Iterative methods for variational and complementarity problems

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.

[1]  H. Samelson,et al.  A partition theorem for Euclidean $n$-space , 1958 .

[2]  С. И. Радауцан.  ИССЛЕДОВАНИЕ НЕКОТО РЫХ СЛОЖНЫХ ПОЛУПРОВОДНИКОВЫХ Т ВДРДЫХ РАСТВОРОВ И СОЕДИНЕНИЙ НА ОСНОВ Е ИНДИЯ , 1962 .

[3]  M. Fiedler,et al.  On matrices with non-positive off-diagonal elements and positive principal minors , 1962 .

[4]  S. Karamardian.  Generalized complementarity problem , 1970 .

[5]  W. Rheinboldt.  On M-functions and their application to nonlinear Gauss-Seidel iterations and to network flows☆ , 1970 .

[6]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[7]  P. Zusman.  Spatial and temporal price and allocation models , 1971 .

[8]  C. Cryer.  The Solution of a Quadratic Programming Problem Using Systematic Overrelaxation , 1971 .

[9]  Louis A. Hageman,et al.  Iterative Solution of Large Linear Systems. , 1971 .

[10]  B. Curtis Eaves,et al.  Homotopies for computation of fixed points on unbounded regions , 1972, Math. Program..

[11]  Arie Tamir,et al.  Minimality and complementarity properties associated with Z-functions and M-functions , 1974, Math. Program..

[12]  S. Venit,et al.  Numerical Analysis: A Second Course. , 1974 .

[13]  E. Peterson.  The Conical Duality and Complementarity of Price and Quality for Multicommodity Spatial and Temporal Network Allocation Problems , 1976 .

[14]  Romesh Saigal,et al.  Extension of the Generalized Complementarity Problem , 1976, Math. Oper. Res..

[15]  Olvi L. Mangasarian,et al.  Linear complementarity problems solvable by A single linear program , 1976, Math. Program..

[16]  O. Mangasarian.  Solution of symmetric linear complementarity problems by iterative methods , 1977 .

[17]  R. S. Sacher,et al.  On the solution of large, structured linear complementarity problems: The block partitioned case , 1977 .

[18]  R. Plemmons.  M-matrix characterizations.I—nonsingular M-matrices , 1977 .

[19]  C. Irwin.  Convergence Properties of a Pies-Type Algorithm for Non-Integrable Functions. , 1977 .

[20]  B. Eaves.  Computing Stationary Points, Again. , 1978 .

[21]  R. Cottle,et al.  A SPECIAL CLASS OF LARGE QUADRATIC PROGRAMS , 1978 .

[22]  R. Cottle,et al.  On solving linear complementarity problems as linear programs , 1978 .

[23]  R. Asmuth.  Traffic network equilibria , 1978 .

[24]  B. Eaves.  A Locally Quadratically Convergent Algorithm for Computing Stationary Points. , 1978 .

[25]  B. Eaves.  Computing stationary points , 1978 .

[26]  B. Ahn.  Computation of market equilibria for policy analysis: the project independence evaluation system approach. , 1978 .

[27]  J. Pang.  Hidden Z-matrices with positive principal minors☆ , 1979 .

[28]  H. Z. Aashtiani.  The multi-modal traffic assignment problem. , 1979 .

[29]  N. Josephy.  Newton's Method for Generalized Equations. , 1979 .

[30]  N. Josephy.  Quasi-Newton methods for generalized equations , 1979 .

[31]  Stephen M. Robinson,et al.  Strongly Regular Generalized Equations , 1980, Math. Oper. Res..

[32]  Stella Dafermos,et al.  Traffic Equilibrium and Variational Inequalities , 1980 .

[33]  J. Pang.  Approaches for Convergence of a Basic Iterative Method for the Linear Complementarity Problem. , 1980 .

[34]  S. Fang.  An iterative method for generalized complementarity problems , 1980 .

[35]  J. Douglas Faires,et al.  Numerical Analysis , 1981 .

[36]  J. Pang.  A Hybrid Method for the Solution of Some Multi-Commodity Spatial Equilibrium Problems , 1981 .

[37]  B. Ahn.  Solution of nonsymmetric linear complementarity problems by iterative methods , 1981 .

[38]  J. Pang.  THE IMPLICIT COMPLEMENTARITY PROBLEM , 1981 .

[39]  Jong-Shi Pang.  A Column Generation Technique for the Computation of Stationary Points , 1981, Math. Oper. Res..

[40]  E. Peterson,et al.  Generalized variational inequalities , 1982 .

[41]  R. Cottle,et al.  On the convergence of a block successive over-relaxation method for a class of linear complementarity problems , 1982 .

[42]  J. S. Pang,et al.  The Generalized Quasi-Variational Inequality Problem , 1982, Math. Oper. Res..

[43]  J. Pang.  On the convergence of a basic iterative method for the implicit complementarity problem , 1982 .

[44]  M. Florian,et al.  The convergence of diagonalization algorithms for asymmetric network equilibrium problems , 1982 .

[45]  Lamberto Cesari,et al.  Optimization-Theory And Applications , 1983 .

[46]  Stella Dafermos,et al.  An iterative scheme for variational inequalities , 1983, Math. Program..