A penalty approach for a box constrained variational inequality problem

Zahira Kebaili; Djamel Benterki

Applications of Mathematics (2018)

  • Volume: 63, Issue: 4, page 439-454
  • ISSN: 0862-7940

Abstract

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We propose a penalty approach for a box constrained variational inequality problem ( BVIP ) . This problem is replaced by a sequence of nonlinear equations containing a penalty term. We show that if the penalty parameter tends to infinity, the solution of this sequence converges to that of BVIP when the function F involved is continuous and strongly monotone and the box C contains the origin. We develop the algorithmic aspect with theoretical arguments properly established. The numerical results tested on some examples are satisfactory and confirm the theoretical approach.

How to cite

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Kebaili, Zahira, and Benterki, Djamel. "A penalty approach for a box constrained variational inequality problem." Applications of Mathematics 63.4 (2018): 439-454. <http://eudml.org/doc/294762>.

@article{Kebaili2018,
abstract = {We propose a penalty approach for a box constrained variational inequality problem $(\rm BVIP)$. This problem is replaced by a sequence of nonlinear equations containing a penalty term. We show that if the penalty parameter tends to infinity, the solution of this sequence converges to that of $\rm BVIP$ when the function $F$ involved is continuous and strongly monotone and the box $C$ contains the origin. We develop the algorithmic aspect with theoretical arguments properly established. The numerical results tested on some examples are satisfactory and confirm the theoretical approach.},
author = {Kebaili, Zahira, Benterki, Djamel},
journal = {Applications of Mathematics},
keywords = {box constrained variational inequality problem; power penalty approach; strongly monotone operator},
language = {eng},
number = {4},
pages = {439-454},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A penalty approach for a box constrained variational inequality problem},
url = {http://eudml.org/doc/294762},
volume = {63},
year = {2018},
}

TY - JOUR
AU - Kebaili, Zahira
AU - Benterki, Djamel
TI - A penalty approach for a box constrained variational inequality problem
JO - Applications of Mathematics
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 4
SP - 439
EP - 454
AB - We propose a penalty approach for a box constrained variational inequality problem $(\rm BVIP)$. This problem is replaced by a sequence of nonlinear equations containing a penalty term. We show that if the penalty parameter tends to infinity, the solution of this sequence converges to that of $\rm BVIP$ when the function $F$ involved is continuous and strongly monotone and the box $C$ contains the origin. We develop the algorithmic aspect with theoretical arguments properly established. The numerical results tested on some examples are satisfactory and confirm the theoretical approach.
LA - eng
KW - box constrained variational inequality problem; power penalty approach; strongly monotone operator
UR - http://eudml.org/doc/294762
ER -

References

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  1. Auslender, A., Optimisation. Méthodes numériques, Maitrise de mathématiques et applications fondamentales. Masson, Paris French (1976). (1976) Zbl0326.90057MR0441204
  2. Censor, Y., Iusem, A. N., Zenios, S. A., 10.1007/BF01580089, Math. Program. 81 (1998), 373-400. (1998) Zbl0919.90123MR1617732DOI10.1007/BF01580089
  3. Facchinei, F., Pang, J.-S., 10.1007/b97543, Springer Series in Operations Research Springer, New York (2003). (2003) Zbl1062.90001MR1955648DOI10.1007/b97543
  4. Facchinei, F., Pang, J.-S., 10.1007/b97544, Springer Series in Operations Research Springer, New York (2003). (2003) Zbl1062.90002MR1955649DOI10.1007/b97544
  5. Fukushima, M., 10.1007/BF01585696, Math. Program., Ser. A 53 (1992), 99-110. (1992) Zbl0756.90081MR1151767DOI10.1007/BF01585696
  6. Hao, Z., Wan, Z., Chi, X., Chen, J., 10.1016/j.cam.2015.05.007, J. Comput. Appl. Math. 290 (2015), 136-149. (2015) Zbl1327.90207MR3370398DOI10.1016/j.cam.2015.05.007
  7. Harker, P. T., Pang, J.-S., 10.1007/BF01582255, Math. Program., Ser. B 48 (1990), 161-220. (1990) Zbl0734.90098MR1073707DOI10.1007/BF01582255
  8. Huang, C., Wang, S., 10.1016/j.orl.2009.09.009, Oper. Res. Lett. 38 (2010), 72-76. (2010) Zbl1182.90090MR2565819DOI10.1016/j.orl.2009.09.009
  9. Huang, C., Wang, S., 10.1016/j.na.2011.08.061, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75 (2012), 588-597. (2012) Zbl1233.65040MR2847442DOI10.1016/j.na.2011.08.061
  10. Kanzow, C., Fukushima, M., 10.1007/BF02680550, Math. Program. 83 (1998), 55-87. (1998) Zbl0920.90134MR1643959DOI10.1007/BF02680550
  11. Ma, C., Kang, T., 10.1016/j.amc.2004.03.018, Appl. Math. Comput. 162 (2005), 1397-1429. (2005) Zbl1068.65089MR2113979DOI10.1016/j.amc.2004.03.018
  12. Noor, M. A., Wang, Y., Xiu, N., 10.1016/S0096-3003(02)00148-0, Appl. Math. Comput. 137 (2003), 423-435. (2003) Zbl1031.65078MR1950108DOI10.1016/S0096-3003(02)00148-0
  13. Sun, D.-F., A projection and contraction method for the nonlinear complementarity problem and its extensions, Chin. J. Numer. Math. Appl. 16 (1994), 73-84 English. Chinese original translation from Math. Numer. Sin. 16 1994 183-194. (1994) Zbl0900.65188MR1459564
  14. Tang, J., Liu, S., 10.1016/j.nonrwa.2009.10.002, Nonlinear Anal., Real World Appl. 11 (2010), 2770-2786. (2010) Zbl1208.90172MR2661943DOI10.1016/j.nonrwa.2009.10.002
  15. Ulji, Chen, G.-Q., 10.1007/BF02466422, Appl. Math. Mech., Engl. Ed. 26 (2005), 1083-1092 Appl. Math. Mech. 26 2005 988-996 Chinese. English, Chinese summary. (2005) Zbl1144.65309MR2169264DOI10.1007/BF02466422
  16. Wang, S., Yang, X., 10.1016/j.orl.2007.06.006, Oper. Res. Lett. 36 (2008), 211-214. (2008) Zbl1163.90762MR2396598DOI10.1016/j.orl.2007.06.006
  17. Wang, Y., Zhu, D., 10.1007/s11401-007-0082-6, Chin. Ann. Math., Ser. B 29 (2008), 273-290. (2008) Zbl1151.49025MR2421761DOI10.1007/s11401-007-0082-6

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