Second-Order Test Data at a Global Quadratic Maximum of a Semiconvex Function

Throughout we work in the setting of Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation, whose notation, including that of Euclidean Space and Real Matrices: Standing Notation on which it rests, is in force in the dimension $N$, a natural number with $1\le N$. In particular $-aI_{N}$ denotes $(-a)I_{N}$, an element of $\mathcal{S}(N)$ for every real $a$, as fixed in Euclidean Space and Real Matrices: Standing Notation (iii), and convergence of a sequence in $\mathbb{R}^{N}$ or in $\mathcal{S}(N)$ refers to $d_{E}$ or to $d_{\mathcal{S}(N)}$ respectively, as fixed in Euclidean Space and Real Matrices: Standing Notation (v). We abbreviate $\lVert z\rVert^{2}=\lVert z\rVert\cdot\lVert z\rVert$, and $\tfrac{s}{2}$ denotes the product of $s\in\mathbb{R}$ with the multiplicative inverse of $2=1+1$, which exists by claim 8 of Elementary Order Arithmetic in an Ordered Field. The set $\mathbb{R}^{N}$ is open, directly from Open Subset of a Metric Space, and is convex, directly from that definition. That a quadruple is approximable by test data from above for a function on an open set is as defined there.

Let $\lambda\in\mathbb{R}$ satisfy $0\le\lambda$, let $f:\mathbb{R}^{N}\to\mathbb{R}$ be semiconvex on $\mathbb{R}^{N}$ with constant $\lambda$, and let $B\in\mathcal{S}(N)$. Assume that

$$f(\xi)-\tfrac{1}{2}\,\xi\cdot(B\xi)\ \le\ f\bigl(0_{\mathbb{R}^{N}}\bigr)\qquad\text{for every }\xi\in\mathbb{R}^{N}.$$

Then the following hold.

1. (An approximating sequence of points of twice differentiability) (i) There exist $X\in\mathcal{S}(N)$ and a sequence $(x_{k})_{k\in\mathbb{N}}$ in $\mathbb{R}^{N}$ such that $f$ is twice differentiable, with gradient $Df(x_{k})$ and Hessian $D^{2}f(x_{k})$, at $x_{k}$ for every $k\in\mathbb{N}$, the sequence $(x_{k})_{k\in\mathbb{N}}$ converges to $0_{\mathbb{R}^{N}}$ in $\mathbb{R}^{N}$, the sequence $\bigl(Df(x_{k})\bigr)_{k\in\mathbb{N}}$ converges to $0_{\mathbb{R}^{N}}$ in $\mathbb{R}^{N}$, the sequence $\bigl(D^{2}f(x_{k})\bigr)_{k\in\mathbb{N}}$ converges to $X$ in $\mathcal{S}(N)$, and

$$-\lambda I_{N}\ \preceq\ X\ \preceq\ B .$$

2. (Test data at the origin) (ii) Let $X\in\mathcal{S}(N)$ and let $(x_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathbb{R}^{N}$ with the properties listed in claim 1. Then the quadruple

$$\bigl(0_{\mathbb{R}^{N}},\,f(0_{\mathbb{R}^{N}}),\,0_{\mathbb{R}^{N}},\,X\bigr)$$

is approximable by test data from above for $f$, the domain being $\mathbb{R}^{N}$. In particular there is $X\in\mathcal{S}(N)$ with $-\lambda I_{N}\preceq X\preceq B$ for which that quadruple is approximable by test data from above for $f$.