Modified projection-type methods for monotone variational inequalities

SIAM Journal on Control and Optimization

Volumn 34, Issue 5, 1996, Pages 1814-1830

  Solodov, Michael V ^a   Tseng, Paul ^b  

^a University of Wisconsin Madison   (United States)

^b UNIVERSITY OF WASHINGTON   (United States)

Author keywords

Error bound; Linear convergence; Monotone variational inequalities; Projection type methods

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; ERRORS; ITERATIVE METHODS; MATRIX ALGEBRA; PROBLEM SOLVING; VARIATIONAL TECHNIQUES;

ERROR BOUND; LINEAR CONVERGENCE; MATRIX SPLITTING METHODS; MONOTONE VARIATIONAL INEQUALITIES; PROJECTION TYPE METHODS;

OPTIMIZATION;

EID: 0030243756     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012994268655     Document Type: Article

References (55)

1. A. A. AUSLENDER, Optimisation: Méthodes Numériques, Masson, Paris, 1976.
2. A. B. BAKUSINSKII AND B. T. POLYAK, On the solution of variational inequalities, Soviet Math. Dok., 15 (1974), pp. 1705-1710.
3. D. P. BERTSEKAS AND E. M. GAFNI, Projection methods for variational inequalities with application to the traffic assignment problem, Math. Programming Study, 17 (1982), pp. 139-159.
4. J. F. BONNANS, Local analysis of Newton-type methods for variational inequalities and nonlinear programming, Appl. Math. Optim., 29 (1994), pp. 161-186.
5. H.-G. CHEN, Forward-Backward Splitting Techniques: Theory and Applications, Ph.D. thesis, Department of Applied Mathematics, University of Washington, Seattle, WA, December 1994.
6. R. W. COTTLE, F. GIANNESSI, AND J.-L. LIONS, eds., Variational Inequalities and Complementarity Problems: Theory and Applications, Wiley, New York, 1980.
7. R. W. COTTLE, J.-S. PANG, AND R. E. STONE, The Linear Complementarity Problem, Academic Press, New York, 1992.
8. S. C. DAFERMOS, An iterative scheme for variational inequalities, Math. Programming, 26 (1983), pp. 40-47.
9. S. C. DAFERMOS AND S. C. MCKELVEY, Partitionable variational inequalities with applications to network and economic equilibria, J. Optim. Theory Appl., 73 (1992), pp. 243-268.
10. J. ECKSTEIN AND M. C. FERRIS, Operator Splitting Methods for Monotone Affine Variational Inequalities, with a Parallel Application to Optimal Control, Tech. Report, Thinking Machines Corporation, Cambridge, MA, and Computer Sciences Department, University of Wisconsin, Madison, WI, December 1994.
11. M. C. FERRIS AND O. L. MANGASARIAN, Error bounds and strong upper semicontinuity for monotone affine variational inequalities, Ann. Oper. Res., 47 (1993), pp. 293-305.
12. M. FUKUSHIMA, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems, Math. Programming, 53 (1992), pp. 99-110.
13. E. M. GAFNI AND D. P. BERTSEKAS, Two-metric projection methods for constrained optimization, SIAM J. Control Optim., 22 (1984), pp. 936-964.
14. D. GAY, Electronic mail distribution of linear programming test problems, Math. Programming Soc. COAL Newslett., December 1985.
15. R. GLOWINSKI, J.-L. LIONS, AND R. TRÉMOLIÉRES, Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.
16. P. HARKER AND J.-S. PANG, A damped-Newton method for the linear complementarity problem, in Computational Solution of Nonlinear Systems of Equations, Lectures in Appl. Math. 26, G. Allgower and K. Georg, eds., American Mathematical Society, Providence, RI, 1990, pp. 265-284.
17. B. HE, A projection and contraction method for a class of linear complementarity problems and its application in convex qudratic programming, Appl. Math. Optim., 25 (1992), pp. 247-262.
18. -, A new method for a class of linear variational inequalities, Math. Programming, 66 (1994), pp. 137-144.
19. _, Solving a class of linear projection equations, Numer. Math., 68 (1994), pp. 71-80.
20. A. N. IUSEM, An iterative algorithm for the variational inequality problems, Mat. Apl. Comput., 13 (1994), pp.103-114.
21. E. N. KHOGOTOV, A modification of the extragradient method for the solution of variational inequalities and some optimization problems, Zh. Vychisl. Mat. i Mat. Fiz., 27 (1987), pp. 1462-1473.
22. G. M. KORPELEVICH, The extragradient method for finding saddle points and other problems, Matecon, 12 (1976), pp. 747-756.
23. X.-D. LUO AND P. TSENG, On global projection-type error bound for the linear complementarity problem, Linear Algebra Appl., to appear.
24. 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.
25. Z.-Q. LUO AND P. TSENG, On the linear convergence of descent methods for convex essentially smooth minimization, SIAM J. Control Optim., 30 (1992), pp. 408-425.
