A new unconstrained differentiable merit function for box constrained variational inequality problems and a damped Gauss-Newton method

SIAM Journal on Optimization

Volumn 9, Issue 2, 1999, Pages 388-413

  Sun, Defeng ^a   Womersley, Robert S ^a  

^a UNIVERSITY OF NEW SOUTH WALES   (Australia)

Author keywords

Box constraints; Gauss Newton method; Merit functions; Superlinear convergence; Variational inequality problems

Indexed keywords

EID: 0033407838     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623496314173     Document Type: Article

References (49)

1. S. C. BILLUPS, Algorithms for complementarity problems and generalized equations, Ph.D. thesis, Computer Sciences Department, University of Wisconsin, Madison, WI, 1995.
2. S. C. BILLUPS, S. P. DIRKSE, AND M. C. FERRIS, A comparison of algorithms for large scale mixed complementarity problems, Comput. Optim. Appl., 7 (1997), pp. 3-25.
3. B. CHEN, X. CHEN, AND C. KANZOW, A Modified Fischer-Burmeister NCP-function, Talk presented at the 1997 International Symposium on Mathematical Programming, EPFL, Lausanne, Switzerland, 1997.
4. X. CHEN, L. QI, AND D. SUN, Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities, Math. Comp., 67 (1998), pp. 519-540.
5. F. H. CLARKE, Optimization and Nonsmooth Analysis, John Wiley, New York, 1983; reprinted by SIAM, Philadelphia, 1990.
6. T. DE LUCA, F. FACCHINEI, AND C. KANZOW, A semismooth equation approach to the solution of nonlinear complementarity problems, Math. Programming, 75 (1996), pp. 407-439.
7. S. P. DIRKSE AND M. C. FERRIS, The PATH solver: A non-monotone stabilization scheme for mixed complementarity problems, Optimization Methods and Software, 5 (1995), pp. 123-156.
8. S. P. DIRKSE AND M. C. FERRIS, MCPLIB: A collection of nonlinear mixed complementarity problems, Optim. Methods Softw., 5 (1995), pp. 319-345.
9. B. C. EAVES, On the basic theorem of complementarity, Math. Programming, 1 (1971), pp. 68-75.
10. F. FACCHINEI, A. FISCHER, AND C. KANZOW, A semsimooth Newton method for variational inequalities: The case of box constraints, in Complementarity and Variational Problems: State of the Art, M. C. Ferris and J. S. Pang, eds., SIAM, Philadelphia, PA, 1997, pp. 76-90.
11. F. FACCHINEI AND C. KANZOW, On unconstrained and constrained stationary points of the implicit Lagrangian, J. Optim. Theory Appl., 92 (1997), pp. 99-115.
12. F. FACCHINEI AND C. KANZOW, A nonsmooth inexact Newton method for the solution of large-scale nonlinear complementarity problems, Math. Programming, 76 (1997), pp. 493-512.
13. F. FACCHINEI AND J. SOARES, Testing a new class of algorithms for nonlinear complementarity problems, in Variational Inequalities and Network Equilibrium Problems, F. Giannessi, ed., Plenum Press, New York, 1995, pp. 69-83.
14. F. FACCHINEI AND J. SOARES, A new merit function for nonlinear complementarity problems and a related algorithm, SIAM J. Optim., 7 (1997), pp. 225-247.
15. M. C. FERRIS AND T. F. RUTHERFORD, Accessing realistic mixed complementarity problems within MATLAB, in Nonlinear Optimization and Applications, G. Di Pillo and F. Giannessi, eds., Plenum Press, New York, 1996, pp. 141-153.
16. A. FISCHER, A special Newton-type optimization method, Optimization, 24 (1992), pp. 269-284.
17. A. FISCHER, An NCP-function and its use for the solution of complementarity problems, in Recent Advances in Nonsmooth Optimization, D. Du, L. Qi, and R. Womersley, eds., World Scientific, River Edge, NJ, 1995, pp. 88-105.
18. A. FISCHER, Solution of monotone complementarity problems with locally Lipschitzian functions, Math. Programming, 76 (1997), pp. 513-532.
19. M. FUKUSHIMA, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Math. Programming, 53 (1992), pp. 99-110.
20. M. FUKUSHIMA, Merit functions for variational inequality and complementarity problems, in Nonlinear Optimization and Applications, G. Di Pillo and F. Giannessi, eds., Plenum Press, New York, 1996, pp. 155-170.
21. C. GEIGER AND C. KANZOW, On the resolution of monotone complementarity problems, Comput. Optim. Appl., 5 (1996), pp. 155-173.
