CKV-type $ B $-matrices and error bounds for linear complementarity problems

Xinnian Song; Lei Gao; Xinnian Song; Lei Gao

doi:10.3934/math.2021630

AIMS Mathematics

2021, Volume 6, Issue 10: 10846-10860. doi: 10.3934/math.2021630

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Research article

CKV-type $ B $-matrices and error bounds for linear complementarity problems

Xinnian Song ,
Lei Gao ^,

School of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji, Shaanxi, 721013, China

Received: 11 May 2021 Accepted: 16 July 2021 Published: 27 July 2021
MSC : 15A24, 15A60, 90C33, 65G50

In this paper, we introduce a new subclass of $ P $-matrices called Cvetković-Kostić-Varga type $ B $-matrices (CKV-type $ B $-matrices), which contains DZ-type-$ B $-matrices as a special case, and present an infinity norm bound for the inverse of CKV-type $ B $-matrices. Based on this bound, we also give an error bound for linear complementarity problems of CKV-type $ B $-matrices. It is proved that the new error bound is better than that provided by Li et al. ^[24] for DZ-type-$ B $-matrices, and than that provided by M. García-Esnaola and J.M. Peña ^[10] for $ B $-matrices in some cases. Numerical examples demonstrate the effectiveness of the obtained results.
- CKV-type $ B $-matrices,
- linear complementarity problems,
- error bounds,
- infinity norm,
- $ P $-matrices
Citation: Xinnian Song, Lei Gao. CKV-type $ B $-matrices and error bounds for linear complementarity problems[J]. AIMS Mathematics, 2021, 6(10): 10846-10860. doi: 10.3934/math.2021630

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Abstract

In this paper, we introduce a new subclass of $ P $-matrices called Cvetković-Kostić-Varga type $ B $-matrices (CKV-type $ B $-matrices), which contains DZ-type-$ B $-matrices as a special case, and present an infinity norm bound for the inverse of CKV-type $ B $-matrices. Based on this bound, we also give an error bound for linear complementarity problems of CKV-type $ B $-matrices. It is proved that the new error bound is better than that provided by Li et al. ^[24] for DZ-type-$ B $-matrices, and than that provided by M. García-Esnaola and J.M. Peña ^[10] for $ B $-matrices in some cases. Numerical examples demonstrate the effectiveness of the obtained results.

