American Institute of Mathematical Sciences

Advanced
doi: 10.3934/naco.2020053

A primal-dual interior point method for $ P_{\ast }\left( \kappa \right) $-HLCP based on a class of parametric kernel functions

Nadia Hazzam and  Zakia Kebbiche

Laboratoire de Mathématiques Fondamentales et Numériques, Université Ferhat Abbas Sétif1. Sétif. Algérie

Received  June 2020 Revised  October 2020 Published  November 2020

In an attempt to improve theoretical complexity of large-update methods, in this paper, we propose a primal-dual interior-point method for $ P_{\ast}\left( \kappa \right) $-horizontal linear complementarity problem. The method is based on a class of parametric kernel functions. We show that the corresponding algorithm has $ O\left( \left( 1+2\kappa \right) p^{2}n^{\frac{2+p}{2\left( 1+p\right) }}\log \frac{n}{\epsilon }\right) $ iteration complexity for large-update methods and we match the best known iteration bounds with special choice of the parameter $ p $ for $ P_{\ast }\left(\kappa \right) $-horizontal linear complementarity problem that is $ O\left(\left( 1+2\kappa \right) \sqrt{n}\log n\log \frac{n}{\epsilon }\right) $. We illustrate the performance of the proposed kernel function by some comparative numerical results that are derived by applying our algorithm on five kernel functions.

Keywords: Horizontal linear complementarity problem, P*(κ)-matrix, interior-point method, kernel function, complexity bound.
Mathematics Subject Classification: Primary: 90C33; Secondary: 90C51.
Citation: Nadia Hazzam, Zakia Kebbiche. A primal-dual interior point method for $ P_{\ast }\left( \kappa \right) $-HLCP based on a class of parametric kernel functions. Numerical Algebra, Control & Optimization, doi: 10.3934/naco.2020053
References:
[1]

M. Achache, Complexity analysis of a weighted-full-Newton step interior-point algorithm for $P_{\ast }\left(\kappa \right)$-LCP, RAIRO-Oper. Res., 50 (2016), 131-143.  doi: 10.1051/ro/2015020.  Google Scholar

[2]

M. Achache and N. Tabchouche, Complexity analysis and numerical implementation of large-update interior-point methods for SDLCP based on a new parametric barrier kernel function, Optimization, 67 (2018), 1211-1230.  doi: 10.1080/02331934.2018.1462356.  Google Scholar

[3]

S. Asadi and H. Mansouri, Polynomial interior-point algorithm for $P_{\ast }\left(\kappa \right)$-horizontal linear complementarity problems, Numer. Algorithms, 63 (2013), 385-398.  doi: 10.1007/s11075-012-9628-0.  Google Scholar

[4]

S. Asadi, H. Mansouri and M. Zangiabadi, A class of path-following interior-point methods for $P_{\ast }\left(\kappa \right)$-horizontal linear complementarity problems, J. Oper Res Soc China, 3 (2015), 17-30.  doi: 10.1007/s40305-015-0070-6.  Google Scholar

[5]

S. Asadi, M. Zangiabadi and H. Mansouri, An infeasible interior-point algorithm with full-Newton steps for $P_{\ast }\left(\kappa \right)$-horizontal linear complementarity problems based on a kernel function, J. Appl. Math. Comput., 50 (2016), 15-37.  doi: 10.1007/s12190-014-0856-4.  Google Scholar

[6]

Y. Q. Bai, M. El Ghami and C. Roos, A new efficient large-update primal-dual interior-point method based on a finite barrier, SIAM J. Optim., 13 (2003), 766-782.  doi: 10.1137/S1052623401398132.  Google Scholar

[7]

Y. Q. Bai, M. El Ghami and C. Roos, A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization, SIAM J. Optim., 15 (2004), 101-128.  doi: 10.1137/S1052623403423114.  Google Scholar

[8]

Y. Q. Bai and C. Roos, A primal-dual interior-point method based on a new kernel function with linear growth rate, Proceedings of the 9th Australian Optimization Day, 2002. Google Scholar

