Generalized Descent Methods for Asymmetric Systems of Equations and Variational Inequalities
Name
OR-137-85.pdf
Size
2.37 MB
Format
Adobe PDF
Checksum (MD5)
8e3bc3939aa6236ea4a6ecd36c9fb715
Author(s)
Hammond, Janice H.
Magnanti, Thomas L.
Date Issued
August 1985
Publisher
Massachusetts Institute of Technology, Operations Research Center
Series/Report no.
Operations Research Center Working Paper;OR 137-85
Abstract
We consider generalizations of the steepest descent algorithm for solving asymmetric systems of equations. We first show that if the system is linear and is defined by a matrix M, then the method converges if M2 is positive definite. We also establish easy to verify conditions on the matrix M that ensure that M is positive definite, and develop a scaling procedure that extends the class of matrices that satisfy the convergence conditions. In addition, we establish a local convergence result for nonlinear systems defined by uniformly monotone maps, and discuss a class of general descent methods. Finally, we show that a variant of the Frank-Wolfe method will solve a certain class of variational inequality problems. All of the methods that we consider reduce to standard nonlinear programming algorithms for equivalent optimization problems when the Jacobian of the underlying problem map is symmetric. We interpret the convergence conditions for the generalized steepest descent algorithms as restricting the degree of asymmetry of the problem map.
MIT Department
Massachusetts Institute of Technology. Operations Research Center
Persistent DSpace Link
