The solution of a quadratic programming problem is straightforward
 because of the fact that the optimality conditions, the counterparts of
 (NL1) and (NL2), for a quadratic program are linear. Applying the Kuhn-
 Tucker theorems, the sufficient, (NL4), and necessary, (NL1), (NL2), and
 (NL3), conditions for an optimal solution can be reduced to:

 (Q3) the Q matrix be positive semi-definite (sufficiency condition)

 and
 T-
 T T-) x= 0
 (Q4) c Qx Ap < 0 and-(c -Qx A

 -T
 (Q5) b Ax i 0 and (b Ax) p = 0

 where x > 0 is the equilibrium set of quantities and p > 0 is the
 Lagranigian and can be interpreted as the equilibrium set of market
 prices in the context of [5].
 The positive semi-definiteness condition, (Q3) is generally met as
long as the demand and supply functions are well-behaved (downward sloping
and upward sloping respectively). The condition (Q5) is essentially the
same as for the linear programming situation and has the same interpre-
tation. The relationships (Q4) are the same as for LP in rows where the
Q matrix coefficients are zero. In the case of non-zero Q coefficients,
the row will have the following form.

 n n
 S- x and (c ..x.)p
 S ji i 1 j=l 13 i1

If the first part holds with equality, it means that the demand (or supply)
function is satisfied with equality and the particular commodity can have
a positive price. If it does not hold with equality, this implies that
there is not a positive price that will clear the market and thus price
has to be zero.
 It should be clear that the economic rationality which prevailed in
the LP problem carries into QP, with the added benefits of endogenous,
market prices and quantities. The quadratic programming approach suffers
a major deficiency, however, as a result of the requirement that the Q
matrix be symmetric. The off diagonal elements of Q represent substitution
and complementary relationships between different commodities in the