Non-vertical supporting hyperplane of a value function ($r=\varphi (b)+{\lambda }^T(u-b)$)

At a point where $\varphi (b)$ is finite, this is an affine plane supporting the epigraph from below: $\varphi (u)\ge \varphi (b)+{\lambda }^T(u-b)$ for every $u$. A nonzero coefficient of the vertical coordinate permits normalization to the displayed graph form. For a constrained value function, the slope is exactly an attained multiplier in the Strong Lagrangian property. When the value function is differentiable at an interior point of its finite domain, a supporting slope equals its gradient: apply the supporting inequality along both positive and negative multiples of each coordinate direction and take limits.