Reformulation of variational inequalities on a simplex and compactification of complementarity problems

SIAM Journal on Optimization

Volumn 10, Issue 3, 2000, Pages 878-895

  Andreani, Roberto ^a   Martínez, José Mario ^b  

^a SÃO PAULO STATE UNIVERSITY UNESP   (Brazil)

^b UNIVERSITY OF CAMPINAS   (Brazil)

Author keywords

Complementarity; Minimization algorithms; Reformulation; Variational inequalities

Indexed keywords

EID: 0039030508     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/S1052623499352826     Document Type: Article

References (46)

1. R. ANDREANI, A. FRIEDLANDER, AND J. M. MARTÍNEZ, On the solution of finite-dimensional variational inequalities using smooth optimization with simple bounds, J. Optim. Theory Appl., 94 (1997), pp. 635-657.
2. R. ANDREANI AND J. M. MARTÍNEZ, Solving complementarity problems by means of a new smooth constrained nonlinear solver, in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999, pp. 1-24.
3. R. ANDREANI AND J. M. MARTÍNEZ, On the solution of the extended linear complementarity problem, Linear Algebra Appl., 281 (1998), pp. 247-257.
4. R. ANDREANI AND J. M. MARTÍNEZ, On the reformulation of nonlinear complementarity problems using the Fischer-Burmeister function, Appl. Math. Lett., 12 (1999), pp. 7-12.
5. R. ANDREANI AND J. M. MARTÍNEZ, On the solution of bounded and unbounded mixed complementarity problems, Optimization, to appear.
6. E. G. BIRGIN, J. M. MARTÍNEZ, AND M. RAYDAN, Nonmonotone spectral projected gradient methods on convex sets, SIAM J. Optim., to appear.
7. R. H. BYRD, P. LU, J. NOCEDAL, AND C. ZHU, A limited memory algorithm for bound constrained optimization, SIAM J. Sci. Comput., 16 (1995), pp. 1190-1208.
8. B. CHEN, X. CHEN, AND C. KANZOW, A penalized Fischer-Burmeister NCP-function: Theoretical investigation and numerical results, Math. Program., to appear.
9. B. CHEN AND P. T. HARKER, Smooth approximations to nonlinear complementarity problems, SIAM J. Optim., 7 (1997), pp. 403-420.
10. A. R. CONN, N. I. M. GOULD, AND PH. L. TOINT, Global convergence of a class of trust region algorithms for optimization with simple bounds, SIAM J. Numer. Anal., 25 (1988), pp. 433-460. See also SIAM J. Numer. Anal., 26 (1989) pp. 764-767.
11. A. R. CONN, N. I. M. GOULD, AND PH. L. TOINT, Global convergence of a class of trust region algorithms for optimization with simple bounds, SIAM J. Numer. Anal., 25 (1988), pp. 433-460. See also SIAM J. Numer. Anal., 26 (1989) pp. 764-767.
12. T. DE LUCA, F. FACCHINEI, AND C. KANZOW, A semismooth equation approach to the solution of nonlinear complementarity problem, Math. Programming, 75 (1996), pp. 407-439.
13. J. E. DENNIS, JR. AND R. B. SCHNABEL, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice-Hall Ser. Comput. Math., Prentice-Hall, Englewood Cliffs, NJ, 1983.
14. S. P. DIRKSE AND M. C. FERRIS, A collection of nonlinear mixed complementarity problems, Optim. Methods Softw., 5 (1995), pp. 319-345.
15. F. FACCHINEI, A. FISCHER, AND C. KANZOW, A semismooth 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.
16. L. FERNANDES, A. FRIEDLANDER, M. GUEDESE, AND J. JÚDICE, Solution of generalized linear complementarity problems using smooth optimization and its application to bilinear programming and LCP, Appl. Math. Optim., to appear.
17. M. C. FERRIS AND C. KANZOW, Complementarity and Related Problems: A Survey, Tech. Report, Comput. Sci. Dept., University of Wisconsin, Madison, WI, 1998.
18. M. C. FERRIS, C. KANZOW, AND T. S. MUNSON, Feasible descent algorithms for mixed complementarity problems, Math. Program., 86 (1999), pp. 475-497.
19. A. FRIEDLANDER AND J. M. MARTÍNEZ, On the numerical solution of bound constrained optimization problems, RAIRO Rech. Opér., 23 (1989), pp. 319-341.
20. A. FRIEDLANDER, J. M. MARTÍNEZ, AND S. A. SANTOS, A new trust region algorithm for bound constrained minimization, Appl. Math. Optim., 30 (1994), pp. 235-266.
