January  2021, 17(1): 101-116. doi: 10.3934/jimo.2019101

A diagonal PRP-type projection method for convex constrained nonlinear monotone equations

Department of Mathematical Sciences, Faculty of Physical Sciences, Bayero University, Kano, Kano, 700241, Nigeria

* Corresponding author: Hassan Mohammad

Received  April 2018 Revised  May 2019 Published  September 2019

Iterative methods for nonlinear monotone equations do not require the differentiability assumption on the residual function. This special property of the methods makes them suitable for solving large-scale nonsmooth monotone equations. In this work, we present a diagonal Polak-Ribi$ \grave{e} $re-Polyak (PRP) conjugate gradient-type method for solving large-scale nonlinear monotone equations with convex constraints. The search direction is a combine form of a multivariate (diagonal) spectral method and a modified PRP conjugate gradient method. Proper safeguards are devised to ensure positive definiteness of the diagonal matrix associated with the search direction. Based on Lipschitz continuity and monotonicity assumptions the method is shown to be globally convergent. Numerical results are presented by means of comparative experiments with recently proposed multivariate spectral Dai-Yuan-type (J. Ind. Manag. Optim. 13 (2017) 283-295) and Wei-Yao-Liu-type (Int. J. Comput. Math. 92 (2015) 2261-2272) conjugate gradient methods.

Citation: Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101
References:
[1]

A. B. Abubakar and P. Kumam, An improved three-term derivative-free method for solving nonlinear equations, Comput. Appl. Math., 37 (2018), 6760-6773.  doi: 10.1007/s40314-018-0712-5.  Google Scholar

[2]

A. B. Abubakar and P. Kumam, A descent Dai-Liao conjugate gradient method for nonlinear equations, Numer. Algorithms, 81 (2019), 197-210.  doi: 10.1007/s11075-018-0541-z.  Google Scholar

[3]

Y. Bing and G. Lin, An efficient implementation of Merrill's method for sparse or partially separable systems of nonlinear equations, SIAM J. Optim., 1 (1991), 206-221.  doi: 10.1137/0801015.  Google Scholar

[4]

W. Cheng, A PRP type method for systems of monotone equations, Math. Comput. Modelling, 50 (2009), 15-20.  doi: 10.1016/j.mcm.2009.04.007.  Google Scholar

[5]

Y. Dai and Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim., 10 (1999), 177-182.  doi: 10.1137/S1052623497318992.  Google Scholar

[6]

E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), 201-213.  doi: 10.1007/s101070100263.  Google Scholar

[7]

X. L. DongH. LiuY. L. Xu and X. M. Yang, Some nonlinear conjugate gradient methods with sufficient descent condition and global convergence, Optim. Lett., 9 (2015), 1421-1432.  doi: 10.1007/s11590-014-0836-5.  Google Scholar

[8]

M. EshaghnezhadS. Effati and A. Mansoori, A neurodynamic model to solve nonlinear pseudo-monotone projection equation and its applications, IEEE Transactions on Cybernetics, 47 (2017), 3050-3062.  doi: 10.1109/TCYB.2016.2611529.  Google Scholar

[9]

M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Math. Programming, 53 (1992), 99-110.  doi: 10.1007/BF01585696.  Google Scholar

[10]

B. GhaddarJ. Marecek and M. Mevissen, Optimal power flow as a polynomial optimization problem, IEEE Transactions on Power Systems, 31 (2016), 539-546.  doi: 10.1109/TPWRS.2015.2390037.  Google Scholar

[11]

B. GuV. S. ShengK. Y. TayW. Romano and S. Li, Incremental support vector learning for ordinal regression, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 1403-1416.  doi: 10.1109/TNNLS.2014.2342533.  Google Scholar

[12]

L. HanG. Yu and L. Guan, Multivariate spectral gradient method for unconstrained optimization, Appl. Math. Comput., 201 (2008), 621-630.  doi: 10.1016/j.amc.2007.12.054.  Google Scholar

[13]

Y. Hu and Z. Wei, Wei–Yao–Liu conjugate gradient projection algorithm for nonlinear monotone equations with convex constraints, Int. J. Comput. Math., 92 (2015), 2261-2272.  doi: 10.1080/00207160.2014.977879.  Google Scholar

[14]

W. La Cruz, A projected derivative-free algorithm for nonlinear equations with convex constraints, Optim. Methods Softw., 29 (2014), 24-41.  doi: 10.1080/10556788.2012.721129.  Google Scholar

[15]

W. La Cruz, A spectral algorithm for large-scale systems of nonlinear monotone equations, Numer. Algorithms, 76 (2017), 1109-1130.  doi: 10.1007/s11075-017-0299-8.  Google Scholar

[16]

W. La CruzJ. Martínez and M. Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Math. Comp., 75 (2006), 1429-1448.  doi: 10.1090/S0025-5718-06-01840-0.  Google Scholar

[17]

J. LiX. LiB. Yang and X. Sun, Segmentation-based image copy-move forgery detection scheme, IEEE Transactions on Information Forensics and Security, 10 (2015), 507-518.   Google Scholar

[18]

Q. Li and D. H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations, IMA J. Numer. Anal., 31 (2011), 1625-1635.  doi: 10.1093/imanum/drq015.  Google Scholar

[19]

J. Liu and X. L. Du, A gradient projection method for the sparse signal reconstruction in compressive sensing, Appl. Anal., 97 (2018), 2122-2131.  doi: 10.1080/00036811.2017.1359556.  Google Scholar

[20]

J. Liu and Y. Duan, Two spectral gradient projection methods for constrained equations and their linear convergence rate, J. Inequal. Appl., 2015 (2015), 13pp. doi: 10.1186/s13660-014-0525-z.  Google Scholar

[21]

J. Liu and Y. Feng, A derivative-free iterative method for nonlinear monotone equations with convex constraints, Numerical Algorithms, 82 (2019), 1-18.  doi: 10.1007/s11075-018-0603-2.  Google Scholar

[22]

J. Liu and S. Li, Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations, J. Ind. Manag. Optim., 13 (2017), 283-295.  doi: 10.3934/jimo.2016017.  Google Scholar

[23]

J. Liu and S. Li, A projection method for convex constrained monotone nonlinear equations with applications, Comput. Math. Appl., 70 (2015), 2442-2453.  doi: 10.1016/j.camwa.2015.09.014.  Google Scholar

[24]

S. Liu, Y. Huang and H. W. Jiao, Sufficient descent conjugate gradient methods for solving convex constrained nonlinear monotone equations, Abstr. Appl. Anal., 2014 (2014), 12pp. doi: 10.1155/2014/305643.  Google Scholar

[25]

F. Ma and C. Wang, Modified projection method for solving a system of monotone equations with convex constraints, J. Appl. Math. Comput., 34 (2010), 47-56.  doi: 10.1007/s12190-009-0305-y.  Google Scholar

[26]

H. Mohammad and A. B. Abubakar, A positive spectral gradient-like method for nonlinear monotone equations, Bull. Comput. Appl. Math., 5 (2017), 99-115.   Google Scholar

[27]

H. Mohammad and S. A. Santos, A structured diagonal Hessian approximation method with evaluation complexity analysis for nonlinear least squares, Comput. Appl. Math., 37 (2018), 6619-6653.  doi: 10.1007/s40314-018-0696-1.  Google Scholar

[28]

Y. Ou and J. Li, A new derivative-free SCG-type projection method for nonlinear monotone equations with convex constraints, J. Appl. Math. Comput., 56 (2018), 195-216.  doi: 10.1007/s12190-016-1068-x.  Google Scholar

[29]

Y. Ou and Y. Liu, Supermemory gradient methods for monotone nonlinear equations with convex constraints, Comput. Appl. Math., 36 (2017), 259-279.  doi: 10.1007/s40314-015-0228-1.  Google Scholar

[30]

Z. Papp and S. Rapajić, FR type methods for systems of large-scale nonlinear monotone equations, Appl. Math. Comput., 269 (2015), 816-823.  doi: 10.1016/j.amc.2015.08.002.  Google Scholar

[31]

E. Polak and G. Ribiere, Note sur la convergence de méthodes de directions conjuguées, Revue française d'informatique et de recherche opérationnelle. Série rouge, 3 (1969), 35–43.  Google Scholar

[32]

B. T. Polyak, The conjugate gradient method in extremal problems, USSR Comp. Math. and Mathem. Physics, 9 (1969), 94-112.  doi: 10.1016/0041-5553(69)90035-4.  Google Scholar

[33]

G. QianD. HanL. Xu and H. Y., Solving nonadditive traffic assignment problems: A self-adaptive projection-auxiliary problem method for variational inequalities, J. Ind. Manag. Optim., 9 (2013), 255-274.  doi: 10.3934/jimo.2013.9.255.  Google Scholar

[34]

M. V. Solodov and B. Svaiter, A globally convergent inexact Newton method for systems of monotone equations, in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Applied Optimization, Springer, 1998,355–369. doi: 10.1007/978-1-4757-6388-1_18.  Google Scholar

[35]

M. Sun and J. Liu, Three derivative-free projection methods for nonlinear equations with convex constraints, J. Appl. Math. Comput., 47 (2015), 265-276.  doi: 10.1007/s12190-014-0774-5.  Google Scholar

[36]

C. Wang and Y. Wang, A superlinearly convergent projection method for constrained systems of nonlinear equations, J. Global Optim., 44 (2009), 283-296.  doi: 10.1007/s10898-008-9324-8.  Google Scholar

[37]

C. WangY. Wang and C. Xu, A projection method for a system of nonlinear monotone equations with convex constraints, Math. Methods Oper. Res., 66 (2007), 33-46.  doi: 10.1007/s00186-006-0140-y.  Google Scholar

[38]

X. WangS. Li and X. Kou, A self-adaptive three-term conjugate gradient method for monotone nonlinear equations with convex constraints, Calcolo, 53 (2016), 133-145.  doi: 10.1007/s10092-015-0140-5.  Google Scholar

[39]

A. J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control, John Wiley & Sons, 2012. Google Scholar

[40]

Y. Xiao and H. Zhu, A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing, J. Math. Anal. Appl., 405 (2013), 310-319.  doi: 10.1016/j.jmaa.2013.04.017.  Google Scholar

