Further application of \(H\)-differentiability to generalized complementarity problems based on generalized Fisher-Burmeister functions. (English) Zbl 1468.90134

Gu, Wei-Zhe; Tawhid, Mohamed A.

Abstr. Appl. Anal. 2014, Article ID 468065, 11 p. (2014).

Summary: We study nonsmooth generalized complementarity problems based on the generalized Fisher-Burmeister function and its generalizations, denoted by GCP(\(f, g\)) where \(f\) and \(g\) are \(H\)-differentiable. We describe \(H\)-differentials of some GCP functions based on the generalized Fisher-Burmeister function and its generalizations, and their merit functions. Under appropriate conditions on the \(H\)-differentials of \(f\) and \(g\), we show that a local/global minimum of a merit function (or a “stationary point” of a merit function) is coincident with the solution of the given generalized complementarity problem. When specializing GCP\((f, g)\) to the nonlinear complementarity problems, our results not only give new results but also extend/unify various similar results proved for \(C^1\), semismooth, and locally Lipschitzian.

MSC:

90C33	Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
90C29	Multi-objective and goal programming
49J52	Nonsmooth analysis

Keywords:

nonsmooth generalized complementarity problem; generalized Fisher-Burmeister function

References:

[1]	Gowda, M. S.; Ravindran, G., Algebraic univalence theorems for nonsmooth functions, Journal of Mathematical Analysis and Applications, 252, 2, 917-935 (2000) · Zbl 0993.49017 · doi:10.1006/jmaa.2000.7171
[2]	Clarke, F. H., Optimization and Nonsmooth Analysis (1990), Philadelphia, Pa, USA: SIAM, Philadelphia, Pa, USA · Zbl 0696.49002 · doi:10.1137/1.9781611971309
[3]	Mifflin, R., Semismooth and semiconvex functions in constrained optimization, SIAM Journal on Control and Optimization, 15, 6, 959-972 (1977) · Zbl 0376.90081 · doi:10.1137/0315061
[4]	Qi, L. Q., Convergence analysis of some algorithms for solving nonsmooth equations, Mathematics of Operations Research, 18, 1, 227-244 (1993) · Zbl 0776.65037 · doi:10.1287/moor.18.1.227
[5]	Qi, L. Q.; Sun, J., A nonsmooth version of Newton’s method, Mathematical Programming, 58, 3, 353-367 (1993) · Zbl 0780.90090 · doi:10.1007/BF01581275
[6]	Qi, L., C-Differential Operators, C-Differentiability and Generalized Newton Methods. C-Differential Operators, C-Differentiability and Generalized Newton Methods, Research Report (1996), Sydney, Australia: School of Mathematics, The University of New South Wales, Sydney, Australia
[7]	Tawhid, M. A., On the local uniqueness of solutions of variational inequalities under \(H\)-differentiability, Journal of Optimization Theory and Applications, 113, 1, 149-164 (2002) · Zbl 1001.49015 · doi:10.1023/A:1014813415372
[8]	Gowda, M. S., Inverse and Implicit Function Theorems for H-Differentiable and Semismooth Functions, Optimization Methods & Software, 19, 5, 443-461 (2004) · Zbl 1099.49018 · doi:10.1080/10556780410001697668
[9]	Song, Y.; Gowda, M. S.; Ravindran, G., On some properties of \(P\)-matrix sets, Linear Algebra and Its Applications, 290, 1-3, 237-246 (1999) · Zbl 0949.15027 · doi:10.1016/S0024-3795(98)10228-8
[10]	Tawhid, M. A., An application of H-differentiability to nonnegative and unrestricted generalized complementarity, Computational Optimization and Applications, 39, 1, 51-74 (2008) · Zbl 1147.90405 · doi:10.1007/s10589-007-9052-x
[11]	Tawhid, M. A., Solution of nonsmooth generalized complementarity problems, Journal of the Operations Research Society of Japan, 54, 1, 12-24 (2011) · Zbl 1228.90131
[12]	Tawhid, M. A., Unconstrained methods for nonsmooth nonlinear complementarity problems, Advanced Modeling and Optimization, 12, 1, 20-35 (2010) · Zbl 1332.90316
