The solution to an LCP problem of the special type in (LCP1), (LCP2)
and (LCP3) with the matrix structure discussed above, will be the optimal
solution to the corresponding quadratic programming problem. Also, if
Q is a zero matrix, the problem reduces to a linear programming problem.
There are a number of algorithms for solving an LCP problem [6], [3], [1].
The reader is referred to [1] for the conditions for solvability of LCP.
The algorithm used in LCRAPD (and RANDQP) is described in Appendix D.
Typically, LCP algorithms'work by creating an initial basic solution
that violates w > 0 or in which a nearly complementary solution is
 2 -
created with the introduction of another artificial vector. Then the
algorithm proceeds by carrying out a series of simplex like pivot opera-
tions until a non-negative fully complementary solution is reached.
 It was mentioned above in the discussion of quadratic programming
that the symmetry restriction on Q sometimes creates a serious problem
in applied research (see chapters 14 and 18 of [5]). It is apparent
that a solution to (LCPI), (LCP2), and (LCP3) with an asymmetric Q
represents a true market equilibrium soltuion in the Marshallian sense
and still preserves all the relationships discussed above. The only
difference is that the solution is not "optimal" in the sense of
quadratic programming, since a welfare function involving producers
and consumers surplus cannot be formed because the integration used to
create such an objective function requires the symmetry restriction
[5, p. 116]. For purposes of analyzing market behavior or making pro-
jections for the future, an objective function is sometimes really
unnecessary and everything which is needed is present in a general LCP
problem. In conclusion, LCP modeling represents a market oriented
approach which:

 1. Allows endogenous prices and quantities,
 2. Meets conditions for a Marshallian market equilibrium,
 3. Has the freedom to include technical and political
 constraints explicitly; also both prices and quantities
 can be constrained in a model which is impossible in
 LP formulation,
 4. Has an efficient solution procedure.