26. _, Error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem, SIAM J. Optim., 2 (1992), pp. 43-54.
27. T. L. MAGNANTI AND G. PERAKIS, On the Convergence of Classical Variational Inequality Algorithms, Working Paper, Operations Research Center, Masachusetts Institute of Technology, Cambridge, MA, 1993.
28. O. L. MANGASARIAN, Convergence of iterates of an inexact matrix splitting algorithm for the symmetric monotone linear complementarity problem, SIAM J. Optim., 1 (1991), pp. 114-122.
29. O.L. MANGASARIAN AND J. REN, New improved error bounds for the linear complementarity problem, Math. Programming, 66 (1994), pp. 241-255.
30. O. L. MANGASARIAN AND M. V. SOLODOV, Nonlinear complementarity as unconstrained and constrained minimization, Math. Programming, 62 (1993), pp. 277-297.
31. P. MARCOTTE, Application of Khobotov's algorithm to variational inequalities and network equilibrium problems, Inform. Systems Oper. Res., 29 (1991), pp. 258-270.
32. P. MARCOTTE AND J.-P. DUSSAULT. A note on a globally convergent Newton method for solving monotone variational inequalities, Oper. Res. Lett., 6 (1987), pp. 35-42.
33. _, A sequential linear programming algorithm for solving monotone variational inequalities, SIAM J. Control Optim., 21 (1989), pp. 1260-1278.
34. P. MARCOTTE AND J.-H. WU, On the convergence of projection methods: Application to the decomposition of affine variational inequalities, J. Optim. Theory Appl., 85 (1995), pp. 347-362.
35. L. MATHIESEN, An algorithm based on a sequence of linear complementarity problems applied to a Walras' equilibrium model: An example, Math. Programming, 37 (1987), pp. 1-18.
36. K. G. MURTY, Linear Complementarity, Linear and Nonlinear Programming, Helderman-Verlag, Berlin, 1988.
37. J. M. ORTEGA AND W. C. RHEINBOLDT, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
38. J.-S. PANG, Asymmetric variational inequality problems over product sets: Applications and iterative methods, Math. Programming, 31 (1985), pp. 206-219.
39. _, A posteriori error bounds for the linearly-constrained variational inequality problem, Math. Oper. Res., 12 (1987), pp. 474-484.
40. J.-S. PANG AND D. CHAN, Iterative methods for variational and complementarity problems, Math. Programming, 24 (1982),pp. 284-313.
41. J.-S. PANG AND S. A. GABRIEL NE/SQP: A robust algorithm for the nonlinear complementarity problem, Math. Programming, 60 (1993), pp. 295-337.
42. P. M. PARDALOS AND N. KOVOOR, An algorithm for a singly constrained class of quadratic programs subject to upper and lower bounds, Math. Programming, 46 (1986), pp. 235-238.
43. S. M. ROBISSON, Some continuity properties of polyhedral multifunctions, Math. Programming Study, 14 (1981), pp. 206-214.
44. R. T. ROCKAFELLAR, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), pp. 877-898.
45. R. T. ROCKAFELLAR AND R. J.-B. WETS, Generalized linear-quadratic problems of deterministic and stochastic optimal control in discrete time, SIAM J. Control Optim., 28 (1990), pp. 810-822.
46. M. SIBONY, Méthodes itératives pour les équations et inéquations aux dérivées partielles nonlinéares de type monotone, Calcolo, 7 (1970), pp. 65-183.
47. D. SUN, A new step-size skill for solving a class of nonlinear projection equations, J. Comput. Math., 13 (1995), pp. 357-368.
48. _, A class of iterative methods for solving nonlinear projection equations, J. Optim. Theory Appl., 91 (1996), to appear.
49. P. TSENG, Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming, Math. Programming, 48 (1990), pp. 249-263.
50. P. TSENG, Applications of a splitting algorithm to decomposition in convex programming and variational inequalities, SIAM J. Control Optim., 29 (1991), pp. 119-138.
51. _, On linear convergence of iterative methods for the variational inequality problem, J. Comput. Appl. Math., 60 (1995), pp. 237-252.
52. N. YAMASHITA AND M. FUKUSHIMA, On stationary points of the implicit Lagrangian for nonlinear complementarity problems, J. Optim. Theory Appl., 84 (1995), pp. 653-663.
53. Y. YE, A new complexity result on minimization of a quadratic function with a sphere constraint, in Recent Advances in Global Optimization, C. Floudas and P. M. Pardalos, eds., Princeton University Press, Princeton, NJ, 1992.
54. L. ZHAO AND S. DAFERMOS, General economic equilibrium and variational inequalities, Oper. Res. Lett., 10 (1991), pp. 369-376.
55. C. ZHU AND R. T. ROCKAFELLAR, Primal-dual projected gradient algorithms for extended linear-quadratic programming, SIAM J. Optim., 3 (1993), pp. 751-783.