[41]

Q. YanX. Z. Peng and D. H. Li, A globally convergent derivative-free method for solving large-scale nonlinear monotone equations, J. Comput. Appl. Math., 234 (2010), 649-657.  doi: 10.1016/j.cam.2010.01.001.  Google Scholar

[42]

G. YuS. Niu and J. Ma, Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints, J. Ind. Manag. Optim., 9 (2013), 117-129.  doi: 10.3934/jimo.2013.9.117.  Google Scholar

[43]

Z. YuJ. LinJ. SunY. H. XiaoL. Liu and Z. H. Li, Spectral gradient projection method for monotone nonlinear equations with convex constraints, Appl. Numer. Math., 59 (2009), 2416-2423.  doi: 10.1016/j.apnum.2009.04.004.  Google Scholar

[44]

N. Yuan, A derivative-free projection method for solving convex constrained monotone equations, SCIENCEASIA, 43 (2017), 195-200.  doi: 10.2306/scienceasia1513-1874.2017.43.195.  Google Scholar

[45]

M. ZhangY. Xiao and H. Dou, Solving nonlinear constrained monotone equations via limited memory BFGS algorithm, J. of Comp. Infor. Syst., 7 (2011), 3995-4006.   Google Scholar

[46]

Y. ZhengB. JeonD. XuQ. M. Wu and H. Zhang, Image segmentation by generalized hierarchical fuzzy c-means algorithm, J. of Int. & Fuzzy Syst., 28 (2015), 961-973.   Google Scholar

[47]

W. Zhou and D. H. Li, Limited memory BFGS method for nonlinear monotone equations, J. Comput. Math., 25 (2007), 89-96.   Google Scholar

[48]

W. Zhou and F. Wang, A PRP-based residual method for large-scale monotone nonlinear equations, Appl. Math. Comput., 261 (2015), 1-7.  doi: 10.1016/j.amc.2015.03.069.  Google Scholar

show all references

References:
[1]

A. B. Abubakar and P. Kumam, An improved three-term derivative-free method for solving nonlinear equations, Comput. Appl. Math., 37 (2018), 6760-6773.  doi: 10.1007/s40314-018-0712-5.  Google Scholar

[2]

A. B. Abubakar and P. Kumam, A descent Dai-Liao conjugate gradient method for nonlinear equations, Numer. Algorithms, 81 (2019), 197-210.  doi: 10.1007/s11075-018-0541-z.  Google Scholar

[3]

Y. Bing and G. Lin, An efficient implementation of Merrill's method for sparse or partially separable systems of nonlinear equations, SIAM J. Optim., 1 (1991), 206-221.  doi: 10.1137/0801015.  Google Scholar

[4]

W. Cheng, A PRP type method for systems of monotone equations, Math. Comput. Modelling, 50 (2009), 15-20.  doi: 10.1016/j.mcm.2009.04.007.  Google Scholar

[5]

Y. Dai and Y. Yuan, A nonlinear conjugate gradient method with a strong global convergence property, SIAM J. Optim., 10 (1999), 177-182.  doi: 10.1137/S1052623497318992.  Google Scholar

[6]

E. D. Dolan and J. J. Moré, Benchmarking optimization software with performance profiles, Math. Program., 91 (2002), 201-213.  doi: 10.1007/s101070100263.  Google Scholar

[7]

X. L. DongH. LiuY. L. Xu and X. M. Yang, Some nonlinear conjugate gradient methods with sufficient descent condition and global convergence, Optim. Lett., 9 (2015), 1421-1432.  doi: 10.1007/s11590-014-0836-5.  Google Scholar

[8]

M. EshaghnezhadS. Effati and A. Mansoori, A neurodynamic model to solve nonlinear pseudo-monotone projection equation and its applications, IEEE Transactions on Cybernetics, 47 (2017), 3050-3062.  doi: 10.1109/TCYB.2016.2611529.  Google Scholar

[9]

M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Math. Programming, 53 (1992), 99-110.  doi: 10.1007/BF01585696.  Google Scholar

[10]

B. GhaddarJ. Marecek and M. Mevissen, Optimal power flow as a polynomial optimization problem, IEEE Transactions on Power Systems, 31 (2016), 539-546.  doi: 10.1109/TPWRS.2015.2390037.  Google Scholar

[11]

B. GuV. S. ShengK. Y. TayW. Romano and S. Li, Incremental support vector learning for ordinal regression, IEEE Trans. Neural Netw. Learn. Syst., 26 (2015), 1403-1416.  doi: 10.1109/TNNLS.2014.2342533.  Google Scholar

[12]

L. HanG. Yu and L. Guan, Multivariate spectral gradient method for unconstrained optimization, Appl. Math. Comput., 201 (2008), 621-630.  doi: 10.1016/j.amc.2007.12.054.  Google Scholar

[13]

Y. Hu and Z. Wei, Wei–Yao–Liu conjugate gradient projection algorithm for nonlinear monotone equations with convex constraints, Int. J. Comput. Math., 92 (2015), 2261-2272.  doi: 10.1080/00207160.2014.977879.  Google Scholar

[14]

W. La Cruz, A projected derivative-free algorithm for nonlinear equations with convex constraints, Optim. Methods Softw., 29 (2014), 24-41.  doi: 10.1080/10556788.2012.721129.  Google Scholar

[15]

W. La Cruz, A spectral algorithm for large-scale systems of nonlinear monotone equations, Numer. Algorithms, 76 (2017), 1109-1130.  doi: 10.1007/s11075-017-0299-8.  Google Scholar

[16]

W. La CruzJ. Martínez and M. Raydan, Spectral residual method without gradient information for solving large-scale nonlinear systems of equations, Math. Comp., 75 (2006), 1429-1448.  doi: 10.1090/S0025-5718-06-01840-0.  Google Scholar

[17]

J. LiX. LiB. Yang and X. Sun, Segmentation-based image copy-move forgery detection scheme, IEEE Transactions on Information Forensics and Security, 10 (2015), 507-518.   Google Scholar

[18]

Q. Li and D. H. Li, A class of derivative-free methods for large-scale nonlinear monotone equations, IMA J. Numer. Anal., 31 (2011), 1625-1635.  doi: 10.1093/imanum/drq015.  Google Scholar

[19]

J. Liu and X. L. Du, A gradient projection method for the sparse signal reconstruction in compressive sensing, Appl. Anal., 97 (2018), 2122-2131.  doi: 10.1080/00036811.2017.1359556.  Google Scholar

[20]

J. Liu and Y. Duan, Two spectral gradient projection methods for constrained equations and their linear convergence rate, J. Inequal. Appl., 2015 (2015), 13pp. doi: 10.1186/s13660-014-0525-z.  Google Scholar

[21]

J. Liu and Y. Feng, A derivative-free iterative method for nonlinear monotone equations with convex constraints, Numerical Algorithms, 82 (2019), 1-18.  doi: 10.1007/s11075-018-0603-2.  Google Scholar

[22]

J. Liu and S. Li, Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations, J. Ind. Manag. Optim., 13 (2017), 283-295.  doi: 10.3934/jimo.2016017.  Google Scholar

[23]

J. Liu and S. Li, A projection method for convex constrained monotone nonlinear equations with applications, Comput. Math. Appl., 70 (2015), 2442-2453.  doi: 10.1016/j.camwa.2015.09.014.  Google Scholar

[24]

S. Liu, Y. Huang and H. W. Jiao, Sufficient descent conjugate gradient methods for solving convex constrained nonlinear monotone equations, Abstr. Appl. Anal., 2014 (2014), 12pp. doi: 10.1155/2014/305643.  Google Scholar

[25]

F. Ma and C. Wang, Modified projection method for solving a system of monotone equations with convex constraints, J. Appl. Math. Comput., 34 (2010), 47-56.  doi: 10.1007/s12190-009-0305-y.  Google Scholar

[26]

H. Mohammad and A. B. Abubakar, A positive spectral gradient-like method for nonlinear monotone equations, Bull. Comput. Appl. Math., 5 (2017), 99-115.   Google Scholar

[27]

H. Mohammad and S. A. Santos, A structured diagonal Hessian approximation method with evaluation complexity analysis for nonlinear least squares, Comput. Appl. Math., 37 (2018), 6619-6653.  doi: 10.1007/s40314-018-0696-1.  Google Scholar

[28]

Y. Ou and J. Li, A new derivative-free SCG-type projection method for nonlinear monotone equations with convex constraints, J. Appl. Math. Comput., 56 (2018), 195-216.  doi: 10.1007/s12190-016-1068-x.  Google Scholar

[29]

Y. Ou and Y. Liu, Supermemory gradient methods for monotone nonlinear equations with convex constraints, Comput. Appl. Math., 36 (2017), 259-279.  doi: 10.1007/s40314-015-0228-1.  Google Scholar

[30]

Z. Papp and S. Rapajić, FR type methods for systems of large-scale nonlinear monotone equations, Appl. Math. Comput., 269 (2015), 816-823.  doi: 10.1016/j.amc.2015.08.002.  Google Scholar

[31]

E. Polak and G. Ribiere, Note sur la convergence de méthodes de directions conjuguées, Revue française d'informatique et de recherche opérationnelle. Série rouge, 3 (1969), 35–43.  Google Scholar

[32]

B. T. Polyak, The conjugate gradient method in extremal problems, USSR Comp. Math. and Mathem. Physics, 9 (1969), 94-112.  doi: 10.1016/0041-5553(69)90035-4.  Google Scholar

[33]

G. QianD. HanL. Xu and H. Y., Solving nonadditive traffic assignment problems: A self-adaptive projection-auxiliary problem method for variational inequalities, J. Ind. Manag. Optim., 9 (2013), 255-274.  doi: 10.3934/jimo.2013.9.255.  Google Scholar

[34]

M. V. Solodov and B. Svaiter, A globally convergent inexact Newton method for systems of monotone equations, in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, Applied Optimization, Springer, 1998,355–369. doi: 10.1007/978-1-4757-6388-1_18.  Google Scholar

[35]