[9]

M. Bouafia, D. Benterki and A. Yassine, An efficient primal–dual interior point method for linear programming problems based on a new kernel function with a trigonometric barrier term, J. Optim. Theory Appl., 170 (2016), 528-545.  doi: 10.1007/s10957-016-0895-0.  Google Scholar

[10]

Ch. Chennouf, Extension d'une Méthode de Point Intérieur au Problème Complémentaire Linéaire avec $P_{\ast }\left(\kappa \right)-$matrice, Mémoire de Master $2, $ Université Ferhat Abbas Sétif1, Algérie, 2018. Google Scholar

[11]

G. M. Cho and M. K. Kim, A new large-update interior point algorithm for $P_{\ast}\left(\kappa \right)$ LCPs based on kernel functions, Appl. Math. Comput., 182 (2006), 1169-1183.  doi: 10.1016/j.amc.2006.04.060.  Google Scholar

[12]

G. M. Cho, A new large–update interior point algorithm for $P_{\ast}\left(\kappa \right)$ linear complementarity problems, J. Comput. Appl. Math., 216 (2008), 256-278.  doi: 10.1016/j.cam.2007.05.007.  Google Scholar

[13]

G. M. Cho, Large–update interior point algorithm for $P_{\ast }$-linear complementarity problem, J. Inequalities Appl., 363 (2014), 1-12.  doi: 10.1186/1029-242X-2014-363.  Google Scholar

[14] R. W. Cottle, J. S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, San Diego, 1992.   Google Scholar
[15]

M. El Ghami and T. Steihaug, Kernel–function based primal-dual algorithms for $P_{\ast }\left(\kappa \right) $ linear comlementarity problem, RAIRO-Oper. Res., 44 (2010), 185-205.  doi: 10.1051/ro/2010014.  Google Scholar

[16]

M. El Ghami and G. Q. Wang, Interior–point methods for $P_{\ast}\left(\kappa \right) $ linear comlementarity problem based on generalized trigonometric barrier function, International Journal of Applied Mathematics, (2017), 11–33. doi: 10.12732/ijam.v30i1.2.  Google Scholar

[17]

S. Fathi-Hafshejani and A. Fakharzadeh Jahromi, An interior point method for $P_{\ast}\left(\kappa \right) $-horizontal linear complementarity problem based on a new proximity function, J. Appl. Math. Comput., 62 (2020), 281-300.  doi: 10.1007/s12190-019-01284-9.  Google Scholar

[18]

S. Fathi-Hafshejani, H. Mansouri and M. Peyghamic, An interior-point algorithm for $P_{\ast }\left(\kappa \right) $-linear complementarity problem based on a new trigonometric kernel functions, Journal of Mathematical Modeling, 5 (2017), 171-197.   Google Scholar

[19]

P. Ji, M. Zhang and X. Li, A Primal-dual large-update interior-point algorithm for $P_{\ast}(\kappa)$-LCP based on a new class of kernel functions, Acta Mathematicae Applicatae Sinica, English Series, 34 (2018), 119-134.  doi: 10.1007/s10255-018-0729-y.  Google Scholar

[20]

M. Kojima, N. Megiddo, T. Noma and A. Yoshise, A unified approach to interior point algorithms for linear complementarity problems: A summary, Operations Research Letters, 10 (1991), 247-254.  doi: 10.1016/0167-6377(91)90010-M.  Google Scholar

[21]

Y. Lee, Y. Cho and G. Cho, Kernel function based interior-point methods for horizontal linear complementarity problems, Journal of Inequalities and Applications, (2013), Article number: 215. doi: 10.1186/1029-242X-2013-215.  Google Scholar

[22]

G. Lesaja and C. Roos, Unified analysis of kernel–based interior–point methods for $P_{\ast } (\kappa) $-LCP, SIAM Journal on Optimization, 20 (2010), 3014-3039.  doi: 10.1137/090766735.  Google Scholar

