On the Optimal Value Function of a Linearly Perturbed Quadratic Program

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 We also present an example showing that, in general, $\varphi$ is not piecewise linear-quadratic on W. The preceding (unpublished) example of Klatte is also discussed.