M. Sun and J. Liu, Three derivative-free projection methods for nonlinear equations with convex constraints, J. Appl. Math. Comput., 47 (2015), 265-276.  doi: 10.1007/s12190-014-0774-5.  Google Scholar

[36]

C. Wang and Y. Wang, A superlinearly convergent projection method for constrained systems of nonlinear equations, J. Global Optim., 44 (2009), 283-296.  doi: 10.1007/s10898-008-9324-8.  Google Scholar

[37]

C. WangY. Wang and C. Xu, A projection method for a system of nonlinear monotone equations with convex constraints, Math. Methods Oper. Res., 66 (2007), 33-46.  doi: 10.1007/s00186-006-0140-y.  Google Scholar

[38]

X. WangS. Li and X. Kou, A self-adaptive three-term conjugate gradient method for monotone nonlinear equations with convex constraints, Calcolo, 53 (2016), 133-145.  doi: 10.1007/s10092-015-0140-5.  Google Scholar

[39]

A. J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control, John Wiley & Sons, 2012. Google Scholar

[40]

Y. Xiao and H. Zhu, A conjugate gradient method to solve convex constrained monotone equations with applications in compressive sensing, J. Math. Anal. Appl., 405 (2013), 310-319.  doi: 10.1016/j.jmaa.2013.04.017.  Google Scholar

[41]

Q. YanX. Z. Peng and D. H. Li, A globally convergent derivative-free method for solving large-scale nonlinear monotone equations, J. Comput. Appl. Math., 234 (2010), 649-657.  doi: 10.1016/j.cam.2010.01.001.  Google Scholar

[42]

G. YuS. Niu and J. Ma, Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints, J. Ind. Manag. Optim., 9 (2013), 117-129.  doi: 10.3934/jimo.2013.9.117.  Google Scholar

[43]

Z. YuJ. LinJ. SunY. H. XiaoL. Liu and Z. H. Li, Spectral gradient projection method for monotone nonlinear equations with convex constraints, Appl. Numer. Math., 59 (2009), 2416-2423.  doi: 10.1016/j.apnum.2009.04.004.  Google Scholar

[44]

N. Yuan, A derivative-free projection method for solving convex constrained monotone equations, SCIENCEASIA, 43 (2017), 195-200.  doi: 10.2306/scienceasia1513-1874.2017.43.195.  Google Scholar

[45]

M. ZhangY. Xiao and H. Dou, Solving nonlinear constrained monotone equations via limited memory BFGS algorithm, J. of Comp. Infor. Syst., 7 (2011), 3995-4006.   Google Scholar

[46]

Y. ZhengB. JeonD. XuQ. M. Wu and H. Zhang, Image segmentation by generalized hierarchical fuzzy c-means algorithm, J. of Int. & Fuzzy Syst., 28 (2015), 961-973.   Google Scholar

[47]

W. Zhou and D. H. Li, Limited memory BFGS method for nonlinear monotone equations, J. Comput. Math., 25 (2007), 89-96.   Google Scholar

[48]

W. Zhou and F. Wang, A PRP-based residual method for large-scale monotone nonlinear equations, Appl. Math. Comput., 261 (2015), 1-7.  doi: 10.1016/j.amc.2015.03.069.  Google Scholar