[23]

J. Peng, C. Roos and T. Terlaky, Self-regular functions and new search directions for linear and semidefinite optimization, Math. Program., Ser., 93 (2002), 129-171.  doi: 10.1007/s101070200296.  Google Scholar

[24]

Z. G. Qian and Y. Q. Bai, Primal-dual interior point algorithm with dynamic step-size based on kernel function for linear programming, Journal of Shanghai University (English Edition), 9 (2005), 391-396.  doi: 10.1007/s11741-005-0021-2.  Google Scholar

[25]

M. Reza Peyghami, S. Fathi Hafshejani and L. Shirvani, Complexity of interior-point methods for linear optimization based on a new trigonometric kernel function, Journal of Computational and Applied Mathematics, 2 (2014), 74-85.  doi: 10.1016/j.cam.2013.04.039.  Google Scholar

[26]

C. Roos, T. Terlaky and J. Ph. Vial, Theory and Algorithms for Linear optimization. An Interior Point Approach, John Wiley and Sons, Chichester, 1997.  Google Scholar

[27]

G. Q. Wang and Y. Q. Bai, Polynomial interior-point algorithm for $P_{\ast}\left(\kappa \right)$ horizontal linear complementarity problem, J. Comput. Appl. Math., 233 (2009), 248-263.  doi: 10.1016/j.cam.2009.07.014.  Google Scholar

[28]

S. J. Wright, Primal-Dual Interior Point Methods, SIAM, 1997. doi: 10.1137/1.9781611971453.  Google Scholar

show all references

References:
[1]

M. Achache, Complexity analysis of a weighted-full-Newton step interior-point algorithm for $P_{\ast }\left(\kappa \right)$-LCP, RAIRO-Oper. Res., 50 (2016), 131-143.  doi: 10.1051/ro/2015020.  Google Scholar

[2]

M. Achache and N. Tabchouche, Complexity analysis and numerical implementation of large-update interior-point methods for SDLCP based on a new parametric barrier kernel function, Optimization, 67 (2018), 1211-1230.  doi: 10.1080/02331934.2018.1462356.  Google Scholar

[3]

S. Asadi and H. Mansouri, Polynomial interior-point algorithm for $P_{\ast }\left(\kappa \right)$-horizontal linear complementarity problems, Numer. Algorithms, 63 (2013), 385-398.  doi: 10.1007/s11075-012-9628-0.  Google Scholar

[4]

S. Asadi, H. Mansouri and M. Zangiabadi, A class of path-following interior-point methods for $P_{\ast }\left(\kappa \right)$-horizontal linear complementarity problems, J. Oper Res Soc China, 3 (2015), 17-30.  doi: 10.1007/s40305-015-0070-6.  Google Scholar

[5]

S. Asadi, M. Zangiabadi and H. Mansouri, An infeasible interior-point algorithm with full-Newton steps for $P_{\ast }\left(\kappa \right)$-horizontal linear complementarity problems based on a kernel function, J. Appl. Math. Comput., 50 (2016), 15-37.  doi: 10.1007/s12190-014-0856-4.  Google Scholar

[6]

Y. Q. Bai, M. El Ghami and C. Roos, A new efficient large-update primal-dual interior-point method based on a finite barrier, SIAM J. Optim., 13 (2003), 766-782.  doi: 10.1137/S1052623401398132.  Google Scholar

[7]

Y. Q. Bai, M. El Ghami and C. Roos, A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization, SIAM J. Optim., 15 (2004), 101-128.  doi: 10.1137/S1052623403423114.  Google Scholar

[8]

Y. Q. Bai and C. Roos, A primal-dual interior-point method based on a new kernel function with linear growth rate, Proceedings of the 9th Australian Optimization Day, 2002. Google Scholar

[9]

M. Bouafia, D. Benterki and A. Yassine, An efficient primal–dual interior point method for linear programming problems based on a new kernel function with a trigonometric barrier term, J. Optim. Theory Appl., 170 (2016), 528-545.  doi: 10.1007/s10957-016-0895-0.  Google Scholar