References

[1]	A. Berman, R. J. Plemmons, Nonnegative Matrix in the Mathematical Sciences, Philadelphia: SIAM Publisher, 1994.
[2]	R. W. Cottle, J. S. Pang, R. E. Stone, The Linear Complementarity Problem, San Diego: Academic Press, 1992.
[3]	X. J. Chen, S. H. Xiang, Computation of error bounds for $P$-matrix linear complementarity problems, Math. Program, Ser., 106 (2006), 513–525. doi: 10.1007/s10107-005-0645-9
[4]	T. T. Chen, W. Li, X. Wu, S. Vong, Error bounds for linear complementarity problems of $MB$-matrices, Numer. Algorithms, 70 (2015), 341–356. doi: 10.1007/s11075-014-9950-9
[5]	D. Lj. Cvetković, L. Cvetković, C. Q. Li, CKV-type matrices with applications, Linear Algebra Appl., 608 (2021), 158–184. doi: 10.1016/j.laa.2020.08.028
[6]	P. F. Dai, Error bounds for linear complementarity problem of DB-matrices Linear Algebra Appl., 434 (2011), 830–840.
[7]	P. F. Dai, Y. T. Li, C. J. Lu, Error bounds for the linear complementarity problem for $SB$-matrices, Numer. Algorithms, 61 (2012), 121–139. doi: 10.1007/s11075-012-9533-6
[8]	P. F. Dai, C. J. Lu, Y. T. Li, New error bounds for the linear complementarity problem for $SB$-matrix, Numer. Algorithms, 64 (2013), 741–757. doi: 10.1007/s11075-012-9691-6
[9]	P. F. Dai, J. C. Li, Y. T. Li, C. Y. Zhang, Error bounds for linear complementarity problem of $QN$-matrices, Calcolo, 53 (2016), 647–657. doi: 10.1007/s10092-015-0167-7
[10]	M. García-Esnaola, J. M. Peña, Error bounds for the linear complementarity problem for $B$-matrices, Appl. Math. Lett., 22 (2009), 1071–1075. doi: 10.1016/j.aml.2008.09.001
[11]	M. García-Esnaola, J. M. Peña, $B$-Nekrasov matrices and error bounds for the linear complementarity problems, Numer. Algorithms, 72 (2016), 435–445. doi: 10.1007/s11075-015-0054-y
[12]	M. García-Esnaola, J. M. Peña, $B_\pi^{R}$-matrices and error bounds for linear complementarity problems, Calcolo, 54 (2017), 813–822. doi: 10.1007/s10092-016-0209-9
[13]	L. Gao, C. Q. Li, Y. T. Li, Parameterized error bounds for linear complementarity problems of $B_\pi^{R}$-matrices and their optimal values, Calcolo, 56 (2019), 31. doi: 10.1007/s10092-019-0328-1
[14]	M. García-Esnaola, J. M. Peña. A comparison of error bounds for linear complementarity problems of $H$-matrices, Linear Algebra Appl., 433 (2010), 956–964. doi: 10.1016/j.laa.2010.04.024
[15]	L. Gao, Y. Q. Wang, C. Q. Li, Y. T. Li, Error bounds for linear complementarity problems of $S$-Nekrasov matrices and $B$-$S$-Nekrasov matrices, J. Comput. Appl. Math., 336 (2018), 147–159. doi: 10.1016/j.cam.2017.12.032
[16]	L. Gao, C. Q. Li, New error bounds for linear complementarity problem of $QN$-matrices, Numer. Algorithms, 80 (2018), 229–242.
[17]	M. García-Esnaola, J. M. Peña, On the asymptotic of error bounds for some linear complementarity problems, Numer. Algorithms, 80 (2019), 521–532. doi: 10.1007/s11075-018-0495-1
[18]	Z. Q. Luo, P. Tseng, Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem, SIAM J. Optimiz., 2 (1992), 43–54. doi: 10.1137/0802004
[19]	Z. Q. Luo, P. Tseng, On the linear convergence of descent methods for convex essentially smooth minimization, SIAM J. Control Optim., 30 (1992), 408–425. doi: 10.1137/0330025
[20]	C. Q. Li, Y. T. Li, Note on error bounds for linear complementarity problems of $B$-matrices, Appl. Math. Lett., 57 (2016), 108–113 doi: 10.1016/j.aml.2016.01.013
[21]	C. Q. Li, P. F. Dai, Y. T. Li, New error bounds for linear complementarity problems of Nekrasov matrices and $B$-Nekrasov matrices, Numer. Algorithms, 74 (2017), 997–1009. doi: 10.1007/s11075-016-0181-0
[22]	C. Q. Li, Y. T. Li, Weakly chained diagonally dominant $B$-matrices and error bounds for linear complementarity problems, Numer. Algorithms, 73 (2016), 985–998. doi: 10.1007/s11075-016-0125-8
[23]	W. Li, H. Zhang, Some new error bounds for linear complementarity problems of $H$-matrices, Numer. Algorithums, 67 (2014), 257–269. doi: 10.1007/s11075-013-9786-8
[24]	C. Q. Li, L. Cvetković, Y. Wei, J. X. Zhao, An infinity norm bound for the inverse of Dashnic-Zusmanovich type matrices with applications, Linear Algebra Appl., 565 (2019), 99–122. doi: 10.1016/j.laa.2018.12.013
[25]	H. B. Li, T. Z. Huang, H. Li, On some subclasses of $P$-matrices, Numer. Linear Algebra Appl., 14 (2007), 391–405. doi: 10.1002/nla.524
[26]	K. G. Murty, Linear Complementarity, Linear and Nonlinear Programming, Berlin: Heldermann Verlag, 1988.
[27]	H. Orera, J. M. Peña, Error bounds for linear complementarity problems of $B_\pi^{R}$-matrices, Comp. Appl. Math., 40 (2021), 94. doi: 10.1007/s40314-021-01491-w
[28]	J. S. Pang, A posteriori error bounds for the linearly-constrained variational inequality problem, Math. Oper. Res., 12 (1987), 474–484. doi: 10.1287/moor.12.3.474
[29]	J. M. Peña, On an alternative to Geršchgorin circle and ovals of Cassini, Numer. Math., 95 (2003), 337–345. doi: 10.1007/s00211-002-0427-8
[30]	J. M. Peña, A class of $P$-matrices with applications to the localization of the eigenvalues of a real matrix, SIAM J. Matrix Anal. Appl., 22 (2001), 1027–1037. doi: 10.1137/S0895479800370342
[31]	P. N. Shivakumar, K. H. Chew, A sufficient condition for nonvanishing of determinants, Proc. Amer. Math. Soc., 43 (1974), 63–66. doi: 10.1090/S0002-9939-1974-0332820-0
[32]	C. L. Sang, Z. Chen, A new error bound for linear complementarity problems of weakly chained diagonally dominant $B$-matrices, Linear Multilinear A., 69 (2021), 1909–1921. doi: 10.1080/03081087.2019.1649995
[33]	J. X. Zhao, Q. L. Liu, C. Q. Li, Y. T. Li, Dashnic-Zusmanovich type matrices: A new subclass of nonsingular $H$-matrices, Linear Algebra Appl., 552 (2018), 277–287. doi: 10.1016/j.laa.2018.04.028
[34]	R. J. Zhao, B. Zheng, M. L. Liang, A new error bound for linear complementarity problems with weakly chained diagonally dominant $B$-matrices, Appl. Math. Compt., 367 (2020), 124788. doi: 10.1016/j.amc.2019.124788
[35]	F. Wang, D. S. Sun, New error bound for linear complementarity problems for $B$-matrices, Linear Multilinear A., 66 (2018), 2154–2167.
[36]	Z. F. Wang, C. Q. Li, Y. T. Li, Infimum of error bounds for linear complementarity problems of $\Sigma$-SDD and $\Sigma_1$-SSD matrices, Linear Algebra Appl., 581 (2019), 285–303. doi: 10.1016/j.laa.2019.07.020

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