Figure 1.  Performance profile with respect to number of iterations (ITER)
Figure 2.  Performance profile with respect to number of function evaluations
Figure 3.  Performance profile with respect to CPU time
Table 1.  The initial points used for the test problems
INITIAL POINT VALUE
$ x_1 $ $ (1, 1, \ldots , 1)^T $
$ x_2 $ $ (0.1, 0.1, \ldots , 0.1)^T $
$ x_3 $ $ \bigl(\frac{1}{2}, \frac{1}{2^2}, \ldots , \frac{1}{2^n}\bigr)^T $
$ x_4 $ $ \bigl(0, 1-\frac{1}{2}, \ldots , 1-\frac{1}{n}\bigr)^T $
$ x_5 $ $ \bigl(0, \frac{1}{n}, \ldots , \frac{n-1}{n}\bigr)^T $
$ x_6 $ $ \bigl(1, \frac{1}{2}, \ldots , \frac{1}{n}\bigr)^T $
$ x_7 $ $ \bigl(n-\frac{1}{n}, n- \frac{2}{n}, \ldots , n-1 \bigr)^T $
$ x_8 $ $ \bigl(\frac{1}{n}, \frac{2}{n}, \ldots , 1\bigr)^T $
INITIAL POINT VALUE
$ x_1 $ $ (1, 1, \ldots , 1)^T $
$ x_2 $ $ (0.1, 0.1, \ldots , 0.1)^T $
$ x_3 $ $ \bigl(\frac{1}{2}, \frac{1}{2^2}, \ldots , \frac{1}{2^n}\bigr)^T $
$ x_4 $ $ \bigl(0, 1-\frac{1}{2}, \ldots , 1-\frac{1}{n}\bigr)^T $
$ x_5 $ $ \bigl(0, \frac{1}{n}, \ldots , \frac{n-1}{n}\bigr)^T $
$ x_6 $ $ \bigl(1, \frac{1}{2}, \ldots , \frac{1}{n}\bigr)^T $
$ x_7 $ $ \bigl(n-\frac{1}{n}, n- \frac{2}{n}, \ldots , n-1 \bigr)^T $
$ x_8 $ $ \bigl(\frac{1}{n}, \frac{2}{n}, \ldots , 1\bigr)^T $
Table 2.  Numerical Results for DPPM, MDYP and WYLP for Problem 1 with given initial points and dimension, $ f $ represents failure
DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 5 14 0.2355 8.79E-08 16 69 0.0200 4.84E-06 2 9 0.0085 0.00E+00
$ x_2 $ 4 11 0.0156 1.06E-08 10 40 0.0126 5.44E-06 4 15 0.0079 0.00E+00
$ x_3 $ 15 35 0.0887 6.01E-06 13 54 0.1723 7.55E-06 5 20 0.0119 0.00E+00
$ x_4 $ 6 16 0.0204 5.47E-06 25 133 0.1284 3.35E-06 2 9 0.0079 0.00E+00
$ x_5 $ 8 20 0.0294 2.47E-06 27 138 0.0757 6.03E-06 2 11 0.0076 0.00E+00
$ x_6 $ 8 19 0.0362 1.95E-06 26 134 0.0350 4.52E-06 4 15 0.0102 0.00E+00
$ x_7 $ 8 20 0.0292 2.48E-06 25 133 0.1248 3.35E-06 2 11 0.0071 0.00E+00
$ x_8 $ 8 20 0.0299 2.49E-06 22 112 0.0317 4.53E-06 2 11 0.0080 0.00E+00
$ x_1 $ 5 14 0.0720 1.71E-07 17 73 0.0820 2.67E-06 2 9 0.0143 0.00E+00
$ x_2 $ 4 11 0.0457 2.31E-08 8 31 0.0563 8.85E-06 4 15 0.0182 0.00E+00
$ x_3 $ 15 35 0.0707 6.05E-06 13 54 1.5146 7.55E-06 5 20 0.0308 0.00E+00
$ x_4 $ 6 16 0.0492 5.61E-06 24 126 0.1178 2.59E-06 2 9 0.0154 0.00E+00
$ x_5 $ 8 20 0.0611 5.54E-06 39 226 0.4130 3.08E-06 2 11 0.0190 0.00E+00
$ x_6 $ 8 19 0.0831 1.95E-06 22 99 0.1434 5.73E-06 4 15 0.0251 0.00E+00
$ x_7 $ 8 20 0.0651 5.54E-06 24 126 0.1656 2.59E-06 2 11 0.0175 0.00E+00
$ x_8 $ 8 20 0.0476 5.55E-06 35 216 0.3063 8.62E-06 2 11 0.0242 0.00E+00
$ x_1 $ 5 14 0.0956 2.37E-07 15 60 0.1038 4.70E-06 2 9 0.0256 0.00E+00
$ x_2 $ 4 11 0.0494 3.25E-08 9 34 0.0697 5.29E-06 4 15 0.0457 0.00E+00
$ x_3 $ 15 35 0.2145 6.05E-06 13 54 3.1315 7.55E-06 5 20 0.0614 0.00E+00
$ x_4 $ 6 16 0.0723 5.75E-06 52 336 0.7500 6.45E-06 2 9 0.0262 0.00E+00
$ x_5 $ 8 20 0.1135 7.84E-06 39 257 0.8268 7.40E-06 2 11 0.0234 0.00E+00
$ x_6 $ 8 19 0.1167 1.95E-06 21 97 0.1880 7.11E-06 4 15 0.0360 0.00E+00
$ x_7 $ 8 20 0.1115 7.84E-06 58 456 0.8633 8.74E-06 2 11 0.0253 0.00E+00
$ x_8 $ 8 20 0.1302 7.84E-06 40 250 0.3797 4.87E-06 2 11 0.0324 0.00E+00
$ x_1 $ 7 23 0.3971 9.16E-11 14 56 0.2812 4.32E-06 3 16 0.1857 0.00E+00
$ x_2 $ 4 11 0.1609 7.25E-08 10 38 0.3246 3.24E-06 4 15 0.1384 0.00E+00
$ x_3 $ 15 35 0.6173 6.05E-06 13 54 25.8148 7.55E-06 5 20 0.1416 0.00E+00
$ x_4 $ 7 23 0.4199 7.98E-10 41 287 2.6315 4.61E-06 2 13 0.1230 0.00E+00
$ x_5 $ 9 23 0.3295 2.21E-06 43 261 7.0197 8.15E-06 2 10 0.0809 0.00E+00
$ x_6 $ 8 19 0.4273 1.95E-06 23 121 0.3488 2.56E-06 4 15 0.1403 0.00E+00
$ x_7 $ 9 23 0.4588 2.21E-06 42 295 2.2382 7.02E-06 3 13 0.1398 0.00E+00
$ x_8 $ 9 23 0.3960 2.21E-06 41 252 2.0126 5.39E-07 3 13 0.1062 0.00E+00
$ x_1 $ 7 27 0.9775 3.55E-07 14 56 0.5772 4.93E-06 3 18 0.5132 0.00E+00
$ x_2 $ 4 11 0.4880 1.03E-07 10 38 0.5872 4.73E-06 4 15 0.2445 0.00E+00
$ x_3 $ 15 35 1.0573 6.05E-06 13 54 87.3280 7.55E-06 5 20 0.3240 0.00E+00
$ x_4 $ 7 27 0.5609 3.63E-06 41 265 3.9792 2.29E-06 3 18 0.4003 0.00E+00
$ x_5 $ 9 24 1.1510 4.99E-06 52 420 13.2180 1.53E-06 2 10 0.2150 0.00E+00
$ x_6 $ 11 25 0.7559 9.20E-08 21 99 1.2739 4.64E-06 4 15 0.2824 0.00E+00
$ x_7 $ 9 24 1.0529 4.99E-06 41 266 3.6878 6.90E-06 3 13 0.1655 0.00E+00
$ x_8 $ 9 24 0.8091 4.99E-06 42 316 15.6351 3.58E-06 3 13 0.2459 0.00E+00
DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 5 14 0.2355 8.79E-08 16 69 0.0200 4.84E-06 2 9 0.0085 0.00E+00
$ x_2 $ 4 11 0.0156 1.06E-08 10 40 0.0126 5.44E-06 4 15 0.0079 0.00E+00
$ x_3 $ 15 35 0.0887 6.01E-06 13 54 0.1723 7.55E-06 5 20 0.0119 0.00E+00
$ x_4 $ 6 16 0.0204 5.47E-06 25 133 0.1284 3.35E-06 2 9 0.0079 0.00E+00
$ x_5 $ 8 20 0.0294 2.47E-06 27 138 0.0757 6.03E-06 2 11 0.0076 0.00E+00
$ x_6 $ 8 19 0.0362 1.95E-06 26 134 0.0350 4.52E-06 4 15 0.0102 0.00E+00
$ x_7 $ 8 20 0.0292 2.48E-06 25 133 0.1248 3.35E-06 2 11 0.0071 0.00E+00
$ x_8 $ 8 20 0.0299 2.49E-06 22 112 0.0317 4.53E-06 2 11 0.0080 0.00E+00
$ x_1 $ 5 14 0.0720 1.71E-07 17 73 0.0820 2.67E-06 2 9 0.0143 0.00E+00
$ x_2 $ 4 11 0.0457 2.31E-08 8 31 0.0563 8.85E-06 4 15 0.0182 0.00E+00
$ x_3 $ 15 35 0.0707 6.05E-06 13 54 1.5146 7.55E-06 5 20 0.0308 0.00E+00
$ x_4 $ 6 16 0.0492 5.61E-06 24 126 0.1178 2.59E-06 2 9 0.0154 0.00E+00
$ x_5 $ 8 20 0.0611 5.54E-06 39 226 0.4130 3.08E-06 2 11 0.0190 0.00E+00
$ x_6 $ 8 19 0.0831 1.95E-06 22 99 0.1434 5.73E-06 4 15 0.0251 0.00E+00
$ x_7 $ 8 20 0.0651 5.54E-06 24 126 0.1656 2.59E-06 2 11 0.0175 0.00E+00
$ x_8 $ 8 20 0.0476 5.55E-06 35 216 0.3063 8.62E-06 2 11 0.0242 0.00E+00
$ x_1 $ 5 14 0.0956 2.37E-07 15 60 0.1038 4.70E-06 2 9 0.0256 0.00E+00
$ x_2 $ 4 11 0.0494 3.25E-08 9 34 0.0697 5.29E-06 4 15 0.0457 0.00E+00
$ x_3 $ 15 35 0.2145 6.05E-06 13 54 3.1315 7.55E-06 5 20 0.0614 0.00E+00
$ x_4 $ 6 16 0.0723 5.75E-06 52 336 0.7500 6.45E-06 2 9 0.0262 0.00E+00
$ x_5 $ 8 20 0.1135 7.84E-06 39 257 0.8268 7.40E-06 2 11 0.0234 0.00E+00
$ x_6 $ 8 19 0.1167 1.95E-06 21 97 0.1880 7.11E-06 4 15 0.0360 0.00E+00
$ x_7 $ 8 20 0.1115 7.84E-06 58 456 0.8633 8.74E-06 2 11 0.0253 0.00E+00
$ x_8 $ 8 20 0.1302 7.84E-06 40 250 0.3797 4.87E-06 2 11 0.0324 0.00E+00
$ x_1 $ 7 23 0.3971 9.16E-11 14 56 0.2812 4.32E-06 3 16 0.1857 0.00E+00