[10]

Ch. Chennouf, Extension d'une Méthode de Point Intérieur au Problème Complémentaire Linéaire avec $P_{\ast }\left(\kappa \right)-$matrice, Mémoire de Master $2, $ Université Ferhat Abbas Sétif1, Algérie, 2018. Google Scholar

[11]

G. M. Cho and M. K. Kim, A new large-update interior point algorithm for $P_{\ast}\left(\kappa \right)$ LCPs based on kernel functions, Appl. Math. Comput., 182 (2006), 1169-1183.  doi: 10.1016/j.amc.2006.04.060.  Google Scholar

[12]

G. M. Cho, A new large–update interior point algorithm for $P_{\ast}\left(\kappa \right)$ linear complementarity problems, J. Comput. Appl. Math., 216 (2008), 256-278.  doi: 10.1016/j.cam.2007.05.007.  Google Scholar

[13]

G. M. Cho, Large–update interior point algorithm for $P_{\ast }$-linear complementarity problem, J. Inequalities Appl., 363 (2014), 1-12.  doi: 10.1186/1029-242X-2014-363.  Google Scholar

[14] R. W. Cottle, J. S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, San Diego, 1992.   Google Scholar
[15]

M. El Ghami and T. Steihaug, Kernel–function based primal-dual algorithms for $P_{\ast }\left(\kappa \right) $ linear comlementarity problem, RAIRO-Oper. Res., 44 (2010), 185-205.  doi: 10.1051/ro/2010014.  Google Scholar

[16]

M. El Ghami and G. Q. Wang, Interior–point methods for $P_{\ast}\left(\kappa \right) $ linear comlementarity problem based on generalized trigonometric barrier function, International Journal of Applied Mathematics, (2017), 11–33. doi: 10.12732/ijam.v30i1.2.  Google Scholar

[17]

S. Fathi-Hafshejani and A. Fakharzadeh Jahromi, An interior point method for $P_{\ast}\left(\kappa \right) $-horizontal linear complementarity problem based on a new proximity function, J. Appl. Math. Comput., 62 (2020), 281-300.  doi: 10.1007/s12190-019-01284-9.  Google Scholar

[18]

S. Fathi-Hafshejani, H. Mansouri and M. Peyghamic, An interior-point algorithm for $P_{\ast }\left(\kappa \right) $-linear complementarity problem based on a new trigonometric kernel functions, Journal of Mathematical Modeling, 5 (2017), 171-197.   Google Scholar

[19]

P. Ji, M. Zhang and X. Li, A Primal-dual large-update interior-point algorithm for $P_{\ast}(\kappa)$-LCP based on a new class of kernel functions, Acta Mathematicae Applicatae Sinica, English Series, 34 (2018), 119-134.  doi: 10.1007/s10255-018-0729-y.  Google Scholar

[20]

M. Kojima, N. Megiddo, T. Noma and A. Yoshise, A unified approach to interior point algorithms for linear complementarity problems: A summary, Operations Research Letters, 10 (1991), 247-254.  doi: 10.1016/0167-6377(91)90010-M.  Google Scholar

[21]

Y. Lee, Y. Cho and G. Cho, Kernel function based interior-point methods for horizontal linear complementarity problems, Journal of Inequalities and Applications, (2013), Article number: 215. doi: 10.1186/1029-242X-2013-215.  Google Scholar

[22]

G. Lesaja and C. Roos, Unified analysis of kernel–based interior–point methods for $P_{\ast } (\kappa) $-LCP, SIAM Journal on Optimization, 20 (2010), 3014-3039.  doi: 10.1137/090766735.  Google Scholar

[23]

J. Peng, C. Roos and T. Terlaky, Self-regular functions and new search directions for linear and semidefinite optimization, Math. Program., Ser., 93 (2002), 129-171.  doi: 10.1007/s101070200296.  Google Scholar

[24]

