MCP is more general than Linear Complementarity Problems (LCP) and Nonlinear Complementarity Problems (NCP).

The form of MCP is as follows:

F(x) ⟂ lb ≤ x ≤ ub

which means

• x = lb, then F(x) ≥ 0   #lb means lower bound
• lb < x < ub, then F(x) = 0
• x = ub, then F(x) ≤ 0   #ub means upper bound

When there is no upper bound ub, and the lower bound lb=0, then it is a regular Nonlinear Complementarity Problem (NCP) of the form:

0 ≤ F(x) ⟂ x ≥ 0

which means, for each   ,  we can get the below expressions:

F(x)' x = 0, F(x) ≥ 0, x ≥ 0

or, we can write:⟂   . 

When F(x) is a linear mapping such as F(x) = Ax + q with matrix A and vector q, then it is a Linear Complementarity Problem (LCP). 

Note: ⟂ denotes perpendicular symbol.

An example from GAMS tutorial doc:

Among which,

where, 

.

Then implement it in GAMS with the codes as:

$TITLE simple mpec example

variable f, x1, x2, y1, y2;
positive variable y1;
y2.lo = -1;
y2.up =  1;

equations cost, g, h1, h2;

* define the function f
cost..  f =E= x1 + x2; 
g..     sqr(x1) + sqr(x2) =L= 1;
h1..    x1 =G= y1 - y2 + 1;
h2..    x2 + y2 =N= 0;

model example / cost, g, h1.y1, h2.y2 /;
solve example using mpec min f;

the results are shown below pic with red mark: