Nonsmooth generalized complementarity as unconstrained optimization

Journal of Industrial and Management Optimization

Volumn 6, Issue 2, 2010, Pages 411-423

  Tawhid, Mohamed Aly ^a  

^a THOMPSON RIVERS UNIVERSITY   (Canada)

Author keywords

GCP function; Generalized complementarity problem; Generalized Jacobian; H differentiability; Locally Lipschitzian; Merit function; Semismooth functions

Indexed keywords

EID: 77952537615     PISSN: 15475816     EISSN: 1553166X     Source Type: Journal    
DOI: 10.3934/jimo.2010.6.411     Document Type: Article

References (35)

1. B. Chen, X. Chen and C. Kanzow, A penalized Fischer-Burmeister GCP-function, Mathematical Programming Ser. A, 88 (2000), 211-216.
2. F. H. Clarke, "Optimization and Nonsmooth Analysis," Wiley, New York, 1983 (reprinted by SIAM, Philadelphia, PA, 1990).
3. R. W. Cottle, J.-S. Pang and R. E. Stone, "The Linear Complementarity Problem," Academic Press, Boston, 1992.
4. R. W. Cottle, F. Giannessi and J.-L. Lions (Eds.), "Variational Inequalities and Complementarity Problems: Theory and Applications," J. Wiley, New York, 1980.
5. F. Facchinei and J. Soares, A new merit function for nonlinear complementarity problems and related algorithm, SIAM Journal on Optimization, 7 (1997), 225-247.
6. F. Facchinei, A. Fischer and C. Kanzow, Regularity properties of a semismooth reformulation of variational inequalities, SIAM Journal on Optimization, 8 (1998), 850-869.
7. M. C. Ferris and J.-S. Pang (Eds.), "Complementarity and Variational problems: State of the Art," SIAM, Philadelphia, 1997.
8. M. C. Ferris and J.-S. Pang, Engineering and economic applications of complementarity problems, SIAM Review, 39 (1997), 669-713.
9. A. Fischer, A special Newton-type optimization method, Optimization, 24 (1992), 269-284.
10. A. Fischer, Solution of monotone complementarity problems with locally lipschitzian functions, Mathematical Programming, 76 (1997), 513-532.
11. C. Geiger and C. Kanzow, On the resolution of monotone complementarity problems, Computational Optimization and Applications, 5 (1996), 155-173.
12. M. S. Gowda, "A Note on H-Differentiable Functions," Department of Mathematics and Statistics, University of Maryland, Baltimore County, Baltimore, Maryland, November, 1998.
13. M. S. Gowda, Inverse and implicit function theorems for H-differentiable and semismooth functions, Optimization Methods and Software, 19 (2004), 443-461.
14. M. S. Gowda and G. Ravindran, Algebraic univalence theorems for nonsmooth functions, Journal of Mathematical Analysis and Applications, 252 (2000), 917-935.
15. P. T. Harker and J.-S. Pang, Finite dimension variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Mathematical programming, 48 (1990), 161-220.
16. G. Isac, "Complementarity Problems," Lecture Notes in Mathematics 1528, Springer Verlag, Berlin, Germany, 1992.
17. H. Jiang and L. Qi, A new nonsmooth equations approach to nonlinear complementarity problems, SIAM Journal on Control and Optimization, 35 (1997), 178-193.
18. H. Jiang, Unconstrained minimization approaches to nonlinear complementarity problems, Journal of Global Optimization, 9 (1996), 169-181.
19. C. Kanzow and H. Kleinmichel, A new class of semismooth newton-type methods for nonlinear complementarity problems, Computational Optimization and Applications, 11 (1998), 227-251.
20. C. Kanzow and M. Fukushima, Equivalence of the generalized complementarity problem to differentiable unconstrained minimization, Journal of Optimization Theory and Applications, 90 (1996), 581-603.
21. T. D. Luca, F. Facchinei and C. Kanzow, A semismooth equation approach to the solution of nonlinear complementarity problems, Mathematical Programming, 75 (1996), 407-439.
22. O. L. Mangasarian, "Nonlinear Programming," McGraw-Hill, New York, 1969.
23. R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM Journal on Control and Optimization, 15 (1977), 952-972.
24. J. J. Moré and W. C. Rheinboldt, On P and S functions and related classes of N-dimensional nonlinear mappings, Linear Algebra and its Applications, 6 (1973), 45-68.
25. M. A. Noor, The quasi-complementarity problem, Journal of Mathematical Analysis and Applications, 130 (1988), 344-353.
26. J. S. Pang, The implicit complementarity problem, "Nonlinear Programming," Edited by O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, Academic Press, New York, 487-518, 1981.
27. G. Di Pillo and F. Giannessi (Eds.), "Nonlinear Optimization and Applications," Plenum Press, New York, 1996.
28. L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations, Mathematics of Operations Research, 18 (1993), 227-244.
29. L. Qi, "C-Differentiability, C-Differential Operators and Generalized Newton Methods," Research Report, School of Mathematics, The University of New South Wales, Sydney, New South Wales 2052, Australia, January 1996.
30. L. Qi and J. Sun, A nonsmooth version of Newton's method, Mathematical Programming, 58 (1993), 353-367.
31. R. T. Rockafellar and R. J.-B. Wets, "Variational Analysis," Grundlehren der Mathematischen Wissenschaften, 317, Springer-Verlag, Berlin, 1998.
32. Y. Song, M. S. Gowda and G. Ravindran, On characterizations of P-and P0- properties in nonsmooth functions, Mathematics of Operations Research, 25 (2000), 400-408.
33. Y. Song, M. S. Gowda and G. Ravindran, On some properties of P-matrix sets, Linear Algebra and its Applications, 290 (1999), 246-273.
34. M. A. Tawhid, On the local uniqueness of solutions of variational inequalities under H-differentiability, Journal of Optimization Theory and Applications, 113 (2002), 149-164.
35. M. A. Tawhid and M. S. Gowda, On two applications of H-differentiability to optimization and complementarity problems, Computational Optimization and Applications, 17 (2000), 279-299.