GAMS — The Solver Manuals - Available Software

MILES
Thomas F. Rutherford, University of Colorado
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
2 The Newton Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
3 Lemke’s Method with Implicit Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . 316
4 The Options File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
5 Log File Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
6 Status File Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
7 Termination Messages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Abstract
MILES is a solver for nonlinear complementarity problems and nonlinear systems of equations. This solver can be
accessed indirectly through GAMS/MPSGE or GAMS/MCP. This paper documents the solution algorithm, user
options, and program output. The purpose of the paper is to provide users of GAMS/MPSGE and GAMS/MCP
an overview of how the MCP solver works so that they can use the program effectively.
1 Introduction
MILES is a Fortran program for solving nonlinear complementarity problems and nonlinear systems of equations.
The solution procedure is a generalized Newton method with a backtracking line search. This code is based
on an algorithm investigated by Mathiesen (1985) who proposed a modeling format and sequential method
for solving economic equilibrium models. The method is closely related to algorithms proposed by Robinson
(1975), Hogan (1977), Eaves (1978) and Josephy (1979). In this implementation, subproblems are solved as
linear complementarity problems (LCPs), using an extension of Lemke’s almost-complementary pivoting scheme
in which upper and lower bounds are represented implicitly. The linear solver employs the basis factorization
package LUSOL, developed by Gill et al. (1991).
The class of problems for which MILES may be applied are referred to as ”generalized” or ”mixed” complementarity
problems, which is defined as follows:
Given: F : R n → R n , l, u ∈ R n
Find:
such that
z, w, v ∈ R n
F (z) = w − v
l ≤ z ≤ u, w ≥ 0, v ≥ 0
w T (z − l) = 0 , v T (u − z) = 0.