A class of smoothing functions for nonlinear and mixed complementarity problems

Computational Optimization and Applications

Volumn 5, Issue 2, 1996, Pages 97-138

  Chen, Chunhui ^a   Mangasarian, O L ^a  

^a University of Wisconsin Madison   (United States)

Author keywords

Complementarity problems; Smoothing

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; COMPUTATIONAL METHODS; CONVERGENCE OF NUMERICAL METHODS; INTEGRATION; MATHEMATICAL PROGRAMMING; NONLINEAR EQUATIONS; PROBABILITY DENSITY FUNCTION; PROBLEM SOLVING; VARIATIONAL TECHNIQUES; VECTORS;

FUNDAMENTAL PLUS FUNCTION; MIXED COMPLIMENTARITY PROBLEMS; NEWTON BASED ALGORITHMS; NONLINEAR COMPLIMENTARITY PROBLEMS; PARAMETRIC SMOOTH FUNCTIONS;

OPTIMIZATION;

EID: 0030104686     PISSN: 09266003     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF00249052     Document Type: Article

References (40)

1. D.P. Bertsekas, "Projected Newton methods for optimization problems with simple constraints," SIAM Journal on Control and Optimization, vol. 20, pp. 221-246, 1982.
2. B. Chen and P.T. Harker, "A non-interior-point continuation method for linear complementarity problems," SIAM Journal on Matrix Analysis and Applications, vol. 14, pp. 1168-1190, 1993.
3. C. Chen and O.L. Mangasarian, "Smoothing methods for convex inequalities and linear complementarity problems," Technical Report 1191, Computer Sciences Department, University of Wisconsin, Madison, Wisconsin, 53706, November 1993.
4. (To appear in Mathematical Programming, vol. 71, 1995.)
5. R.W. Cottle, F. Giannessi, and J.-L. Lions, Variational Inequalities and Complementarity Problems, John Wiley & Sons: New York, 1980.
6. J.E. Dennis and R.B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall: Englewood Cliffs, N.J., 1983.
7. S.P. Dirkse and M.C. Ferns, "The path solver: A non-monotone stabilization scheme for mixed complementarity problems," Optimization Methods and Software, vol. 5, pp. 123-156, 1995.
8. S.P. Dirkse and M.C. Ferris, "MCPLIB: A collection of nonlinear mixed complementarity problems," Optimization Methods and Software, vol. 5, pp. 319-345, 1995.
9. S.P. Dirkse, M.C. Ferris, P.V. Preckel, and T. Rutherford, "The GAMS callable program library for variational and complementarity solvers," Manuscript, University of Wisconsin, 1993.
10. M.S. Gowda, "On the extended linear complementarity problem," Technical Report, Department of Mathematics & Statistics, University of Maryland Baltimore Country, Baltimore, Maryland, 1994, Mathematical Programming, to appear.
11. P.T. Harker and J.-S. Pang, "Finite-dimensional variational inequality and nonlinear complementarty problems: A survey of theory, algorithms and applications," Mathematical Programming, vol. 48, pp. 161-220, 1990.
12. P.T. Harker and B. Xiao, "Newton's method for the nonlinear complementarity problem: A B-differentiable equation approach," Mathematical Programming, vol. 48, pp. 339-357, 1990.
13. J. Hertz, A. Krogh, and R.G. Palmer, Introduction to the Theory of Neural Computation, Addison-Wesley: Redwood City, California, 1991.
14. J.-B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I, Springer-Verlag: Berlin, 1993.
15. N.H. Josephy, "Newton's method for generalized equations," Technical Summary Report 1965, Mathematics Research Center, University of Wisconsin, Madison, Wisconsin, 1979.
16. C. Kanzow, "Some tools allowing interior-point methods to become noninterior," Technical Report, Institute of Applied Mathematics, University of Hamburg, Germany, 1994.
17. M. Kojima and N. Megiddo, "The relation between the path of centers and smale's regularization of the linear programming problem," Linear Algebra and Its Applications, vol. 152, pp. 135-139, 1991.
