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April  2017, 13(2): 659-680. doi: 10.3934/jimo.2016039

The bundle scheme for solving arbitrary eigenvalue optimizations

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.

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. AlizadehJ.-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. 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

[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. FriedlandJ. 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. 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

[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. KojimaM. 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. SunS. BoydL. 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. AlizadehJ.-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. 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

[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. FriedlandJ. 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. 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

[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. KojimaM. 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. SunS. BoydL. 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
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
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]

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