Quadratic Test Functions: a Global $C^2$ Majorant and Minorant Matching a $C^2$ Function to Second Order at a Point

In the setting of Real Hilbert Spaces: Standing Notation and Background, let $H$ be a real Hilbert space with inner product $\langle\cdot,\cdot\rangle$, norm $|\cdot|$ and distance $d$, and let $\mathrm{Sym}(H)$ be the set of bounded symmetric bilinear forms on $H$, with its norm $\lVert\cdot\rVert$, its sums and scalar multiples and its identity form $I=I_{H}$, as fixed there. Let $U\subseteq H$ be open in $(H,d)$, let $\varphi$ belong to the class $C^{2}(U)$, with gradient $D\varphi(x)\in H$ and Hessian $D^{2}\varphi(x)\in\mathrm{Sym}(H)$ at $x\in U$, let $\hat{x}\in U$, and let $\eta\in\mathbb{R}$. Let $T:H\to\mathbb{R}$ be the function

$$T(x)=\varphi(\hat{x})+\langle D\varphi(\hat{x}),x-\hat{x}\rangle+\tfrac{1}{2}\,D^{2}\varphi(\hat{x})(x-\hat{x},x-\hat{x})+\eta\,|x-\hat{x}|^{2},$$

where $\tfrac12$ is $2^{-1}$ and $D^{2}\varphi(\hat{x})(z,z)$ is the value of the form $D^{2}\varphi(\hat{x})$ at the pair $(z,z)$. Local maxima and local minima relative to a subset of $H$ are as fixed there. Then the following hold.

1. (The quadratic is of class $C^{2}$)(i) $T\in C^{2}(H)$, and for every $U'\subseteq H$ open in $(H,d)$ the restriction of $T$ to $U'$ belongs to $C^{2}(U')$, with the same gradients and Hessians at the points of $U'$. Moreover

$$T(\hat{x})=\varphi(\hat{x}),\qquad DT(\hat{x})=D\varphi(\hat{x}),\qquad D^{2}T(x)=D^{2}\varphi(\hat{x})+2\eta I\quad\text{for every }x\in H,$$

and consequently $\lVert D^{2}T(x)-D^{2}\varphi(\hat{x})\rVert\le 2\,|\eta|$ for every $x\in H$.

2. (A majorant when $\eta$ is positive)(ii) Suppose $0<\eta$. Then there is a positive $\rho\in\mathbb{R}$ such that every $x\in H$ with $|x-\hat{x}|<\rho$ satisfies $x\in U$ and $\varphi(x)\le T(x)$.

3. (A minorant when $\eta$ is negative)(iii) Suppose $\eta<0$. Then there is a positive $\rho\in\mathbb{R}$ such that every $x\in H$ with $|x-\hat{x}|<\rho$ satisfies $x\in U$ and $T(x)\le\varphi(x)$.

4. (Transfer of a local maximum)(iv) Suppose $0<\eta$, let $A\subseteq U$ satisfy $\hat{x}\in A$, and let $g:A\to\mathbb{R}$. If the function $A\to\mathbb{R}$ with value $g(x)-\varphi(x)$ at $x$ has a local maximum at $\hat{x}$ relative to $A$, then the function $A\to\mathbb{R}$ with value $g(x)-T(x)$ at $x$ has a local maximum at $\hat{x}$ relative to $A$.

5. (Transfer of a local minimum)(v) Suppose $\eta<0$, let $A\subseteq U$ satisfy $\hat{x}\in A$, and let $g:A\to\mathbb{R}$. If the function $A\to\mathbb{R}$ with value $g(x)-\varphi(x)$ at $x$ has a local minimum at $\hat{x}$ relative to $A$, then the function $A\to\mathbb{R}$ with value $g(x)-T(x)$ at $x$ has a local minimum at $\hat{x}$ relative to $A$.