$ x_2 $ 4 11 0.1609 7.25E-08 10 38 0.3246 3.24E-06 4 15 0.1384 0.00E+00
$ x_3 $ 15 35 0.6173 6.05E-06 13 54 25.8148 7.55E-06 5 20 0.1416 0.00E+00
$ x_4 $ 7 23 0.4199 7.98E-10 41 287 2.6315 4.61E-06 2 13 0.1230 0.00E+00
$ x_5 $ 9 23 0.3295 2.21E-06 43 261 7.0197 8.15E-06 2 10 0.0809 0.00E+00
$ x_6 $ 8 19 0.4273 1.95E-06 23 121 0.3488 2.56E-06 4 15 0.1403 0.00E+00
$ x_7 $ 9 23 0.4588 2.21E-06 42 295 2.2382 7.02E-06 3 13 0.1398 0.00E+00
$ x_8 $ 9 23 0.3960 2.21E-06 41 252 2.0126 5.39E-07 3 13 0.1062 0.00E+00
$ x_1 $ 7 27 0.9775 3.55E-07 14 56 0.5772 4.93E-06 3 18 0.5132 0.00E+00
$ x_2 $ 4 11 0.4880 1.03E-07 10 38 0.5872 4.73E-06 4 15 0.2445 0.00E+00
$ x_3 $ 15 35 1.0573 6.05E-06 13 54 87.3280 7.55E-06 5 20 0.3240 0.00E+00
$ x_4 $ 7 27 0.5609 3.63E-06 41 265 3.9792 2.29E-06 3 18 0.4003 0.00E+00
$ x_5 $ 9 24 1.1510 4.99E-06 52 420 13.2180 1.53E-06 2 10 0.2150 0.00E+00
$ x_6 $ 11 25 0.7559 9.20E-08 21 99 1.2739 4.64E-06 4 15 0.2824 0.00E+00
$ x_7 $ 9 24 1.0529 4.99E-06 41 266 3.6878 6.90E-06 3 13 0.1655 0.00E+00
$ x_8 $ 9 24 0.8091 4.99E-06 42 316 15.6351 3.58E-06 3 13 0.2459 0.00E+00
Table 3.  Numerical Results for DPPM, MDYP and WYLP for Problem 2 with given initial points and dimension, $ f $ represents failure
DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 9 26 0.0452 4.37E-06 13 40 0.0794 3.39E-06 19 93 0.3206 9.75E-07
$ x_2 $ 7 19 0.0244 7.69E-06 7 22 0.0300 5.13E-06 15 73 0.0934 4.17E-07
$ x_3 $ 8 22 0.0703 3.51E-06 11 36 0.1053 3.48E-06 15 73 0.1950 4.96E-07
$ x_4 $ 9 26 0.0401 6.71E-06 14 46 0.0217 7.56E-06 19 93 0.0969 9.62E-07
$ x_5 $ 10 29 0.0461 3.13E-06 14 46 0.0238 7.56E-06 19 93 0.1020 4.80E-07
$ x_6 $ 9 25 0.0377 6.09E-06 10 32 0.0135 8.15E-06 16 78 0.0441 6.75E-07
$ x_7 $ 10 29 0.0764 3.13E-06 14 46 0.0165 7.56E-06 19 93 0.0604 4.80E-07
$ x_8 $ 10 29 0.0419 3.13E-06 14 46 0.0264 7.53E-06 19 93 0.0568 4.81E-07
$ x_1 $ 9 26 0.1909 9.84E-06 13 40 0.0641 7.50E-06 20 98 0.4133 8.51E-07
$ x_2 $ 8 22 0.0677 3.39E-06 8 25 0.0360 4.94E-06 15 73 0.1504 9.03E-07
$ x_3 $ 8 22 0.1055 3.51E-06 11 36 1.2062 8.57E-06 15 73 0.1302 4.89E-07
$ x_4 $ 10 29 0.1083 2.25E-06 35 157 0.1346 7.40E-06 20 98 0.1267 8.49E-07
$ x_5 $ 10 29 0.1217 7.01E-06 31 138 0.4007 4.24E-06 20 98 0.1494 4.20E-07
$ x_6 $ 9 25 0.0721 6.02E-06 10 32 0.0487 8.13E-06 16 78 0.1823 6.64E-07
$ x_7 $ 10 29 0.1134 7.01E-06 31 138 0.2469 4.24E-06 20 98 0.1852 4.20E-07
$ x_8 $ 10 29 0.0780 7.01E-06 22 88 0.3009 7.25E-06 20 98 0.1700 4.21E-07
$ x_1 $ 10 29 0.1738 2.79E-06 14 43 0.1073 2.89E-06 21 103 0.5148 4.80E-07
$ x_2 $ 8 22 0.0853 4.78E-06 8 25 0.0799 6.92E-06 16 78 0.2554 5.09E-07
$ x_3 $ 8 22 0.0803 3.51E-06 11 36 3.0274 9.89E-06 15 73 0.1792 4.88E-07
$ x_4 $ 10 29 0.1861 2.99E-06 16 54 0.4752 4.83E-06 21 103 0.3229 4.79E-07
$ x_5 $ 10 29 0.2444 9.91E-06 16 54 0.1676 4.83E-06 20 98 0.3376 5.93E-07
$ x_6 $ 9 25 0.2425 6.02E-06 10 32 0.0720 8.13E-06 16 78 0.2158 6.63E-07
$ x_7 $ 10 29 0.2000 9.91E-06 16 54 0.1314 4.83E-06 20 98 0.2923 5.93E-07
$ x_8 $ 10 29 0.2440 9.92E-06 16 54 0.0871 4.69E-06 20 98 0.3689 5.93E-07
$ x_1 $ 12 39 1.0114 2.30E-06 14 43 0.4292 6.46E-06 23 116 2.6259 6.55E-07
$ x_2 $ 9 25 0.6191 2.13E-06 9 28 0.3785 1.05E-06 17 83 0.9079 4.53E-07
$ x_3 $ 8 22 0.3316 3.51E-06 12 39 24.3728 3.64E-06 15 73 0.4278 4.88E-07
$ x_4 $ 12 39 0.6189 2.30E-06 f f f f 23 116 1.4482 6.55E-07
$ x_5 $ 11 32 0.8394 4.43E-06 f f f f 21 103 1.7160 5.29E-07
$ x_6 $ 9 25 0.5946 6.01E-06 10 32 1.1865 8.12E-06 16 78 0.7065 6.62E-07
$ x_7 $ 11 32 0.7496 4.43E-06 f f f f 21 103 1.1586 5.29E-07
$ x_8 $ 11 32 0.8790 4.44E-06 f f f f 21 103 0.8051 5.29E-07
$ x_1 $ 12 45 2.1717 2.90E-06 14 43 1.1046 9.13E-06 25 130 3.6530 6.38E-07
$ x_2 $ 9 25 1.3327 3.02E-06 9 28 0.7541 1.49E-06 17 81 1.5444 1.60E-11
$ x_3 $ 8 22 0.9087 3.51E-06 12 41 94.8845 3.66E-06 15 73 1.2575 4.88E-07
$ x_4 $ 12 45 1.7092 3.05E-06 f f f f 25 130 1.9217 6.38E-07
$ x_5 $ 13 40 1.2000 2.81E-06 f f f f 22 110 2.0242 6.94E-07
$ x_6 $ 9 25 1.2705 6.01E-06 10 32 0.8082 8.12E-06 16 78 1.4797 6.62E-07
$ x_7 $ 13 40 1.9988 2.81E-06 f f f f 22 110 1.7161 6.94E-07
$ x_8 $ 13 40 1.7253 2.81E-06 f f f f 22 110 2.3099 6.94E-07
DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 9 26 0.0452 4.37E-06 13 40 0.0794 3.39E-06 19 93 0.3206 9.75E-07
$ x_2 $ 7 19 0.0244 7.69E-06 7 22 0.0300 5.13E-06 15 73 0.0934 4.17E-07
$ x_3 $ 8 22 0.0703 3.51E-06 11 36 0.1053 3.48E-06 15 73 0.1950 4.96E-07
$ x_4 $ 9 26 0.0401 6.71E-06 14 46 0.0217 7.56E-06 19 93 0.0969 9.62E-07
$ x_5 $ 10 29 0.0461 3.13E-06 14 46 0.0238 7.56E-06 19 93 0.1020 4.80E-07
$ x_6 $ 9 25 0.0377 6.09E-06 10 32 0.0135 8.15E-06 16 78 0.0441 6.75E-07
$ x_7 $ 10 29 0.0764 3.13E-06 14 46 0.0165 7.56E-06 19 93 0.0604 4.80E-07
$ x_8 $ 10 29 0.0419 3.13E-06 14 46 0.0264 7.53E-06 19 93 0.0568 4.81E-07
$ x_1 $ 9 26 0.1909 9.84E-06 13 40 0.0641 7.50E-06 20 98 0.4133 8.51E-07
$ x_2 $ 8 22 0.0677 3.39E-06 8 25 0.0360 4.94E-06 15 73 0.1504 9.03E-07
$ x_3 $ 8 22 0.1055 3.51E-06 11 36 1.2062 8.57E-06 15 73 0.1302 4.89E-07
$ x_4 $ 10 29 0.1083 2.25E-06 35 157 0.1346 7.40E-06 20 98 0.1267 8.49E-07
$ x_5 $ 10 29 0.1217 7.01E-06 31 138 0.4007 4.24E-06 20 98 0.1494 4.20E-07
$ x_6 $ 9 25 0.0721 6.02E-06 10 32 0.0487 8.13E-06 16 78 0.1823 6.64E-07
$ x_7 $ 10 29 0.1134 7.01E-06 31 138 0.2469 4.24E-06 20 98 0.1852 4.20E-07
$ x_8 $ 10 29 0.0780 7.01E-06 22 88 0.3009 7.25E-06 20 98 0.1700 4.21E-07
$ x_1 $ 10 29 0.1738 2.79E-06 14 43 0.1073 2.89E-06 21 103 0.5148 4.80E-07
$ x_2 $ 8 22 0.0853 4.78E-06 8 25 0.0799 6.92E-06 16 78 0.2554 5.09E-07
$ x_3 $ 8 22 0.0803 3.51E-06 11 36 3.0274 9.89E-06 15 73 0.1792 4.88E-07
$ x_4 $ 10 29 0.1861 2.99E-06 16 54 0.4752 4.83E-06 21 103 0.3229 4.79E-07
$ x_5 $ 10 29 0.2444 9.91E-06 16 54 0.1676 4.83E-06 20 98 0.3376 5.93E-07
$ x_6 $ 9 25 0.2425 6.02E-06 10 32 0.0720 8.13E-06 16 78 0.2158 6.63E-07
$ x_7 $ 10 29 0.2000 9.91E-06 16 54 0.1314 4.83E-06 20 98 0.2923 5.93E-07
$ x_8 $ 10 29 0.2440 9.92E-06 16 54 0.0871 4.69E-06 20 98 0.3689 5.93E-07
$ x_1 $ 12 39 1.0114 2.30E-06 14 43 0.4292 6.46E-06 23 116 2.6259 6.55E-07
$ x_2 $ 9 25 0.6191 2.13E-06 9 28 0.3785 1.05E-06 17 83 0.9079 4.53E-07
$ x_3 $ 8 22 0.3316 3.51E-06 12 39 24.3728 3.64E-06 15 73 0.4278 4.88E-07
$ x_4 $ 12 39 0.6189 2.30E-06 f f f f 23 116 1.4482 6.55E-07
