We consider the parametric programming problem (Q
p
) of minimizing the quadratic function f(x,p):=x
T
Ax+b
T
x subject to the constraint Cx≤d, where x∈ℝn, A∈ℝn×n, b∈ℝn, C∈ℝm×n, d∈ℝm, and p:=(A,b,C,d) is the parameter. Here, the matrix A is not assumed to be positive semidefinite. The set of the global minimizers and the set of the local minimizers to (Q
p
) are denoted by M(p) and M
loc(p), respectively. It is proved that if the point-to-set mapping M
loc(·) is lower semicontinuous at p then M
loc(p) is a nonempty set which consists of at most ?
m,n
points, where ?
m,n
=\(\binom{m}{{\text{min}}\{[m/2],n\}}\) is the maximal cardinality of the antichains of distinct subsets of {1,2,...,m} which have at most n elements. It is proved also that the lower semicontinuity of M(·) at p implies that M(p) is a singleton. Under some regularity assumption, these necessary conditions become the sufficient ones.