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PATH 4.6 471
$include walras.dat
positive variables p(i), y(j);
variables d(i);
equations S(i), L(j), demand(i);
demand(i)..
d(i) =e= c(i)*sum(k, g(k)*p(k)) / p(i) ;
S(i).. b(i) + sum(j, A(i,j)*y(j)) - d(i) =g= 0 ;
L(j).. -sum(i, p(i)*A(i,j)) =g= 0 ;
model walras / demand.d, S.p, L.y /;
solve walras using mcp;
Figure 28.3: Walrasian equilibrium as an MCP, walras2.gms
1.2.3 Mixed Complementarity Problem
A mixed complementarity problem (MCP) is specified by three pieces of data, namely the lower bounds l, the
upper bounds u and the function F .
(MCP) Given lower bounds l ∈ {R ∪ {−∞}} n , upper bounds u ∈ {R ∪ {∞}} n and a function F : R n → R n ,
find z ∈ R n such that precisely one of the following holds for each i ∈ {1, . . . , n}:
F i (z) = 0 and l i ≤ z i ≤ u i
F i (z) > 0 and z i = l i
F i (z) < 0 and z i = u i .
These relationships define a general MCP (sometimes termed a rectangular variational inequality [25]). We will
write these conditions compactly as
l ≤ x ≤ u ⊥ F (x).
Note that the nonlinear complementarity problem of Section 1.1.3 is a special case of the MCP. For example, to
formulate an NCP in the GAMS/MCP format we set
or declare
z.lo(I) = 0;
positive variable z;
Another special case is a square system of nonlinear equations
(NE) Given a function F : R n → R n find z ∈ R n such that
F (z) = 0.
In order to formulate this in the GAMS/MCP format we just declare
free variable z;
In both the above cases, we must not modify the lower and upper bounds on the variables later (unless we wish
to drastically change the problem under consideration).