A parametric procedure for linear programming is shown to solve the bilinear complementarity problem, of which a special case is the problem of computing a competitive equilibrium of a piecewise linear economic model. THE PROBLEM OF COMPUTING an equilibrium of an economic model requires ordinarily the construction of a fixed point of a correspondence (H. Scarf [15]). R. Mantel [11] has shown, however, that for piecewise linear models the problem can be reduced to finding a "coordinate-wise value-preserving activity," which is an instance of the "bilinear complementarity problem" studied here. In this note, I present a constructive computational theory for the "regular" case in which certain nondegeneracy assumptions are satisfied. The algorithm can be implemented as a parametric linear programming procedure. An alternative algorithm for computing solutions has been proposed recently by G. Dantzig, B. C. Eaves, and D. Gale [3].
[1]
A. Sard,et al.
The measure of the critical values of differentiable maps
,
1942
.
[2]
G. Debreu.
The Coefficient of Resource Utilization
,
1951
.
[3]
H. Scarf.
Some Examples of Global Instability of the Competitive Equilibrium
,
1960
.
[4]
J. Milnor.
Topology from the differentiable viewpoint
,
1965
.
[5]
C. E. Lemke,et al.
Bimatrix Equilibrium Points and Mathematical Programming
,
1965
.
[6]
G. Dantzig,et al.
COMPLEMENTARY PIVOT THEORY OF MATHEMATICAL PROGRAMMING
,
1968
.
[7]
B. Eaves.
The Linear Complementarity Problem
,
1971
.
[8]
K. Arrow,et al.
General Competitive Analysis
,
1971
.
[9]
S. Smale.
Convergent process of price adjust-ment and global newton methods
,
1976
.
[10]
George B. Dantzig,et al.
An algorithm for a piecewise linear model of trade and production with negative prices and bankruptcy
,
1979,
Math. Program..