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Journal of Global Optimization

, Volume 32, Issue 1, pp 119–134 | Cite as

On the Optimal Value Function of a Linearly Perturbed Quadratic Program

  • G. M. Lee
  • N. N. Tam
  • N. D. YenEmail author
Article

Abstract

The optimal value function \((c, b)\mapsto \varphi (c, b)\) of the quadratic program \(\min \{ {1\over 2} x^{T}Dx + c^{T}x : Ax \geq b\}\), where \(D \in R_{S}^{n \times n}\) is a given symmetric matrix, \(A \in R^{m \times n}\) a given matrix, \(c \in R^{n}\) and \(b \in R^{m}\) are the linear perturbations, is considered. It is proved that \(\varphi\) is directionally differentiable at any point \(\bar{w} = (\bar{c}, \bar{b} )\) in its effective domain \(W : =\{w = (c, b) \in R^{n} \times R^{m} :-\infty < \varphi (c, b) < + \infty\}\). Formulae for computing the directional derivative \(\varphi' (\bar{w}; z)\) of \(\varphi\) at \(\bar{w}\) in a direction \(z = (u, v) \in R^{n} \times R^{m}\) are obtained. We also present an example showing that, in general, \(\varphi\) is not piecewise linear-quadratic on W. The preceding (unpublished) example of Klatte is also discussed.

Keywords

Directional differentiability Linear perturbation Nonconvex quadratic programming problem Optimal value function Piecewise linear-quadratic property 

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Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of Applied MathematicsPukyong National UniversityPusanKorea
  2. 2.Department of MathematicsHanoi Pedagogical Institute No. 2Xuan Hoa, Me LinhVietnam
  3. 3.Institute of MathematicsHanoiVietnam

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