22. L. GRIPPO, F. L. LAMPARIELLO, AND S. LUCIDI, A nonmonotone linesearch technique for Newton's method, SIAM J. Numer. Anal., 23 (1986), pp. 707-716.
23. P. T. HARKER AND J. S. PANG, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Programming, 48 (1990), pp. 161-220.
24. H. JIANG, Unconstrained minimization approaches to nonlinear complementarity problems, J. Global Optim., 9 (1996), pp. 169-181.
25. H. JIANG AND L. QI, A new nonsmooth equations approach to nonlinear complementarity problems, SIAM J. Control Optim., 35 (1997), pp. 178-193.
26. C. KANZOW, Nonlinear complementarity as unconstrained optimization, J. Optim. Theory Appl., 88 (1996), pp. 139-155.
27. C. KANZOW AND M. FUKUSHIMA, Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities, Math. Programming, 83 (1998), pp. 55-87.
28. Z. -Q. LUO, O. L. MANGASARIAN, J. REN, AND M. V. SOLODOV, New error bounds for the linear complementarity problem, Math. Oper. Res., 19 (1994), pp. 880-892.
29. Z. -Q. LUO AND P. TSENG, A new class of merit functions for the nonlinear complementarity problem, in Complementarity and Variational Problems: State of the Art, M. C. Ferris and J. S. Pang, eds., SIAM, Philadelphia, PA, 1997, pp. 204-225.
30. O. L. MANGASARIAN AND M. V. SOLODOV, Nonlinear complementarity as unconstrained and constrained minimization, Math. Programming, 62 (1993), pp. 277-297.
31. R. MIFFLIN, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 15 (1977), pp. 959-972.
32. J. S. PANG, Newton's method for B-differentiable equations, Math. Oper. Res., 15 (1990), pp. 331-341.
33. J. S. PANG, A B-differentiable equation based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems, Math. Programming, 51 (1991), pp. 101-131.
34. J. S. PANG, Complementarity problems, in Handbook of Global Optimization, R. Horst and P. Pardalos eds., Kluwer Academic Publishers, Norwell, MA, 1995, pp. 271-338.
35. J. S. PANG AND S. A. GABRIEL, NE/SQP: A robust algorithm for the nonlinear complementarity problem, Math. Programming, 60 (1993), pp. 295-337.
36. J. S. PANG AND L. QI, Nonsmooth equations: Motivation and algorithms, SIAM J. Optim., 3 (1993), pp. 443-465.
37. J. M. PENG, Equivalence of variational inequality problems to unconstrained optimization, Math. Programming, 78 (1997), pp. 347-355.
38. J. M. PENG AND Y. YUAN, Unconstrained methods for generalized complementarity problems, J. Comput. Math., 15 (1997), pp. 253-264.
39. L. QI, Convergence analysis of some algorithms for solving nonsmooth equations, Math. Oper. Res., 18 (1993), pp. 227-244.
40. L. QI AND J. SUN, A nonsmooth version of Newton's method, Math. Programming, 58 (1993), pp. 353-367.
41. D. RALPH, Global convergence of damped Newton's method for nonsmooth equations via the path search, Math. Oper. Res., 19 (1994), pp. 352-389.
42. S. M. ROBINSON, Generalized equations, in Mathematical Programming: The State of the Art, A. Bachem, M. Grötschel, and B. Korte, eds., Springer-Verlag, Berlin, 1983, pp. 346-367.
43. D. SUN, A class of iterative methods for solving nonlinear projection equations, J. Optim. Theory Appl., 91 (1996), pp. 123-140.
44. D. SUN, M. FUKUSHIMA, AND L. QI, A computable generalized Hessian of the D-gap function and Newton-type methods for variational inequality problems, in Complementarity and Variational Problems: State of the Art, M. C. Ferris and J. S. Pang, eds., SIAM, Philadelphia, PA, 1997, pp. 452-473.
45. P. TSENG, Global behaviour of a class of merit functions for the nonlinear complementarity problem, J. Optim. Theory Appl., 89 (1996), pp. 17-37.
46. P. TSENG, N. YAMASHITA, AND M. FUKUSHIMA, Equivalence of complementarity problems to differentiable minimization: A unified approach, SIAM J. Optim., 6 (1996), pp. 446-460.
47. N. YAMASHITA AND M. FUKUSHIMA, On stationary points of implicit Lagrangian for nonlinear complementarity problems, J. Optim. Theory Appl., 84 (1995), pp. 653-663.
48. N. YAMASHITA AND M. FUKUSHIMA, Equivalent unconstrained minimization and global error bounds for variational inequality problems, SIAM J. Control Optim., 35 (1997), pp. 273-284.
49. N. YAMASHITA, K. TAJI, AND M. FUKUSHIMA, Unconstrained optimization reformulations of variational inequality problems, J. Optim. Theory Appl., 92 (1997), pp. 439-456.