Advertisement

Journal of Global Optimization

, Volume 32, Issue 1, pp 119–134

# On the Optimal Value Function of a Linearly Perturbed Quadratic Program

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We also present an example showing that, in general, $$\varphi$$ is not piecewise linear-quadratic on W. The preceding (unpublished) example of Klatte is also discussed.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

## Preview

Unable to display preview. Download preview PDF.

## References

1. 1.
Auslender, A., Coutat, P. 1996Sensitivity analysis for generalized linear-quadratic problemsJournal of Optimization Theory and Applications88541559Google Scholar
2. 2.
Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K. 1982Non-Linear Parametric OptimizationAkademie-VerlagBerlinGoogle Scholar
3. 3.
Best, M.J., Chakravarti, N. 1990Stability of linearly constrained convex quadratic programsJournal of Optimizaiton Theory and Applications644353Google Scholar
4. 4.
Blum, E., Oettli, W. 1972Direct proof of the existence theorem for quadratic programmingOperations Research20165167Google Scholar
5. 5.
Clarke, F.H. 1975Generalized gradients and applicationsTransactions of the American Mathematics Society205247262Google Scholar
6. 6.
Clarke, F.H. 1983Optimization and Nonsmooth AnalysisJohn Wiley SonsNew YorkGoogle Scholar
7. 7.
Cottle, R.W., Pang, J.-S., Stone, R.E. 1992The Linear Complementarity ProblemAcademic PressNew YorkGoogle Scholar
8. 8.
Klatte, D. 1985On the Lipschitz behavior of optimal solutions in parametric problems of quadratic optimization and linear complementarityOptimization16819831Google Scholar
9. 9.
Rockafellar, R.T. 1970Convex AnalysisPrinceton University PressPrinceton, New JerseyGoogle Scholar
10. 10.
Rockafellar, R.T. 1988First- and second-order epi-differentiability in nonlinear programmingTransactions of the American Mathematics Society30775108Google Scholar
11. 11.
Rockafellar, R.T., Wets, R.J.-B. 1998Variational AnalysisSpringerBerlin-HeidelbergGoogle Scholar
12. 12.
Sun, J. (1986), Ph.D. Thesis, University of WashingtonGoogle Scholar
13. 13.
Tam, N.N. 2001Directional differentiability of the optimal value function in indefinite quadratic programmingActa Mathematica Vietnamica26377394Google Scholar
14. 14.
Tam, N.N. 2002Continuity of the optimal value function in indefinite quadratic programmingJournal of Global Optimizaiton234361Google Scholar

© 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