Advertisement

Mathematical Programming

, Volume 89, Issue 3, pp 385–394 | Cite as

On the stability of solutions to quadratic programming problems

  • H.X. Phu
  • N.D. Yen

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 Cxd, 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.

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • H.X. Phu
    • 1
  • N.D. Yen
    • 1
  1. 1.Institute of Mathematics, P.O. Box 631 Bo Ho, Hanoi, VietnamVN

Personalised recommendations