## Abstract

In this paper, we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems. Included among such methods are the Newton and several successive overrelaxation algorithms. For the most part, the study is concerned with the family of linear approximation methods. These are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems. Convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches. The paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria. Finally, several convergence results are obtained for some nonlinear approximation methods.

### Similar content being viewed by others

## References

H.Z. Aashtiani, “The multi-modal traffic assignment problem”, Ph.D. dissertation, Alfred P. Sloan School of Management, Massachusetts Institute of Technology (May 1979).

M. Aganagic, “Variational inequalities and generalized complementarity problems”, Tech. Rept. SOL 78-11, Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford CA (1978).

B.-H. Ahn,

*Computation of market equilibria for policy analysis: The project independence evaluation systems (PIES) approach*(Garland, New York, 1979).B.-H. Ahn, “Computation of asymmetric linear complementarity problems by iterative methods”,

*Journal of Optimization Theory and Applications*33 (1981) 175–185.R.L. Asmuth, “Traffic network equilibria”, Tech. Rept. SOL 78-2, Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, CA (1978).

D. Chan and J.S. Pang, “The generalized quasi-variational inequality problem”,

*Mathematics of Operations Research*, to appear.R.W. Cottle and M.S. Goheen, “A special class of large quadratic programs”, in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.,

*Nonlinear programming*3 (Academic Press, New York, 1978) pp. 361–390.R.W. Cottle, G.H. Golub and R.S. Sacher, “On the solution of large, structured linear complementarity problems: The block partitioned case”,

*Applied Mathematics and Optimization*4 (1978) 347–363.R.W. Cottle and J.S. Pang, “On solving linear complementarity problems as linear programs”,

*Mathematical Programming Study*7 (1978) 88–107.R.W. Cottle and J.S. Pang, “On the convergence of a block successive overrelaxation method for a class of linear complementarity problems”,

*Mathematical Programming Study*17 (1982) 126–138.C.W. Cryer, “The solution of quadratic programming problems using systematic overrelaxation”,

*SIAM Journal on Control*9 (1971) 385–392.S. Dafermos, “Traffic equilibrium and variational inequalities”,

*Transportation Science*14 (1980) 42–54.S. Dafermos, “An iterative scheme for variational inequalities”, Division of Applied Mathematics, Brown University (May 1981).

B.C. Eaves, “Computing stationary points”,

*Mathematical Programming Study*7 (1978) 1–14.B.C. Eaves, “Computing stationary points, again”, in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.,

*Nonlinear programming*3 (Academic Press, New York, 1978) pp. 391–405.B.C. Eaves, “A locally quadratically convergent algorithm for computing stationary points”, Tech. Rept., Department of Operations Research, Stanford University, Stanford, CA (May 1978).

B.C. Eaves and R. Saigal, “Homotopies for the computation of fixed points on unbounded regions”,

*Mathematical Programming*3 (1972) 225–237.S.C. Fang, “Traffic equilibria on multiclass-user transportation networks analysed via variational inequalities”, Mathematics Research Report 79-13, Department of Mathematics, University of Maryland, Baltimore County (November 1979).

S.C. Fang, “An iterative method for generalized complementarity problems”, Mathematics Research Report 79-11, Department of Mathematics, University of Maryland, Baltimore County (October 1979).

S.C. Fang and E.L. Peterson, “Generalized variational inequalities”, Mathematics Research Report 79-10, Department of Mathematics, University of Maryland, Baltimore County (October 1979).

S.C. Fang and E.L. Peterson, “Economic equilibria on networks”, Mathematics Research Report 80-13, Department of Mathematics, University of Maryland, Baltimore County (May 1980).

M. Fiedler and V. Ptak, “On matrices with nonpositive off-diagonal elements and positive principal minors”,

*Czechoslovak Journal of Mathematics*12 (1962) 382–400.M. Florian and H. Spiess, “The convergence of diagonalization algorithms for asymmetric network equilibrium problems”,

*Transportation Research*, to appear.W.W. Hogan, “Project independence evaluation system: Structure and algorithms”,

*Proceedings of Symposia in Applied Mathematics of the American Mathematical Society*21 (1977) 121–137.C.L. Irwin, “Convergence properties of a PIES-type algorithm for non-integrable functions”, Tech. Rept. SOL 77-33, Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, CA (December 1977).

C.L. Irwin, “Analysis of a PIES-algorithm”, Symposium papers: Energy modelling and net energy analysis, Institute of Gas Technology (1978) 471–483.