$ x_5 $ 11 32 0.8394 4.43E-06 f f f f 21 103 1.7160 5.29E-07
$ x_6 $ 9 25 0.5946 6.01E-06 10 32 1.1865 8.12E-06 16 78 0.7065 6.62E-07
$ x_7 $ 11 32 0.7496 4.43E-06 f f f f 21 103 1.1586 5.29E-07
$ x_8 $ 11 32 0.8790 4.44E-06 f f f f 21 103 0.8051 5.29E-07
$ x_1 $ 12 45 2.1717 2.90E-06 14 43 1.1046 9.13E-06 25 130 3.6530 6.38E-07
$ x_2 $ 9 25 1.3327 3.02E-06 9 28 0.7541 1.49E-06 17 81 1.5444 1.60E-11
$ x_3 $ 8 22 0.9087 3.51E-06 12 41 94.8845 3.66E-06 15 73 1.2575 4.88E-07
$ x_4 $ 12 45 1.7092 3.05E-06 f f f f 25 130 1.9217 6.38E-07
$ x_5 $ 13 40 1.2000 2.81E-06 f f f f 22 110 2.0242 6.94E-07
$ x_6 $ 9 25 1.2705 6.01E-06 10 32 0.8082 8.12E-06 16 78 1.4797 6.62E-07
$ x_7 $ 13 40 1.9988 2.81E-06 f f f f 22 110 1.7161 6.94E-07
$ x_8 $ 13 40 1.7253 2.81E-06 f f f f 22 110 2.3099 6.94E-07
Table 4.  Numerical Results for DPPM, MDYP and WYLP for Problem 3 with given initial points and dimension, $ f $ represents failure
DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 4 8 0.8195 6.47E-09 12 39 0.0871 9.82E-06 2 8 0.0309 0.00E+00
$ x_2 $ 3 6 0.0912 8.25E-08 10 32 0.0564 9.57E-06 2 8 0.0068 0.00E+00
$ x_3 $ 13 27 0.0953 8.81E-06 9 29 0.2011 2.35E-06 13 51 0.0271 3.85E-07
$ x_4 $ 4 8 0.0192 1.42E-06 37 210 0.1077 2.98E-06 17 69 0.0250 7.21E-07
$ x_5 $ 6 12 0.0705 5.56E-06 37 210 0.0689 2.98E-06 17 73 0.0373 8.83E-07
$ x_6 $ 5 10 0.0919 6.68E-11 16 56 0.0211 7.32E-06 15 59 0.0292 8.78E-07
$ x_7 $ 6 12 0.0203 5.56E-06 37 210 0.0639 2.98E-06 17 73 0.0332 8.83E-07
$ x_8 $ 8 16 0.1217 4.35E-08 19 85 0.0470 6.92E-06 17 73 0.0263 8.83E-07
$ x_1 $ 4 8 0.0746 1.45E-08 13 42 0.0529 3.54E-06 2 8 0.0133 0.00E+00
$ x_2 $ 3 6 0.0249 1.84E-07 11 35 0.0574 3.71E-06 2 8 0.0097 0.00E+00
$ x_3 $ 13 27 0.1299 8.81E-06 9 29 1.1781 2.35E-06 13 51 0.0781 3.85E-07
$ x_4 $ 4 8 0.0397 1.28E-06 36 209 0.4912 3.49E-06 18 72 0.0891 3.15E-07
$ x_5 $ 13 26 0.0749 7.27E-13 36 209 0.3499 3.49E-06 18 76 0.1789 6.06E-07
$ x_6 $ 5 10 0.0320 6.70E-11 21 81 0.1014 7.63E-06 16 62 0.0643 9.17E-07
$ x_7 $ 13 26 0.0643 7.27E-13 36 209 0.3509 3.49E-06 18 76 0.1192 6.08E-07
$ x_8 $ 9 18 0.0524 8.51E-09 20 84 0.1209 8.44E-06 18 76 0.0923 6.07E-07
$ x_1 $ 4 8 0.3372 2.04E-08 13 42 0.0841 5.00E-06 2 8 0.0186 0.00E+00
$ x_2 $ 3 6 0.0404 2.61E-07 11 35 0.0801 5.24E-06 2 8 0.0165 0.00E+00
$ x_3 $ 13 27 0.1017 8.81E-06 9 29 3.2921 2.35E-06 13 51 0.0892 3.85E-07
$ x_4 $ 4 8 0.0452 1.30E-06 31 177 0.4489 1.82E-06 15 59 0.0737 8.90E-07
$ x_5 $ 11 22 0.1231 1.43E-10 31 177 0.3576 1.82E-06 18 76 0.0882 8.57E-07
$ x_6 $ 5 10 0.0408 6.70E-11 18 72 0.1468 8.19E-06 f f f f
$ x_7 $ 11 22 0.0840 1.43E-10 31 177 0.3460 1.82E-06 18 76 0.1498 8.57E-07
$ x_8 $ 12 24 0.0726 1.21E-11 29 153 0.3801 6.17E-06 18 76 0.1678 8.55E-07
$ x_1 $ 5 15 0.1876 1.58E-06 14 45 0.3743 4.52E-06 3 14 0.2255 0.00E+00
$ x_2 $ 3 6 0.1362 5.83E-07 12 38 0.3290 4.63E-06 2 8 0.1014 0.00E+00
$ x_3 $ 13 27 0.4447 8.81E-06 9 29 22.4652 2.35E-06 13 51 0.3770 3.85E-07
$ x_4 $ 6 17 0.3568 2.93E-10 40 278 8.4671 7.10E-06 18 73 0.6564 6.79E-07
$ x_5 $ 17 36 0.6444 7.27E-06 40 278 9.2475 7.10E-06 f f f f
$ x_6 $ 5 10 0.1576 6.71E-11 21 80 4.8421 9.53E-06 f f f f
$ x_7 $ 17 36 0.4210 7.27E-06 40 278 8.9672 7.10E-06 f f f f
$ x_8 $ 17 36 0.4061 7.27E-06 36 222 0.7206 4.06E-06 f f f f
$ x_1 $ 6 21 0.9043 8.13E-09 14 45 0.3008 6.39E-06 4 21 0.2761 0.00E+00
$ x_2 $ 3 6 0.2392 8.25E-07 12 38 0.6761 6.55E-06 2 8 0.1008 0.00E+00
$ x_3 $ 13 27 0.5491 8.81E-06 9 29 72.1280 2.35E-06 13 51 0.5589 3.85E-07
$ x_4 $ 6 21 0.4396 1.81E-08 34 224 3.5081 4.61E-06 19 80 1.1189 6.33E-07
$ x_5 $ 18 40 1.4053 7.38E-06 34 224 2.4146 4.61E-06 f f f f
$ x_6 $ 5 10 0.3521 6.71E-11 19 71 7.7563 4.00E-06 f f f f
$ x_7 $ 18 40 1.0043 7.38E-06 34 224 3.4369 4.61E-06 f f f f
$ x_8 $ 18 40 1.2596 7.38E-06 37 225 3.3577 4.58E-06 f f f f
DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 4 8 0.8195 6.47E-09 12 39 0.0871 9.82E-06 2 8 0.0309 0.00E+00
$ x_2 $ 3 6 0.0912 8.25E-08 10 32 0.0564 9.57E-06 2 8 0.0068 0.00E+00
$ x_3 $ 13 27 0.0953 8.81E-06 9 29 0.2011 2.35E-06 13 51 0.0271 3.85E-07
$ x_4 $ 4 8 0.0192 1.42E-06 37 210 0.1077 2.98E-06 17 69 0.0250 7.21E-07
$ x_5 $ 6 12 0.0705 5.56E-06 37 210 0.0689 2.98E-06 17 73 0.0373 8.83E-07
$ x_6 $ 5 10 0.0919 6.68E-11 16 56 0.0211 7.32E-06 15 59 0.0292 8.78E-07
$ x_7 $ 6 12 0.0203 5.56E-06 37 210 0.0639 2.98E-06 17 73 0.0332 8.83E-07
$ x_8 $ 8 16 0.1217 4.35E-08 19 85 0.0470 6.92E-06 17 73 0.0263 8.83E-07
$ x_1 $ 4 8 0.0746 1.45E-08 13 42 0.0529 3.54E-06 2 8 0.0133 0.00E+00
$ x_2 $ 3 6 0.0249 1.84E-07 11 35 0.0574 3.71E-06 2 8 0.0097 0.00E+00
$ x_3 $ 13 27 0.1299 8.81E-06 9 29 1.1781 2.35E-06 13 51 0.0781 3.85E-07
$ x_4 $ 4 8 0.0397 1.28E-06 36 209 0.4912 3.49E-06 18 72 0.0891 3.15E-07
$ x_5 $ 13 26 0.0749 7.27E-13 36 209 0.3499 3.49E-06 18 76 0.1789 6.06E-07
$ x_6 $ 5 10 0.0320 6.70E-11 21 81 0.1014 7.63E-06 16 62 0.0643 9.17E-07
$ x_7 $ 13 26 0.0643 7.27E-13 36 209 0.3509 3.49E-06 18 76 0.1192 6.08E-07
$ x_8 $ 9 18 0.0524 8.51E-09 20 84 0.1209 8.44E-06 18 76 0.0923 6.07E-07
$ x_1 $ 4 8 0.3372 2.04E-08 13 42 0.0841 5.00E-06 2 8 0.0186 0.00E+00
$ x_2 $ 3 6 0.0404 2.61E-07 11 35 0.0801 5.24E-06 2 8 0.0165 0.00E+00
$ x_3 $ 13 27 0.1017 8.81E-06 9 29 3.2921 2.35E-06 13 51 0.0892 3.85E-07
$ x_4 $ 4 8 0.0452 1.30E-06 31 177 0.4489 1.82E-06 15 59 0.0737 8.90E-07
$ x_5 $ 11 22 0.1231 1.43E-10 31 177 0.3576 1.82E-06 18 76 0.0882 8.57E-07
$ x_6 $ 5 10 0.0408 6.70E-11 18 72 0.1468 8.19E-06 f f f f
$ x_7 $ 11 22 0.0840 1.43E-10 31 177 0.3460 1.82E-06 18 76 0.1498 8.57E-07
$ x_8 $ 12 24 0.0726 1.21E-11 29 153 0.3801 6.17E-06 18 76 0.1678 8.55E-07
$ x_1 $ 5 15 0.1876 1.58E-06 14 45 0.3743 4.52E-06 3 14 0.2255 0.00E+00
$ x_2 $ 3 6 0.1362 5.83E-07 12 38 0.3290 4.63E-06 2 8 0.1014 0.00E+00
$ x_3 $ 13 27 0.4447 8.81E-06 9 29 22.4652 2.35E-06 13 51 0.3770 3.85E-07
$ x_4 $ 6 17 0.3568 2.93E-10 40 278 8.4671 7.10E-06 18 73 0.6564 6.79E-07
$ x_5 $ 17 36 0.6444 7.27E-06 40 278 9.2475 7.10E-06 f f f f
$ x_6 $ 5 10 0.1576 6.71E-11 21 80 4.8421 9.53E-06 f f f f
$ x_7 $ 17 36 0.4210 7.27E-06 40 278 8.9672 7.10E-06 f f f f
$ x_8 $ 17 36 0.4061 7.27E-06 36 222 0.7206 4.06E-06 f f f f
$ x_1 $ 6 21 0.9043 8.13E-09 14 45 0.3008 6.39E-06 4 21 0.2761 0.00E+00
$ x_2 $ 3 6 0.2392 8.25E-07 12 38 0.6761 6.55E-06 2 8 0.1008 0.00E+00