Z. G. Qian and Y. Q. Bai, Primal-dual interior point algorithm with dynamic step-size based on kernel function for linear programming, Journal of Shanghai University (English Edition), 9 (2005), 391-396.  doi: 10.1007/s11741-005-0021-2.  Google Scholar

[25]

M. Reza Peyghami, S. Fathi Hafshejani and L. Shirvani, Complexity of interior-point methods for linear optimization based on a new trigonometric kernel function, Journal of Computational and Applied Mathematics, 2 (2014), 74-85.  doi: 10.1016/j.cam.2013.04.039.  Google Scholar

[26]

C. Roos, T. Terlaky and J. Ph. Vial, Theory and Algorithms for Linear optimization. An Interior Point Approach, John Wiley and Sons, Chichester, 1997.  Google Scholar

[27]

G. Q. Wang and Y. Q. Bai, Polynomial interior-point algorithm for $P_{\ast}\left(\kappa \right)$ horizontal linear complementarity problem, J. Comput. Appl. Math., 233 (2009), 248-263.  doi: 10.1016/j.cam.2009.07.014.  Google Scholar

[28]

S. J. Wright, Primal-Dual Interior Point Methods, SIAM, 1997. doi: 10.1137/1.9781611971453.  Google Scholar

Table 1.  Some kernel functions
${\psi _1}(t) = p\left( {\frac{{{t^2} - 1}}{2}} \right) + \frac{4}{\pi }\left( {{{\tan }^p}\left( {\frac{\pi }{{2t + 2}}} \right) - 1} \right),t > 0,p \ge \sqrt 2 ,$
$\alpha_{1}=\frac{1}{(1+2 \kappa)\left(9 p+4 \pi p^{2}\right)\left(\frac{8}{p}+2\right)^{\frac{p+2}{p+1}}}.$
$\psi_{2}(t)=\frac{t^{2}-1}{2}-\log t+\frac{1}{8} \tan ^{2}\left(\frac{\pi(1-t)}{2+4 t}\right), t>0,[25]$
$\alpha_{2}=\frac{1}{(1+2 \kappa) 5020 \delta^{\frac{4}{3}}}.$
$\psi_{3}(t)=t-1+\frac{t^{1-q}-1}{q-1}, t>0, q \geq 2,[8]$
$\alpha_{3}=\frac{1}{(1+2 \kappa) q(2 \delta+1)^{\frac{1}{q}}(4 \delta+1)}.$
$\psi_{4}(t)=\frac{t^{2}-1}{2}+\frac{q^{\frac{1}{t}-1}}{q \log q}-\frac{q-1}{q}(t-1), q>1,[2]$
$\alpha_{4}=\frac{1}{(1+2 \kappa)(\log q+2)(1+4 \delta)\left(2+\frac{\log (1+4 \delta)}{\log q}\right)}.$
$\psi_{5}(t)=\frac{t^{2}-1}{2}+\frac{4}{p \pi}\left(\tan ^{p}\left(\frac{\pi}{2 t+2}\right)-1\right), t>0, p \geq 2,[9]$
$\alpha_{5}=\frac{1}{(1+2 \kappa)(9+4 p \pi)(8 \delta+2)^{\frac{p+2}{p+1}}}.$
Table Options

