# On the stability of solutions to quadratic programming problems

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## Abstract.

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*∈ℝ^{ n }, *A*∈ℝ^{ n×n }, *b*∈ℝ^{ n }, *C*∈ℝ^{ m×n }, *d*∈ℝ^{ m }, 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.

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