Elementary Calculus of the Subdifferential of a Convex Function

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the real numbers and sequences, and the Euclidean norm $\lVert\,\cdot\,\rVert$, dot product, distance $d_{E}$, topology, the notions of open and bounded subsets, and the closed balls $\bar{B}(x,r)$, are as fixed there.

Let $U\subseteq\mathbb{R}^{n}$ be open and convex and let $f:U\to\mathbb{R}$ be convex on $U$, with subdifferential $\partial_{U}f$. Then the following hold.

1. (Monotonicity) (i) For all $y,y'\in U$, all $q\in\partial_{U}f(y)$ and all $q'\in\partial_{U}f(y')$ one has $0\le(q-q')\cdot(y-y')$.

2. (Local bound on subgradients) (ii) Let $y_{0}\in U$ and let $r,M\in\mathbb{R}$ with $0<r$ and $0\le M$ be such that $\bar{B}(y_{0},2r)\subseteq U$ and

$$|f(z)-f(w)|\le M\,\lVert z-w\rVert\qquad\text{for all }z,w\in\bar{B}(y_{0},2r).$$

Then $\lVert q\rVert\le M$ for every $y\in\bar{B}(y_{0},r)$ and every $q\in\partial_{U}f(y)$.

3. (Closed graph) (iii) Let $(y_{m})_{m\in\mathbb{N}}$ be a sequence in $U$ converging to a point $y\in U$, for each $m$ let $q_{m}\in\partial_{U}f(y_{m})$, and suppose that $(q_{m})_{m\in\mathbb{N}}$ converges to $q\in\mathbb{R}^{n}$. Then $q\in\partial_{U}f(y)$.

4. (At a point of differentiability) (iv) Let $y\in U$ and suppose that $f$ is differentiable at $y$ with derivative matrix the real matrix $A$ with one row and $n$ columns. Let $g\in\mathbb{R}^{n}$ be the point whose $i$th coordinate is $A_{1i}$, so that $Ah$ has single coordinate $g\cdot h$ for every $h\in\mathbb{R}^{n}$. Then $\partial_{U}f(y)=\{g\}$.

5. (Continuity where the subgradient is unique) (v) Let $y\in U$ and suppose that $\partial_{U}f(y)=\{p\}$ for some $p\in\mathbb{R}^{n}$. Then for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there is $\delta\in\mathbb{R}$ with $0<\delta$ such that every $y'\in U$ with $\lVert y'-y\rVert<\delta$ and every $q\in\partial_{U}f(y')$ satisfy $\lVert q-p\rVert<\varepsilon$.