Download as excel

${\psi _1}(t) = p\left( {\frac{{{t^2} - 1}}{2}} \right) + \frac{4}{\pi }\left( {{{\tan }^p}\left( {\frac{\pi }{{2t + 2}}} \right) - 1} \right),t > 0,p \ge \sqrt 2 ,$
$\alpha_{1}=\frac{1}{(1+2 \kappa)\left(9 p+4 \pi p^{2}\right)\left(\frac{8}{p}+2\right)^{\frac{p+2}{p+1}}}.$
$\psi_{2}(t)=\frac{t^{2}-1}{2}-\log t+\frac{1}{8} \tan ^{2}\left(\frac{\pi(1-t)}{2+4 t}\right), t>0,[25]$
$\alpha_{2}=\frac{1}{(1+2 \kappa) 5020 \delta^{\frac{4}{3}}}.$
$\psi_{3}(t)=t-1+\frac{t^{1-q}-1}{q-1}, t>0, q \geq 2,[8]$
$\alpha_{3}=\frac{1}{(1+2 \kappa) q(2 \delta+1)^{\frac{1}{q}}(4 \delta+1)}.$
$\psi_{4}(t)=\frac{t^{2}-1}{2}+\frac{q^{\frac{1}{t}-1}}{q \log q}-\frac{q-1}{q}(t-1), q>1,[2]$
$\alpha_{4}=\frac{1}{(1+2 \kappa)(\log q+2)(1+4 \delta)\left(2+\frac{\log (1+4 \delta)}{\log q}\right)}.$
$\psi_{5}(t)=\frac{t^{2}-1}{2}+\frac{4}{p \pi}\left(\tan ^{p}\left(\frac{\pi}{2 t+2}\right)-1\right), t>0, p \geq 2,[9]$
$\alpha_{5}=\frac{1}{(1+2 \kappa)(9+4 p \pi)(8 \delta+2)^{\frac{p+2}{p+1}}}.$
Table 2.   
$ \psi _{1}\left( t\right) $ $ \psi _{2}\left( t\right) $ $ \psi _{3}\left( t\right) $ $ \psi _{4}\left( t\right) $ $ \psi _{5}\left( t\right) $
Inn $ 5 $ $ 282 $ $ 10 $ $ 16 $ $ 5 $
Time $ 0.07 $ $ 0.18 $ $ 0.062 $ $ 0.06 $ $ 0.06 $
Table Options

Download as excel

$ \psi _{1}\left( t\right) $ $ \psi _{2}\left( t\right) $ $ \psi _{3}\left( t\right) $ $ \psi _{4}\left( t\right) $ $ \psi _{5}\left( t\right) $
Inn $ 5 $ $ 282 $ $ 10 $ $ 16 $ $ 5 $
Time $ 0.07 $ $ 0.18 $ $ 0.062 $ $ 0.06 $ $ 0.06 $
Table 3.   
$ \psi _{1}\left( t\right) $ $ \psi _{2}\left( t\right) $ $ \psi _{3}\left( t\right) $ $ \psi _{4}\left( t\right) $ $ \psi _{5}\left( t\right) $
Inn $ 7 $ $ 167 $ $ 12 $ $ 19 $ $ 10 $
Time $ 0.06 $ $ 0.10 $ $ 0.06 $ $ 0.063 $ $ 0.07 $
Table Options

Download as excel

$ \psi _{1}\left( t\right) $ $ \psi _{2}\left( t\right) $ $ \psi _{3}\left( t\right) $ $ \psi _{4}\left( t\right) $ $ \psi _{5}\left( t\right) $
Inn $ 7 $ $ 167 $ $ 12 $ $ 19 $ $ 10 $
Time $ 0.06 $ $ 0.10 $ $ 0.06 $ $ 0.063 $ $ 0.07 $
Table 4.   
$ n=3 $ $ n=100 $ $ n=150 $
Inn Time Inn Time Inn Time
$ \psi _{1}\left( t\right) $ $ 4 $ $ 0.06 $ $ 12 $ $ 0.45 $ $ 18 $ $ 3.06 $
$ \psi _{2}\left( t\right) $ $ 179 $ $ 0.11 $ $ 70 $ $ 9.07 $ $ 1447 $ $ 11.6 $
$ \psi _{3}\left( t\right) $ $ 15 $ $ 0.06 $ $ 52 $ $ 0.19 $ $ 77 $ $ 0.91 $
$ \psi _{4}\left( t\right) $ $ 14 $ $ 0.063 $ $ - $ $ - $ $ - $ $ - $
$ \psi _{5}\left( t\right) $ $ 8 $ $ 0.068 $ $ 23 $ $ 0.63 $ $ 28 $ $ 2.30 $
Table Options