[13]	Tawhid, M. A., Nonsmooth generalized complementarity as unconstrained optimization, Journal of Industrial and Management Optimization, 6, 2, 411-423 (2010) · Zbl 1204.65071 · doi:10.3934/jimo.2010.6.411
[14]	Tawhid, M. A.; Goffin, J. L., On Minimizin g Some Merit Functions for Non-linear Complementarity Problems under H-differentiability, Journal of Optimization Theory and Applications, 139, 1, 127-140 (2008) · Zbl 1189.90174 · doi:10.1007/s10957-008-9409-z
[15]	Tawhid, M. A.; Gowda, M. S., On two applications of H-differentiability to optimization and complementarity problems, Computational Optimization and Applications, 17, 2-3, 279-299 (2000) · Zbl 1017.90040 · doi:10.1023/A:1008720430012
[16]	Hyers, D. H.; Isac, G.; Rassias, T. M., Topics in Nonlinear Analysis & Applications (1997), Singapore: World Scientific, Singapore · Zbl 0878.47040 · doi:10.1142/9789812830432
[17]	Isac, G., Complementarity Problems. Complementarity Problems, Lecture Notes in Mathematics, 1528 (1992), Berlin, Germany: Springer, Berlin, Germany · Zbl 0761.90091
[18]	Noor, M. A., The quasi-complementarity problem, Journal of Mathematical Analysis and Applications, 130, 2, 344-353 (1988) · Zbl 0645.90086 · doi:10.1016/0022-247X(88)90310-1
[19]	Pang, J. S.; Mangasarian, O. L.; Meyer, R. R.; Robinson, S. M., The implicit complementarity problem, Nonlinear Programming, 487-518 (1981), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0534.90090
[20]	Moré, J. J.; Rheinboldt, W. C., On P- and S-functions and related classes of n-dimensional nonlinear mappings, Linear Algebra and Its Applications, 6, 45-68 (1973) · Zbl 0247.65038 · doi:10.1016/0024-3795(73)90006-2
[21]	Tseng, P., Growth behavior of a class of merit functions for the nonlinear complementarity problem, Journal of Optimization Theory and Applications, 89, 1, 17-37 (1996) · Zbl 0866.90127 · doi:10.1007/BF02192639
[22]	Chen, J.-S.; Pan, S.-H., A family of NCP functions and a descent method for the nonlinear complementarity problem, Computational Optimization and Applications, 40, 3, 389-404 (2008) · Zbl 1153.90542 · doi:10.1007/s10589-007-9086-0
[23]	Chen, J., On some NCP-functions based on the generalized Fischer-Burmeister function, Asia-Pacific Journal of Operational Research, 24, 3, 401-420 (2007) · Zbl 1141.90557 · doi:10.1142/S0217595907001292
[24]	Song, Y.; Gowda, M. S.; Ravindran, G., On characterizations of P- and \(P_0\)-properties in nonsmooth functions, Mathematics of Operations Research, 25, 400-408 (2000) · Zbl 1073.90570
[25]	Wei-Zhe, G.; Tawhid, M. A., Generalized complementarity problems based on generalized Fisher-Burmeister functions as unconstrained optimization, Advanced Modeling and Optimization, 16, 2, 269-284 (2014) · Zbl 1413.90285
[26]	Kanzow, C.; Fukushima, M., Equivalence of the generalized complementarity problem to differentiable unconstrained minimization, Journal of Optimization Theory and Applications, 90, 3, 581-603 (1996) · Zbl 0866.90124 · doi:10.1007/BF02189797
[27]	Facchinei, F.; Soares, J., A new merit function for nonlinear complementarity problems and a related algorithm, SIAM Journal on Optimization, 7, 1, 225-247 (1997) · Zbl 0873.90096 · doi:10.1137/S1052623494279110
[28]	Fischer, A., Solution of monotone complementarity problems with locally Lipschitzian functions, Mathematical Programming, 76, 3, Ser. B, 513-532 (1997) · Zbl 0871.90097 · doi:10.1016/S0025-5610(96)00059-7
[29]	Geiger, C.; Kanzow, C., On the resolution of monotone complementarity problems, Computational Optimization and Applications, 5, 2, 155-173 (1996) · Zbl 0859.90113 · doi:10.1007/BF00249054
[30]	Jiang, H., Unconstrained minimization approaches to nonlinear complementarity problems, Journal of Global Optimization, 9, 2, 169-181 (1996) · Zbl 0868.90122 · doi:10.1007/BF00121662