| Title:
|
A penalty approach for a box constrained variational inequality problem (English) |
| Author:
|
Kebaili, Zahira |
| Author:
|
Benterki, Djamel |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
63 |
| Issue:
|
4 |
| Year:
|
2018 |
| Pages:
|
439-454 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
box constrained variational inequality problem |
| Keyword:
|
power penalty approach |
| Keyword:
|
strongly monotone operator |
| MSC:
|
47J20 |
| MSC:
|
65J15 |
| MSC:
|
65K10 |
| MSC:
|
65K15 |
| MSC:
|
90C33 |
| idZBL:
|
Zbl 06945741 |
| idMR:
|
MR3842962 |
| DOI:
|
10.21136/AM.2018.0334-17 |
| . |
| Date available:
|
2018-07-30T11:29:23Z |
| Last updated:
|
2020-09-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147320 |
| . |
| Reference:
|
[1] Auslender, A.: Optimisation. Méthodes numériques.Maitrise de mathématiques et applications fondamentales. Masson, Paris French (1976). Zbl 0326.90057, MR 0441204 |
| Reference:
|
[2] Censor, Y., Iusem, A. N., Zenios, S. A.: An interior point method with Bregman functions for the variational inequality problem with paramonotone operators.Math. Program. 81 (1998), 373-400. Zbl 0919.90123, MR 1617732, 10.1007/BF01580089 |
| Reference:
|
[3] Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Vol. I.Springer Series in Operations Research Springer, New York (2003). Zbl 1062.90001, MR 1955648, 10.1007/b97543 |
| Reference:
|
[4] Facchinei, F., Pang, J.-S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Vol. II.Springer Series in Operations Research Springer, New York (2003). Zbl 1062.90002, MR 1955649, 10.1007/b97544 |
| Reference:
|
[5] Fukushima, M.: Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems.Math. Program., Ser. A 53 (1992), 99-110. Zbl 0756.90081, MR 1151767, 10.1007/BF01585696 |
| Reference:
|
[6] Hao, Z., Wan, Z., Chi, X., Chen, J.: A power penalty method for second-order cone nonlinear complementarity problems.J. Comput. Appl. Math. 290 (2015), 136-149. Zbl 1327.90207, MR 3370398, 10.1016/j.cam.2015.05.007 |
| Reference:
|
[7] Harker, P. T., Pang, J.-S.: Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications.Math. Program., Ser. B 48 (1990), 161-220. Zbl 0734.90098, MR 1073707, 10.1007/BF01582255 |
| Reference:
|
[8] Huang, C., Wang, S.: A power penalty approach to a nonlinear complementarity problem.Oper. Res. Lett. 38 (2010), 72-76. Zbl 1182.90090, MR 2565819, 10.1016/j.orl.2009.09.009 |
| Reference:
|
[9] Huang, C., Wang, S.: A penalty method for a mixed nonlinear complementarity problem.Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 75 (2012), 588-597. Zbl 1233.65040, MR 2847442, 10.1016/j.na.2011.08.061 |
| Reference:
|
[10] Kanzow, C., Fukushima, M.: Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities.Math. Program. 83 (1998), 55-87. Zbl 0920.90134, MR 1643959, 10.1007/BF02680550 |
| Reference:
|
[11] Ma, C., Kang, T.: A Jacobian smoothing method for box constrained variational inequality problems.Appl. Math. Comput. 162 (2005), 1397-1429. Zbl 1068.65089, MR 2113979, 10.1016/j.amc.2004.03.018 |
| Reference:
|
[12] Noor, M. A., Wang, Y., Xiu, N.: Some new projection methods for variational inequalities.Appl. Math. Comput. 137 (2003), 423-435. Zbl 1031.65078, MR 1950108, 10.1016/S0096-3003(02)00148-0 |
| Reference:
|
[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. Zbl 0900.65188, MR 1459564 |
| Reference:
|
[14] Tang, J., Liu, S.: A new smoothing Broyden-like method for solving the mixed complementarity problem with a $P_0$-function.Nonlinear Anal., Real World Appl. 11 (2010), 2770-2786. Zbl 1208.90172, MR 2661943, 10.1016/j.nonrwa.2009.10.002 |
| Reference:
|
[15] Ulji, Chen, G.-Q.: New simple smooth merit function for box constrained variational inequalities and damped Newton type method.Appl. Math. Mech., Engl. Ed. 26 (2005), 1083-1092 Appl. Math. Mech. 26 2005 988-996 Chinese. English, Chinese summary. Zbl 1144.65309, MR 2169264, 10.1007/BF02466422 |
| Reference:
|
[16] Wang, S., Yang, X.: A power penalty method for linear complementarity problems.Oper. Res. Lett. 36 (2008), 211-214. Zbl 1163.90762, MR 2396598, 10.1016/j.orl.2007.06.006 |
| Reference:
|
[17] Wang, Y., Zhu, D.: An affine scaling interior trust region method via optimal path for solving monotone variational inequality problem with linear constraints.Chin. Ann. Math., Ser. B 29 (2008), 273-290. Zbl 1151.49025, MR 2421761, 10.1007/s11401-007-0082-6 |
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