21. A. FRIEDLANDER, J. M. MARTÍNEZ, AND S. A. SANTOS, On the resolution of linearly constrained convex minimization problems, SIAM J. Optim., 4 (1994), pp. 331-339.
22. A. FRIEDLANDER, J. M. MARTÍNEZ, AND S. A. SANTOS, Solution of linear complementarity problems using minimization with simple bounds, J. Global Optim., 6 (1995), pp. 253-267.
23. A. FRIEDLANDER, J. M. MARTÍNEZ, AND S. A. SANTOS, A new strategy for solving variational inequalities on bounded polytopes, Numer. Funct. Anal. Optim., 16 (1995), pp. 653-668.
24. C. GEIGER AND C. KANZOW, On the resolution of monotone complementarity problems, Comput. Optim. Appl., 5 (1996), pp. 155-173.
25. M. S. GOWDA, On the extended linear complementarity problem, Math. Programming, 72 (1996), pp. 33-50.
26. L. GRIPPO, F. LAMPARIELLO, AND S. LUCIDI, A nonmonotone line search technique for Newton's method, SIAM J. Numer. Anal., 23 (1986), pp. 707-716.
27. P. T. HARKER AND J. S. PANG, Finite-dimensional variational inequality and nonlinear complementarity problem: A survey of theory, algorithms and applications, Math. Programming, 48 (1990), pp. 161-220.
28. H. JIANG AND L. QI, A new nonsmooth equations approach to nonlinear complementarity problems, SIAM J. Control Optim., 35 (1997), pp. 178-193.
29. C. KANZOW, An unconstrained optimization technique for large-scale linearly constrained convex minimization problems, Computing, 53 (1994), pp. 101-117.
30. C. KANZOW, A new approach to continuation methods for complementarity problems with uniform P-functions, Oper. Res. Lett., 20 (1997), pp. 85-92.
31. C. KANZOW, N. YAMASHITA, AND M. FUKUSHIMA, New NCP-functions and properties, J. Optim. Theory Appl., 94 (1997), pp. 115-135.
32. D. N. KOZAKEVICH, J. M. MARTÍNEZ, AND S. A. SANTOS, Solving nonlinear systems of equations with simple constraints, Comput. Appl. Math., 16 (1997), pp. 215-235.
33. J. J. JÚDICE, Algorithms for linear complementarity problems, in Algorithms for Continuous Optimization, E. Spedicato, ed., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994, pp. 435-474.
34. 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.
35. O. L. MANGASARIAN AND J. S. PANG, The extended linear complementarity problem, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 359-368.
36. O. L. MANGASARIAN AND M. V. SOLODOV, Nonlinear complementarity problem as unconstrained and constrained minimization, Math. Programming, 62 (1993), pp. 277-297.
37. J. M. MARTÍNEZ, Quasi-inexact-Newton methods with global convergence for solving constrained nonlinear systems, Nonlinear Anal., 30 (1997), pp. 1-8.
38. J. J. MORÉ, Global methods for nonlinear complementarity problems, Math. Oper. Res., 21 (1996), pp. 589-614.
39. J. M. ORTEGA AND W. C. RHEINBOLDT, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
40. L. QI AND J. SUN, A nonsmooth version of Newton's method, Math. Programming, 58 (1993), pp. 353-368.
41. M. RAYDAN, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM J. Optim., 7 (1997), pp. 26-33.
42. H. SCHWETLICK, Numerische Lōsung nichtlinearer Gleinchungen, Deutscher Verlag der Wissenschaften, Berlin, 1978.
43. M. V. SOLODOV, Some optimization reformulations of the extended linear complementarity problem, Comput. Optim. Appl., 13 (1999), pp. 187-200.
44. M. V. SOLODOV AND B. F. SVAITER, A globally convergent inexact Newton method for systems of monotone equations, in Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods, M. Fukushima and L. Qi, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 1999, pp. 355-369.
45. M. V. SOLODOV AND B. F. SVAITER, A truly globally convergent Newton-type method for the monotone nonlinear complementarity problem, SIAM J. Optim., 10 (2000), pp. 605-625.
46. M. V. SOLODOV AND P. TSENG, Two methods based on the D-gap function for solving monotone variational inequalities, Comput. Optim. Appl., to appear.