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On the stability of solutions to quadratic programming problems


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 Cxd, 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.

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Received: November 5, 1997 / Accepted: September 12, 2000¶Published online November 17, 2000

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Phu, H., Yen, N. On the stability of solutions to quadratic programming problems. Math. Program. 89, 385–394 (2001).

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  • Key words: quadratic programming – local minimizer set – global minimizer set – lower semicontinuity