Abstract

The optimal value function \((c, b) \mapsto \varphi(c,b)\) of the quadratic program \(\min \{\frac12 x^{T}Dx + c^{T}x : Ax \geq b\}\), where \(D \in \mathbb R_{S}^{n \times n}\) is a given symmetric matrix, \(A \in \mathbb R^{m \times n}\) a given matrix, \(c \in \mathbb R^{n}\) and \(b \in \mathbb R^{m}\) are the linear perturbations, is considered. It is proved that \(\varphi\) is directionally differentiable at any point \(\overline{w} = (\overline{c},\overline{b})\) in its effective domain \(W := \{w=(c,b) \in \mathbb R^{n} \times \mathbb R^{m} : -\infty < \varphi(c,b) < +\infty\}\). Formulae for computing the directional derivative \(\varphi’(\bar{w};z)\) of \(\varphi\) at \(\overline{w}\) in a direction \(z = (u, v) \in \mathbb R^{n} \times \mathbb R^{m}\) 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 The preceding (unpublished) example of Klatte is also discussed.