Download as excel

$ n=3 $ $ n=100 $ $ n=150 $
Inn Time Inn Time Inn Time
$ \psi _{1}\left( t\right) $ $ 4 $ $ 0.06 $ $ 12 $ $ 0.45 $ $ 18 $ $ 3.06 $
$ \psi _{2}\left( t\right) $ $ 179 $ $ 0.11 $ $ 70 $ $ 9.07 $ $ 1447 $ $ 11.6 $
$ \psi _{3}\left( t\right) $ $ 15 $ $ 0.06 $ $ 52 $ $ 0.19 $ $ 77 $ $ 0.91 $
$ \psi _{4}\left( t\right) $ $ 14 $ $ 0.063 $ $ - $ $ - $ $ - $ $ - $
$ \psi _{5}\left( t\right) $ $ 8 $ $ 0.068 $ $ 23 $ $ 0.63 $ $ 28 $ $ 2.30 $
Table 5.   
$ n=3 $ $ n=50 $ $ n=150 $
Inn Time Inn Time Inn Time
$ \psi _{1}\left( t\right) $ $ 3 $ $ 0.07 $ $ 10 $ $ 0.12 $ $ 19 $ $ 1.73 $
$ \psi _{2}\left( t\right) $ $ 259 $ $ 0.15 $ $ 615 $ $ 0.67 $ $ 1301 $ $ 13.96 $
$ \psi _{3}\left( t\right) $ $ 16 $ $ 0.05 $ $ 215 $ $ 0.45 $ $ 71 $ $ 0.64 $
$ \psi _{4}\left( t\right) $ $ 18 $ $ 0.05 $ $ - $ $ - $ $ - $ $ - $
$ \psi _{5}\left( t\right) $ $ 12 $ $ 0.07 $ $ 18 $ $ 0.15 $ $ 39 $ $ 2.47 $
Table Options

Download as excel

$ n=3 $ $ n=50 $ $ n=150 $
Inn Time Inn Time Inn Time
$ \psi _{1}\left( t\right) $ $ 3 $ $ 0.07 $ $ 10 $ $ 0.12 $ $ 19 $ $ 1.73 $
$ \psi _{2}\left( t\right) $ $ 259 $ $ 0.15 $ $ 615 $ $ 0.67 $ $ 1301 $ $ 13.96 $
$ \psi _{3}\left( t\right) $ $ 16 $ $ 0.05 $ $ 215 $ $ 0.45 $ $ 71 $ $ 0.64 $
$ \psi _{4}\left( t\right) $ $ 18 $ $ 0.05 $ $ - $ $ - $ $ - $ $ - $
$ \psi _{5}\left( t\right) $ $ 12 $ $ 0.07 $ $ 18 $ $ 0.15 $ $ 39 $ $ 2.47 $
[1]

Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006
[2]

Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271
[3]

Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247
[4]

Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617
[5]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056
[6]

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
[7]

Jong Yoon Hyun, Yoonjin Lee, Yansheng Wu. Connection of $ p $-ary $ t $-weight linear codes to Ramanujan Cayley graphs with $ t+1 $ eigenvalues. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020133
[8]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021014
[9]

Vakhtang Putkaradze, Stuart Rogers. Numerical simulations of a rolling ball robot actuated by internal point masses. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 143-207. doi: 10.3934/naco.2020021
[10]

Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933
[11]

Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265
[12]

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
[13]

Hyeong-Ohk Bae, Hyoungsuk So, Yeonghun Youn. Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth. Networks & Heterogeneous Media, 2018, 13 (3) : 479-491. doi: 10.3934/nhm.2018021
[14]

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
[15]

Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022
[16]

Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021
[17]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[18]

Guillaume Bal, Wenjia Jing. Homogenization and corrector theory for linear transport in random media. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1311-1343. doi: 10.3934/dcds.2010.28.1311
[19]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203
[20]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

 Impact Factor: 

Tools

Metrics

  • PDF downloads (47)
  • HTML views (144)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences