(0.3177,0.5355)$ 0.6810 2.7525e-07 4.2062e-07
9 13 24 284 $1.0e-06 *(0.4139,0.0045)$ 0.7899 2.2820e-07 3.4873e-07
10 15 22 302 $1.0e-06 *(0.0770,0.2213)$ 0.7927 1.2711e-07 1.9425e-07
11 16 26 256 $1.0e-06 *(0.1415,0.1247)$ 0.9039 9.1817e-08 1.4031e-07
12 15 24 274 $1.0e-06 *(0.3637,0.0449)$ 0.6943 2.2398e-07 3.4228e-07
13 14 23 270 $1.0e-06 *(0.3033,0.4745)$ 0.7462 2.4475e-07 3.7402e-07
14 18 25 292 $1.0e-06 *(0.1657,0.1121)$ 0.9307 1.0812e-07 1.6522e-07
15 15 22 268 $1.0e-06 *(0.4380,0.0246)$ 0.7305 2.5366e-07 3.8764e-07
16 13 24 284 $1.0e-06 *(0.0836,0.2209)$ 0.7983 1.2407e-07 1.8960e-07
17 14 23 270 $1.0e-06 *(0.3016,0.4746)$ 0.7458 2.4466e-07 3.7388e-07
18 22 16 242 $1.0e-06 *(0.5804,0.7083)$ 0.5951 3.9904e-07 6.0980e-07
19 28 23 354 $1.0e-06 *(0.5475,0.7792)$ 1.1864 4.0963e-07 6.2599e-07
20 17 24 288 $1.0e-06 *(0.3219,0.5632)$ 0.7114 2.8989e-07 4.4300e-07
Table 2.  Comparison results of Algorithm 4.1 and Generalized cutting plane method (GCPM)
Algorithm 4.1 GCPM
n $\#f/g$ Time Res $\#f/g$ Time Res
5 214 0.6762 4.7140e-06 338 1.4614 4.7709e-06
10 246 0.8603 4.4696e-06 308 1.2572 4.8573e-06
15 234 0.7433 3.8621e-06 420 1.5076 4.8546e-06
20 266 0.8157 4.2591e-06 418 1.4537 4.6413e-06
25 235 0.7141 3.8623e-06 398 1.4743 4.7922e-06
30 290 0.7934 4.2803e-06 406 1.3006 5.4616e-06
35 258 0.7645 4.1979e-06 414 1.2542 4.8415e-06
40 290 0.8635 4.4795e-06 398 1.3216 5.8507e-06
45 238 0.7330 3.9895e-06 566 1.9189 4.5721e-06
50 282 0.8268 4.5536e-06 532 1.5479 6.6442e-06
Table Options

Download as excel

Algorithm 4.1 GCPM
n $\#f/g$ Time Res $\#f/g$ Time Res
5 214 0.6762 4.7140e-06 338 1.4614 4.7709e-06
10 246 0.8603 4.4696e-06 308 1.2572 4.8573e-06
15 234 0.7433 3.8621e-06 420 1.5076 4.8546e-06
20 266 0.8157 4.2591e-06 418 1.4537 4.6413e-06
25 235 0.7141 3.8623e-06 398 1.4743 4.7922e-06
30 290 0.7934 4.2803e-06 406 1.3006 5.4616e-06
35 258 0.7645 4.1979e-06 414 1.2542 4.8415e-06
40 290 0.8635 4.4795e-06 398 1.3216 5.8507e-06
45 238 0.7330 3.9895e-06 566 1.9189 4.5721e-06
50 282 0.8268 4.5536e-06 532 1.5479 6.6442e-06
[1]

Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028
[2]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023
[3]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[4]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024
[5]

Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021008
[6]

Reza Lotfi, Yahia Zare Mehrjerdi, Mir Saman Pishvaee, Ahmad Sadeghieh, Gerhard-Wilhelm Weber. A robust optimization model for sustainable and resilient closed-loop supply chain network design considering conditional value at risk. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 221-253. doi: 10.3934/naco.2020023
[7]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184
[8]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212
[9]

Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems & Imaging, 2012, 6 (4) : 565-598. doi: 10.3934/ipi.2012.6.565
[10]

Nikolaz Gourmelon. Generation of homoclinic tangencies by $C^1$-perturbations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 1-42. doi: 10.3934/dcds.2010.26.1
[11]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024
[12]

Meiqiao Ai, Zhimin Zhang, Wenguang Yu. First passage problems of refracted jump diffusion processes and their applications in valuing equity-linked death benefits. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021039
[13]

Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709
[14]

Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129
[15]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397
[16]

Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[17]

Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133
[18]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597
[19]

Ralf Hielscher, Michael Quellmalz. Reconstructing a function on the sphere from its means along vertical slices. Inverse Problems & Imaging, 2016, 10 (3) : 711-739. doi: 10.3934/ipi.2016018
[20]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (85)
  • HTML views (368)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences