﻿ 一类广义二阶锥线性互补问题的低阶罚函数算法 A Lower Order Penalty Method for a Kind of Generalized Second-Order Cone Linear Complementarity Problems

Pure Mathematics
Vol.06 No.03(2016), Article ID:17701,10 pages
10.12677/PM.2016.63042

A Lower Order Penalty Method for a Kind of Generalized Second-Order Cone Linear Complementarity Problems

Wenyu Zhao, Xiaojun Ma, Jun Ma

School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan Ningxia

Received: May 8th, 2016; accepted: May 27th, 2016; published: May 30th, 2016

ABSTRACT

For a kind of generalized second-order cone linear complementary problem, using the ideas of lower order penalty function algorithm, it is converted to lower order penalty equations. We prove that the solution sequence of the lower order penalty equations converges to the solution of the generalized second-order cone complementarity problems at an exponential rate under particular conditions.

Keywords:Second-Order Cone Complementarity Problem, Low Order Penalty Method, Exponential Convergence Rate

1. 引言

(1.1)

(1.2)

(1.3)

(1.4)

(1.5)

(1.6)

2. 预备知识

(2.1)

(2.2)

(2.3)

1)

2)

3)

4)是非负的(正的)当且仅当

5)

(2.4)

(2.5)

1)

2)对任意的标量

(2.6)

3. 低阶罚方法及收敛性分析

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

。但此时

(3.7)

(3.8)

。但此时

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

(3.19)

(3.20)

，则对一些正常数。因此

(3.21)

(3.22)

(3.23)

(3.24)

(3.25)

(3.26)

(3.27)

(3.28)

(3.29)

(3.30)

(3.31)

。假设的特征值分解为。因为对任意的，当时，都连续。所以由(3.28)得当时，。特别地，当充分大时有。因此，由(3.26)得

(3.32)

4. 结论

A Lower Order Penalty Method for a Kind of Generalized Second-Order Cone Linear Complementarity Problems[J]. 理论数学, 2016, 06(03): 278-287. http://dx.doi.org/10.12677/PM.2016.63042

1. 1. Alizadeh, F. and Goldfarb, D. (2003) Second-Order Cone Programming. Mathematical Programming, 95, 3-51. http://dx.doi.org/10.1007/s10107-002-0339-5

2. 2. Lobo, M.S., Vandenberghe, L., Boyd, S., et al. (1998) Appli-cations of Second-Order Cone Programming. Linear Algebra and Its Applications, 284, 193-228. http://dx.doi.org/10.1016/S0024-3795(98)10032-0

3. 3. Facchinei, F. and Pang, J.S. (2003) Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Science & Business Media.

4. 4. Nemirovski, A. and Scheinberg, K. (1996) Extension of Karmarkar’s Algorithm onto Convex Quadratically Constrained Quadratic Problems. Mathematical Programming, 72, 273-289. http://dx.doi.org/10.1007/BF02592093

5. 5. Chen, X.D., Sun, D.F. and Sun, J. (2003) Complementarity Functions and Numerical Experiments for Second-Order Cone Complementarity Problems. Computational Optimization and Applications, 25, 39-56. http://dx.doi.org/10.1023/A:1022996819381

6. 6. Kanzow, C., Ferenczi, I. and Fukushima M. (2009) On the Local Convergence of Semismooth Newton Methods for Linear and Nonlinear Second-Order Cone Programs without Strict Complementarity. SIAM Journal on Optimization, 20, 297-320. http://dx.doi.org/10.1137/060657662

7. 7. Pan, S. and Chen, J.S. (2009) A Damped Gauss-Newton Method for the Second-Order Cone Complementarity Problem. Applied Mathematics and Optimization, 59, 293-318. http://dx.doi.org/10.1007/s00245-008-9054-9

8. 8. Hayashi, S., Yamaguchi, T., Yamashita, N., et al. (2005) A Matrix-Splitting Method for Symmetric Affine Second- Order Cone Complementarity Problems. Journal of Computational and Applied Mathematics, 175, 335-353. http://dx.doi.org/10.1016/j.cam.2004.05.018

9. 9. Chen, J.S. and Tseng, P. (2005) An Unconstrained Smooth Mi-nimization Reformulation of the Second-Order Cone Complementarity Problem. Mathematical Programming, 104, 293-327. http://dx.doi.org/10.1007/s10107-005-0617-0

10. 10. Chen, J.S. (2006) Two Classes of Merit Functions for These Cond-Order Cone Complementarity Problem. Mathematical Methods of Operations Research, 64, 495-519. http://dx.doi.org/10.1007/s00186-006-0098-9

11. 11. Hao, Z., Wan, Z. and Chi, X. (2015) A Power Penalty Method for Second-Order Cone Linear Complementarity Problems. Operations Research Letter, 43, 137-142. http://dx.doi.org/10.1016/j.orl.2014.12.012

12. 12. Hao, Z., Wan, Z. and Chi, X. (2015) A Power Penalty Method for Second-Order Cone Nolinear Complementarity Problems. Journal of Computational and Applied Mathematics, 290, 136-149. http://dx.doi.org/10.1016/j.cam.2015.05.007

13. 13. Zangwill, W.I. (1967) Non-Linear Programming via Penalty Functions. Management Science, 13, 344-358. http://dx.doi.org/10.1287/mnsc.13.5.344

14. 14. Luo, Z.Q., Pang, J.S. and Ralph, D. (1996) Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511983658

15. 15. Wang, S. and Yang, X. (2008) A Power Penalty Method for Linear Complementarity Problems. Operations Research Letters, 36, 211-214. http://dx.doi.org/10.1016/j.orl.2007.06.006

16. 16. Huang, C. and Wang, S. (2010) A Power Penalty Approach to a Nonlinear Complementarity Problem. Operations Research Letters, 38, 72-76. http://dx.doi.org/10.1016/j.orl.2009.09.009

17. 17. Chen, J.S. (2006) The Convex and Monotone Functions Asso-ciated with Second-Order Cone. Optimization, 55, 363- 385. http://dx.doi.org/10.1080/02331930600819514

18. 18. Faraut, J. and KorSnyi, A. (1994) Analysis on Symmetric Cones. Oxford University Press, Oxford.

19. 19. Fukushima, M., Luo, Z.Q. and Tseng, P. (2002) Smoothing Functions for Second-Order-Cone Complementarity Problems. SIAM Journal on Optimization, 12, 436-460. http://dx.doi.org/10.1137/S1052623400380365