18. M. Kojima, S. Mizuno, and A. Yoshise, "A polynomial-time algorithms for a class of linear complementarity problems," Mathematical Programming, vol. 44, pp. 1-27, 1989.
19. J. Kreimer and R.Y. Rubinstein, "Nondifferentiable optimization via smooth approximation: General analytical approach," Annals of Operations Research, vol. 39, pp. 97-119, 1992.
20. K. Madsen and H.B. Nielsen, "A finite smoothing algorithm for linear l1 estimation," SIAM on Optimization, vol. 3, no. 2, pp. 223-235, May 1993.
21. O.L. Mangasarian, "Mathematical programming in neural networks," ORSA Journal on Computing, vol. 5, no. 4, pp. 349-360, 1993.
22. O.L. Mangasarian and J.-S. Pang, "The extended linear complementarity problems," SIAM Journal on Matrix Analysis and Applications, vol. 16, pp. 359-368, 1995.
23. Jorge J. Moré, "Global methods for nonlinear complementarity problems," Technical Report Preprint MCS-P429-0494, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois, 1994.
24. B.A. Murtagh and M.A. Saunders, "MINOS 5.0 user's guide," Technical Report SOL 83.20, Stanford University, December 1983.
25. K.G. Murty, Linear Complementarity, Linear and Nonlinear Programming, Helderman-Verlag: Berlin, 1988.
26. J.M. Ortega, Numerical Analysis, A Second Course, Academic Press, 1972.
27. J.-S. Pang, "Newton's method for B-differentiable equations," Mathematics of Operations Research, vol. 15, pp. 331-341, 1990.
28. J.-S. Pang, "A B-differentiable equation based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems," Mathematical Programming, vol. 51, pp. 101-131, 1991.
29. J.-S. Pang and S.A. Gabriel, "NE/SQP: A robust algorithm for the nonlinear complementarity problem," Mathematical Programming, vol. 60, pp. 295-337, 1993.
30. M.C. Pinar and S.A. Zenios "On smoothing exact penalty functions for convex constrained optimization," SIAM Journal on Optimization, vol. 4, pp. 486-511, 1994.
31. D. Ralph, "Global convergence of damped Newton's method for nonsmooth equations via the path search," Mathematics of Operations Research, vol. 19, pp. 352-389, 1994.
32. J. Ren, "Computable error bounds in mathematical programming," Technical Report 1173, Computer Sciences Department, University of Wisconsin, Madison, Wisconsin 53706, August 1993.
33. S.M. Robinson, "Generalized equations and their solution: Part I: Basic theory," Mathematical Programming Study, vol. 10, pp. 128-140, 1979.
34. S.M. Robinson, "Strongly regular generalized equations," Mathematics of Operations Research, vol. 5, pp. 43-62, 1980.
35. S.M. Robinson, "Newton's method for a class of nonsmooth functions," Working Paper, Department of Industrial Engineering, University of Wisconsin, Madison, Wisconsin, 1988.
36. T.F. Rutherford, "MILES: A mixed inequality and nonlinear equation solver," Working Paper, Department of Economics, University of Colorado, Boulder, 1993.
37. S. Smale, "Algorithms for solving equations," In Proceedings of the International Congress of Mathematicians, pp. 172-195, Ameri. Math. Soc., Providence, 1987.
38. B. Xiao, "Global Newtons methods for nonlinear programs and variational inequalities," Technical Report, Ph.D. Thesis, Department of Decision Sciences, The Wharton School, University of Pennsylvania, Philadelphia, PA, 1990.
39. Y. Ye, "A fully polynomial-time approximation algorithm for computing a stationary point of the general linear complementarity problem," Mathematics of Operations Research, vol. 18, pp. 334-345, 1993.
40. I. Zang, "A smoothing-out technique for min-max optimization," Mathematical Programming, vol. 19, pp. 61-77, 1980.