C.L. Irwin and C.W. Yang, “Iteration and sensitivity for a nonlinear spatial equilibrium problem”, manuscript of a paper (August 1980).

N.H. Josephy, “Newton's method for generalized equations”, Tech. Rept. 1965, Mathematics Research Center, University of Wisconsin, Madison, WI (June 1979).

N.H. Josephy, “Quasi-Newton methods for generalized equations”, Tech. Rept. 1966, Mathematics Research Center, University of Wisconsin, Madison, WI (June 1979).

S. Karamardian, “Generalized complementarity problem”,

*Journal of Optimization Theory and Applications*8 (1971) 161–168.O.L. Mangasarian, “Linear complementarity problems solvable by a linear program”,

*Mathematical Programming*10 (1976) 263–270.O.L. Mangasarian, “Solution of symmetric linear complementarity problems by iterative methods”,

*Journal of Optimization Theory and Applications*22 (1977) 465–485.J.M. Ortega,

*Numerical analysis; A second course*(Academic Press, New York, 1972).J.M. Ortega and W.C. Rheinboldt,

*Iterative solution of nonlinear-equations in several variables*(Academic Press, New York, 1970).J.S. Pang, “Hidden

*Z*-matrices with positive principal minors”,*Linear Algebra and its Applications*23 (1979) 201–215.J.S. Pang, “A hybrid method for the solution of some multi-commodity spatial equilibrium problems”,

*Management Science*27 (1981) 1142–1157.J.S. Pang, “A column generation technique for the computation of stationary points”,

*Mathematics of Operations Research*6 (1981) 213–224.J.S. Pang, “The implicit complementarity problem”, in: O.L. Mangasarian, R.R. Meyer and S.M. Robinson, eds.,

*Nonlinear programming*4 (Academic Press, New York, 1981) pp. 487–518.J.S. Pang, “On the convergence of a basic iterative method for the implicit complementarity problem”,

*Journal of Optimization Theory and Applications*, to appear.J.S. Pang, “Approaches for convergence of a basic iterative method for the linear complementarity problem”, M.S.R.R. 446, GSIA, Carnegie-Mellon University, Pittsburgh, PA (November 1980).

E.L. Peterson, “The conical duality and complementarity of price and quantity for multicommodity spatial and temporal network allocation problems”, Discussion paper 207, Center for Mathematical Studies in Economics and Management Science, Northwestern University (March 1976).

R.J. Plemmons, “

*M*-matrix characterization*I*: Nonsingular*M*-matrices”,*Linear Algebra and its Applications*18 (1977) 175–188.W. Rheinboldt, “On

*M*-functions and their application to nonlinear Gauss—Seidel iterations and to network flows”,*Journal of Mathematical Analysis and Applications*32 (1970) 274–307.S.M. Robinson, “Strongly regular generalized equations”,

*Mathematics of Operations Research*5 (1980) 43–62.R.T. Rockafellar,

*Convex analysis*(Princeton University Press, Princeton, NJ, 1970).R. Saigal, “Extension of the generalized complementarity problem”,

*Mathematics of Operations Research*1 (1976) 260–266.H. Samelson, R.M. Thrall and O. Wesler, “A partition theorem for Euclidean

*n*-space”,*Proceedings American Mathematical Society*9 (1958) 805–807.G. Stampacchia, “Variational inequalities”, in: A. Ghizzetti, ed.,

*Theory and applications of monotone operator*, Proceedings of the NATO Advanced Study Institute, Venice, Italy (1968) 101–192.T. Takayama and G.G. Judge,

*Spatial and temporal price and allocation models*(North-Holland, Amsterdam, 1971).A. Tamir, “Minimality and complementarity properties associated with

*Z*-functions and*M*-functions”,*Mathematical Programming*7 (1974) 17–31.R. Thrasher, “Notes on convergence of PIES-like iterative processes”, mimeographed, New Mexico State University (February 1978).

D. Young,

*Iterative solution of large linear systems*(Academic Press, New York, 1971).

## Author information

### Authors and Affiliations

## Additional information

This research was based on work supported by the National Science Foundation under grant ECS-7926320.

## Rights and permissions

## About this article

### Cite this article

Pang, J.S., Chan, D. Iterative methods for variational and complementarity problems.
*Mathematical Programming* **24**, 284–313 (1982). https://doi.org/10.1007/BF01585112

Received:

Revised:

Issue Date:

DOI: https://doi.org/10.1007/BF01585112