In this problem, we have an unknown vector of slack variables formed by

wl and w2; a constant vector formed by the right hand side values, -b;
and the linear part of the objective function, c; a square matrix made
up of the A matrix, the negative transpose of A and the Q matrix; and
finally another unknown vector composed of the prices p, and quantities
x. This type of problem is known as a linear complementarity programming
problem, and has a wide range of applications.
 The general LCP problem is stated as follows.

 Find w, z

 such that

 (LCP4) w = q + Mz

 (LCP5) w'z = 0
 and
 (LCP6) w > 0 and z > 0.

 In such a problem w and z are unknown vectors (of dimension n), q is
a fixed vector, and M is a known square matrix. The correspondence
between (LCP1) and (LCP2) and between (LCP2) and (LCP5) should be clear.
In terms of the model approach used here,


q [ M= T and z =
 c -A -Q x


with w a slack vector. It is called a complementarity programming pro-
blem because a "complementary" relationship must hold between the unknown
vectors, w and z. In particular, w.z = 0 for all i. wi and z. are said
to be complements of each other. Only one of them can be positive. If
the reader will recall our previous arguments, it will be noted that this
is exactly the kind of relationship expressed by the second part of
conditions (L3) and (L4) or (Q4) and (Q5). Aside from this being a purely
mathematical curiosity, it was pointed out previously how this kind of
relationship is inherent in an equilibrium solution to an economic model-
ing problem.