$ x_3 $ 13 27 0.5491 8.81E-06 9 29 72.1280 2.35E-06 13 51 0.5589 3.85E-07
$ x_4 $ 6 21 0.4396 1.81E-08 34 224 3.5081 4.61E-06 19 80 1.1189 6.33E-07
$ x_5 $ 18 40 1.4053 7.38E-06 34 224 2.4146 4.61E-06 f f f f
$ x_6 $ 5 10 0.3521 6.71E-11 19 71 7.7563 4.00E-06 f f f f
$ x_7 $ 18 40 1.0043 7.38E-06 34 224 3.4369 4.61E-06 f f f f
$ x_8 $ 18 40 1.2596 7.38E-06 37 225 3.3577 4.58E-06 f f f f
Table 5.  Numerical Results for DPPM, MDYP and WYLP for Problem 4 with given initial points and dimension, $ f $ represents failure
DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 10 29 0.1975 5.56E-06 13 41 0.0277 2.04E-06 17 83 0.0574 5.78E-07
$ x_2 $ 10 29 0.0528 8.48E-06 13 41 0.0191 3.11E-06 19 89 0.0732 7.34E-07
$ x_3 $ 10 28 0.0530 1.66E-14 13 41 0.0182 3.22E-06 f f f f
$ x_4 $ 10 28 0.0521 1.31E-14 13 41 0.0229 2.65E-06 f f f f
$ x_5 $ 10 28 0.0362 1.23E-14 13 41 0.0206 2.65E-06 20 92 0.0418 6.37E-07
$ x_6 $ 10 28 0.0370 1.12E-14 13 41 0.0291 3.22E-06 17 83 0.0544 7.43E-07
$ x_7 $ 10 28 0.0345 1.23E-14 13 41 0.0273 2.65E-06 20 92 0.0623 6.37E-07
$ x_8 $ 10 28 0.0388 1.16E-14 13 41 0.0181 2.65E-06 20 92 0.0504 6.37E-07
$ x_1 $ 11 32 0.1438 2.49E-06 13 41 0.2237 4.56E-06 f f f f
$ x_2 $ 10 30 0.1277 0.00E+00 13 41 0.2464 6.96E-06 657 2006 2.2801 4.71E-07
$ x_3 $ 11 34 0.1592 9.60E-06 13 41 0.0676 7.22E-06 21 104 0.2050 3.40E-08
$ x_4 $ 11 32 0.1546 2.49E-06 13 41 0.0814 5.94E-06 17 83 0.1518 8.10E-07
$ x_5 $ 11 34 0.0902 7.90E-06 13 41 0.1253 5.94E-06 21 105 0.1327 6.96E-07
$ x_6 $ 11 34 0.2370 9.60E-06 13 41 0.0804 7.22E-06 19 94 0.1391 1.91E-07
$ x_7 $ 11 34 0.1372 7.90E-06 13 41 0.0817 5.94E-06 21 105 0.1985 6.96E-07
$ x_8 $ 11 34 0.1759 7.90E-06 13 41 0.0618 5.94E-06 21 105 0.1921 6.96E-07
$ x_1 $ 10 30 0.3047 8.88E-16 13 41 0.1117 6.46E-06 248 777 2.4396 7.71E-07
$ x_2 $ 12 41 0.4791 8.80E-06 13 41 0.2053 9.84E-06 321 999 3.1750 7.52E-07
$ x_3 $ 11 37 0.2859 1.33E-15 14 44 0.2424 4.02E-06 f f f f
$ x_4 $ 11 34 0.2346 8.59E-06 13 41 0.1677 8.41E-06 27 116 0.1964 4.43E-07
$ x_5 $ 12 39 0.3669 4.87E-06 13 41 0.1120 8.41E-06 22 111 0.4084 6.30E-07
$ x_6 $ 12 41 0.2340 9.13E-06 14 44 0.1076 4.02E-06 240 756 2.1246 6.20E-07
$ x_7 $ 12 39 0.3153 4.87E-06 13 41 0.1737 8.41E-06 22 111 0.5055 6.30E-07
$ x_8 $ 12 39 0.3989 4.87E-06 13 41 0.1488 8.40E-06 22 111 0.3394 6.30E-07
$ x_1 $ 9 38 0.8360 0.00E+00 14 44 0.6011 5.69E-06 22 107 1.0331 4.71E-07
$ x_2 $ 13 66 1.4044 9.93E-14 14 44 0.7478 8.67E-06 23 121 1.4131 8.76E-07
$ x_3 $ 13 69 1.1444 9.93E-16 14 44 0.7075 9.00E-06 29 140 1.7453 9.92E-07
$ x_4 $ 10 41 1.0490 2.03E-14 14 44 0.5255 7.40E-06 f f f f
$ x_5 $ 11 53 1.4307 4.86E-14 14 44 0.6108 7.40E-06 277 878 5.8666 8.74E-07
$ x_6 $ 14 72 1.6761 7.15E-15 14 44 0.6519 9.00E-06 f f f f
$ x_7 $ 11 53 1.0887 4.86E-14 14 44 0.5013 7.40E-06 277 878 5.8247 8.74E-07
$ x_8 $ 11 53 0.9404 4.86E-14 14 44 0.6483 7.40E-06 23 116 1.6068 8.75E-07
$ x_1 $ 13 61 2.9900 1.40E-13 14 44 1.2238 8.04E-06 21 114 2.6706 4.32E-07
$ x_2 $ 15 92 4.1893 1.40E-13 15 47 1.0116 2.14E-06 24 140 3.1769 7.35E-07
$ x_3 $ 15 98 3.8911 1.40E-13 15 47 1.2457 2.23E-06 29 158 2.5033 8.14E-07
$ x_4 $ 11 55 2.2741 1.72E-14 15 47 1.3337 1.83E-06 21 114 1.5601 6.03E-07
$ x_5 $ 13 77 3.0852 9.15E-14 15 47 1.4839 1.83E-06 75 284 7.6840 8.99E-07
$ x_6 $ 15 98 3.7006 1.33E-13 15 47 1.2636 2.23E-06 567 1772 31.9028 7.18E-07
$ x_7 $ 13 77 3.3259 9.15E-14 15 47 1.3969 1.83E-06 75 284 5.1793 8.99E-07
$ x_8 $ 13 77 2.3221 9.15E-14 15 47 1.4001 1.83E-06 75 284 5.6734 8.99E-07
DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 10 29 0.1975 5.56E-06 13 41 0.0277 2.04E-06 17 83 0.0574 5.78E-07
$ x_2 $ 10 29 0.0528 8.48E-06 13 41 0.0191 3.11E-06 19 89 0.0732 7.34E-07
$ x_3 $ 10 28 0.0530 1.66E-14 13 41 0.0182 3.22E-06 f f f f
$ x_4 $ 10 28 0.0521 1.31E-14 13 41 0.0229 2.65E-06 f f f f
$ x_5 $ 10 28 0.0362 1.23E-14 13 41 0.0206 2.65E-06 20 92 0.0418 6.37E-07
$ x_6 $ 10 28 0.0370 1.12E-14 13 41 0.0291 3.22E-06 17 83 0.0544 7.43E-07
$ x_7 $ 10 28 0.0345 1.23E-14 13 41 0.0273 2.65E-06 20 92 0.0623 6.37E-07
$ x_8 $ 10 28 0.0388 1.16E-14 13 41 0.0181 2.65E-06 20 92 0.0504 6.37E-07
$ x_1 $ 11 32 0.1438 2.49E-06 13 41 0.2237 4.56E-06 f f f f
$ x_2 $ 10 30 0.1277 0.00E+00 13 41 0.2464 6.96E-06 657 2006 2.2801 4.71E-07
$ x_3 $ 11 34 0.1592 9.60E-06 13 41 0.0676 7.22E-06 21 104 0.2050 3.40E-08
$ x_4 $ 11 32 0.1546 2.49E-06 13 41 0.0814 5.94E-06 17 83 0.1518 8.10E-07
$ x_5 $ 11 34 0.0902 7.90E-06 13 41 0.1253 5.94E-06 21 105 0.1327 6.96E-07
$ x_6 $ 11 34 0.2370 9.60E-06 13 41 0.0804 7.22E-06 19 94 0.1391 1.91E-07
$ x_7 $ 11 34 0.1372 7.90E-06 13 41 0.0817 5.94E-06 21 105 0.1985 6.96E-07
$ x_8 $ 11 34 0.1759 7.90E-06 13 41 0.0618 5.94E-06 21 105 0.1921 6.96E-07
$ x_1 $ 10 30 0.3047 8.88E-16 13 41 0.1117 6.46E-06 248 777 2.4396 7.71E-07
$ x_2 $ 12 41 0.4791 8.80E-06 13 41 0.2053 9.84E-06 321 999 3.1750 7.52E-07
$ x_3 $ 11 37 0.2859 1.33E-15 14 44 0.2424 4.02E-06 f f f f
$ x_4 $ 11 34 0.2346 8.59E-06 13 41 0.1677 8.41E-06 27 116 0.1964 4.43E-07
$ x_5 $ 12 39 0.3669 4.87E-06 13 41 0.1120 8.41E-06 22 111 0.4084 6.30E-07
$ x_6 $ 12 41 0.2340 9.13E-06 14 44 0.1076 4.02E-06 240 756 2.1246 6.20E-07
$ x_7 $ 12 39 0.3153 4.87E-06 13 41 0.1737 8.41E-06 22 111 0.5055 6.30E-07
$ x_8 $ 12 39 0.3989 4.87E-06 13 41 0.1488 8.40E-06 22 111 0.3394 6.30E-07
$ x_1 $ 9 38 0.8360 0.00E+00 14 44 0.6011 5.69E-06 22 107 1.0331 4.71E-07
$ x_2 $ 13 66 1.4044 9.93E-14 14 44 0.7478 8.67E-06 23 121 1.4131 8.76E-07
$ x_3 $ 13 69 1.1444 9.93E-16 14 44 0.7075 9.00E-06 29 140 1.7453 9.92E-07
$ x_4 $ 10 41 1.0490 2.03E-14 14 44 0.5255 7.40E-06 f f f f
$ x_5 $ 11 53 1.4307 4.86E-14 14 44 0.6108 7.40E-06 277 878 5.8666 8.74E-07
$ x_6 $ 14 72 1.6761 7.15E-15 14 44 0.6519 9.00E-06 f f f f
$ x_7 $ 11 53 1.0887 4.86E-14 14 44 0.5013 7.40E-06 277 878 5.8247 8.74E-07
$ x_8 $ 11 53 0.9404 4.86E-14 14 44 0.6483 7.40E-06 23 116 1.6068 8.75E-07
$ x_1 $ 13 61 2.9900 1.40E-13 14 44 1.2238 8.04E-06 21 114 2.6706 4.32E-07
$ x_2 $ 15 92 4.1893 1.40E-13 15 47 1.0116 2.14E-06 24 140 3.1769 7.35E-07
$ x_3 $ 15 98 3.8911 1.40E-13 15 47 1.2457 2.23E-06 29 158 2.5033 8.14E-07
$ x_4 $ 11 55 2.2741 1.72E-14 15 47 1.3337 1.83E-06 21 114 1.5601 6.03E-07
$ x_5 $ 13 77 3.0852 9.15E-14 15 47 1.4839 1.83E-06 75 284 7.6840 8.99E-07
$ x_6 $ 15 98 3.7006 1.33E-13 15 47 1.2636 2.23E-06 567 1772 31.9028 7.18E-07
$ x_7 $ 13 77 3.3259 9.15E-14 15 47 1.3969 1.83E-06 75 284 5.1793 8.99E-07
$ x_8 $ 13 77 2.3221 9.15E-14 15 47 1.4001 1.83E-06 75 284 5.6734 8.99E-07
Table 6.  Numerical Results for DPPM, MDYP and WYLP for Problem 5 with given initial points and dimension, $ f $ represents failure
DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 5 12 0.2244 3.26E-06 11 38 0.2095 3.21E-06 2 9 0.0260 0.00E+00
$ x_2 $ 4 8 0.1030 7.57E-08 9 29 0.0294 8.71E-06 2 8 0.0143 0.00E+00
$ x_3 $ 5 11 0.0745 1.28E-07 12 38 0.1401 1.70E-06 4 14 0.0322 2.22E-16
$ x_4 $ 7 16 0.0238 1.92E-06 21 79 0.0319 6.66E-06 12 45 0.0305 7.78E-08
$ x_5 $ 9 19 0.0954 6.02E-06 21 79 0.0333 6.66E-06 14 54 0.0385 3.41E-07
$ x_6 $ 9 19 0.0316 4.09E-06 11 36 0.0131 3.32E-06 14 54 0.0249 2.67E-07
$ x_7 $ 9 19 0.0215 6.02E-06 21 79 0.0265 6.66E-06 14 54 0.0204 3.42E-07
$ x_8 $ 9 19 0.0286 5.97E-06 17 61 0.0164 6.86E-06 14 54 0.0198 3.92E-07
$ x_1 $ 5 12 0.0353 7.29E-06 11 38 0.0340 7.19E-06 2 9 0.0136 0.00E+00
$ x_2 $ 4 8 0.0247 1.69E-07 10 32 0.0334 8.33E-06 2 8 0.0197 0.00E+00
$ x_3 $ 5 11 0.0330 1.28E-07 12 38 1.4578 1.70E-06 4 14 0.0172 2.22E-16
$ x_4 $ 7 16 0.0494 1.79E-06 19 69 0.0617 2.22E-06 12 47 0.0369 9.52E-07
$ x_5 $ 9 20 0.0565 2.04E-06 19 69 0.0724 2.22E-06 15 59 0.0548 3.37E-07
$ x_6 $ 9 19 0.0677 4.92E-06 11 36 0.0424 3.32E-06 14 54 0.0385 6.25E-07
$ x_7 $ 9 20 0.0583 2.04E-06 19 69 0.0908 2.22E-06 15 59 0.0642 3.32E-07
$ x_8 $ 9 20 0.0440 2.04E-06 27 122 0.1471 3.72E-06 15 59 0.0605 3.36E-07
$ x_1 $ 6 14 0.0693 3.67E-10 12 41 0.0766 4.72E-06 2 9 0.0312 0.00E+00
$ x_2 $ 4 8 0.0665 2.39E-07 11 35 0.0800 1.49E-06 2 8 0.0131 0.00E+00
$ x_3 $ 5 11 0.0537 1.28E-07 12 38 2.5950 1.70E-06 4 14 0.0239 2.22E-16
$ x_4 $ 8 18 0.1273 3.07E-07 17 62 0.1395 9.13E-06 12 47 0.0824 7.20E-07
$ x_5 $ 9 20 0.0858 2.89E-06 17 62 0.1122 9.13E-06 15 59 0.0614 4.78E-07
$ x_6 $ 9 19 0.1316 4.92E-06 11 36 0.0749 3.32E-06 14 54 0.0617 7.36E-07
$ x_7 $ 9 20 0.0844 2.89E-06 17 62 0.1038 9.13E-06 15 59 0.0613 4.75E-07
$ x_8 $ 9 20 0.1005 2.89E-06 27 103 0.4661 2.52E-06 15 59 0.0524 4.75E-07
$ x_1 $ 7 20 0.4618 3.27E-08 13 44 0.3079 7.07E-07 2 10 0.0474 0.00E+00
$ x_2 $ 4 8 0.1123 5.35E-07 11 35 0.3012 3.33E-06 2 8 0.0923 0.00E+00
$ x_3 $ 5 11 0.1031 1.28E-07 12 38 23.8270 1.70E-06 4 14 0.1142 2.22E-16
$ x_4 $ 7 20 0.4022 3.43E-08 18 65 0.5567 8.62E-06 10 42 0.2322 7.89E-08
$ x_5 $ 9 21 0.3592 4.85E-06 18 65 0.3972 8.62E-06 16 65 0.4585 8.55E-07
$ x_6 $ 9 19 0.3959 4.92E-06 11 36 0.2964 3.32E-06 f f f f
$ x_7 $ 9 21 0.3742 4.85E-06 18 65 0.4319 8.62E-06 16 65 0.1987 8.49E-07
$ x_8 $ 9 21 0.3703 4.85E-06 18 65 0.5524 8.97E-06 16 65 0.5407 8.39E-07
$ x_1 $ 7 23 0.7369 4.77E-06 13 44 0.7618 1.00E-06 3 16 0.1932 0.00E+00
$ x_2 $ 4 8 0.2658 7.57E-07 11 35 0.4434 4.71E-06 2 8 0.1158 0.00E+00
$ x_3 $ 5 11 0.3353 1.28E-07 12 38 107.4606 1.70E-06 4 14 0.2615 2.22E-16
$ x_4 $ 7 23 0.7591 4.81E-06 19 68 0.4613 2.35E-06 14 59 0.6242 5.49E-07
$ x_5 $ 9 22 0.7743 8.65E-06 19 68 1.1533 2.35E-06 f f f f
$ x_6 $ 9 19 0.6254 4.92E-06 11 36 0.5491 3.32E-06 f f f f
$ x_7 $ 9 22 0.7055 8.65E-06 19 68 1.0239 2.35E-06 f f f f
$ x_8 $ 9 22 0.5172 8.66E-06 19 68 0.9601 2.41E-06 f f f f
DPPM MDYP WYLP
DIMENSION INITIAL POINT ITER FVAL TIME NORM ITER FVAL TIME NORM ITER FVAL TIME NORM
1000 $ x_1 $ 5 12 0.2244 3.26E-06 11 38 0.2095 3.21E-06 2 9 0.0260 0.00E+00
$ x_2 $ 4 8 0.1030 7.57E-08 9 29 0.0294 8.71E-06 2 8 0.0143 0.00E+00
$ x_3 $ 5 11 0.0745 1.28E-07 12 38 0.1401 1.70E-06 4 14 0.0322 2.22E-16
$ x_4 $ 7 16 0.0238 1.92E-06 21 79 0.0319 6.66E-06 12 45 0.0305 7.78E-08
$ x_5 $ 9 19 0.0954 6.02E-06 21 79 0.0333 6.66E-06 14 54 0.0385 3.41E-07
$ x_6 $ 9 19 0.0316 4.09E-06 11 36 0.0131 3.32E-06 14 54 0.0249 2.67E-07
$ x_7 $ 9 19 0.0215 6.02E-06 21 79 0.0265 6.66E-06 14 54 0.0204 3.42E-07
$ x_8 $ 9 19 0.0286 5.97E-06 17 61 0.0164 6.86E-06 14 54 0.0198 3.92E-07
$ x_1 $ 5 12 0.0353 7.29E-06 11 38 0.0340 7.19E-06 2 9 0.0136 0.00E+00
$ x_2 $ 4 8 0.0247 1.69E-07 10 32 0.0334 8.33E-06 2 8 0.0197 0.00E+00
$ x_3 $ 5 11 0.0330 1.28E-07 12 38 1.4578 1.70E-06 4 14 0.0172 2.22E-16
$ x_4 $ 7 16 0.0494 1.79E-06 19 69 0.0617 2.22E-06 12 47 0.0369 9.52E-07
$ x_5 $ 9 20 0.0565 2.04E-06 19 69 0.0724 2.22E-06 15 59 0.0548 3.37E-07
$ x_6 $ 9 19 0.0677 4.92E-06 11 36 0.0424 3.32E-06 14 54 0.0385 6.25E-07
$ x_7 $ 9 20 0.0583 2.04E-06 19 69 0.0908 2.22E-06 15 59 0.0642 3.32E-07
$ x_8 $ 9 20 0.0440 2.04E-06 27 122 0.1471 3.72E-06 15 59 0.0605 3.36E-07
$ x_1 $ 6 14 0.0693 3.67E-10 12 41 0.0766 4.72E-06 2 9 0.0312 0.00E+00
$ x_2 $ 4 8 0.0665 2.39E-07 11 35 0.0800 1.49E-06 2 8 0.0131 0.00E+00
$ x_3 $ 5 11 0.0537 1.28E-07 12 38 2.5950 1.70E-06 4 14 0.0239 2.22E-16
$ x_4 $ 8 18 0.1273 3.07E-07 17 62 0.1395 9.13E-06 12 47 0.0824 7.20E-07
$ x_5 $ 9 20 0.0858 2.89E-06 17 62 0.1122 9.13E-06 15 59 0.0614 4.78E-07
$ x_6 $ 9 19 0.1316 4.92E-06 11 36 0.0749 3.32E-06 14 54 0.0617 7.36E-07
$ x_7 $ 9 20 0.0844 2.89E-06 17 62 0.1038 9.13E-06 15 59 0.0613 4.75E-07
$ x_8 $ 9 20 0.1005 2.89E-06 27 103 0.4661 2.52E-06 15 59 0.0524 4.75E-07
$ x_1 $ 7 20 0.4618 3.27E-08 13 44 0.3079 7.07E-07 2 10 0.0474 0.00E+00
$ x_2 $ 4 8 0.1123 5.35E-07 11 35 0.3012 3.33E-06 2 8 0.0923 0.00E+00
$ x_3 $ 5 11 0.1031 1.28E-07 12 38 23.8270 1.70E-06 4 14 0.1142 2.22E-16
$ x_4 $ 7 20 0.4022 3.43E-08 18 65 0.5567 8.62E-06 10 42 0.2322 7.89E-08
$ x_5 $ 9 21 0.3592 4.85E-06 18 65 0.3972 8.62E-06 16 65 0.4585 8.55E-07
$ x_6 $ 9 19 0.3959 4.92E-06 11 36 0.2964 3.32E-06 f f f f
$ x_7 $ 9 21 0.3742 4.85E-06 18 65 0.4319 8.62E-06 16 65 0.1987 8.49E-07
$ x_8 $ 9 21 0.3703 4.85E-06 18 65 0.5524 8.97E-06 16 65 0.5407 8.39E-07
$ x_1 $ 7 23 0.7369 4.77E-06 13 44 0.7618 1.00E-06 3 16 0.1932 0.00E+00
$ x_2 $ 4 8 0.2658 7.57E-07 11 35 0.4434 4.71E-06 2 8 0.1158 0.00E+00
$ x_3 $ 5 11 0.3353 1.28E-07 12 38 107.4606 1.70E-06 4 14 0.2615 2.22E-16
$ x_4 $ 7 23 0.7591 4.81E-06 19 68 0.4613 2.35E-06 14 59 0.6242 5.49E-07
$ x_5 $ 9 22 0.7743 8.65E-06 19 68 1.1533 2.35E-06 f f f f
$ x_6 $ 9 19 0.6254 4.92E-06 11 36 0.5491 3.32E-06 f f f f
$ x_7 $ 9 22 0.7055 8.65E-06 19 68 1.0239 2.35E-06 f f f f
$ x_8 $ 9 22 0.5172 8.66E-06 19 68 0.9601 2.41E-06 f f f f
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