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In "Nonsmooth Optimization" by Mäkela and Neittaanmäki the definition of the generalized directional derivative is given as follows:

Definition 3.1.1 (Clarke). Let $f: \mathbf{R}^{n} \rightarrow \mathbf{R}$ be locally Lipschitz at a point $x \in \mathrm{R}^{n} .$ The generalized directional derivative of $f$ at $x$ in the direction of $v \in \mathrm{R}^{n}$ is defined by $$ f^{\circ}(x ; v)=\limsup _{y \rightarrow x, t \rightarrow 0} \frac{f(y+t v)-f(y)}{t} \, . $$

Then in Theorem 3.1.2. on page 30 they proof that $f^{\circ}(x ; v)$ is upper semicontinuous as a function of $(x ; v)$.

The proof starts with:

Let $\left(x_{i}\right),\left(v_{i}\right) \subset \mathbf{R}^{n}$ be sequences such that $x_{i} \rightarrow x$ and $v_{i} \rightarrow v$. By definition of upper limit, there exist sequences $\left(y_{i}\right) \subset \mathbf{R}^{n}$ and $\left(t_{i}\right) \subset \mathrm{R}$ such that $t_{i}>0$ \begin{equation} \left\|y_{i}-x_{i}\right\|+t_{i}< \frac{1}{i} \end{equation} and \begin{equation} f^{\circ}\left(x_{i} ; v_{i}\right) - \frac{1}{i}\leq \frac{f\left(y_{i}+t_{i} v_{i}\right)-f\left(y_{i}\right)}{t_{i}} \end{equation} for all $i \in \mathrm{N}$.

Why do such sequences $\left(y_{i}\right) \subset \mathbf{R}^{n}$ and $\left(t_{i}\right) \subset \mathrm{R}$ exist?

I mean by the above definition of the generalized directional derivative we have

\begin{align} f^{\circ}\left(x_{i} ; v_{i}\right) - \frac{1}{i} &= \limsup _{y \rightarrow x_{i}, t \rightarrow 0} \frac{f(y+t v_{i})-f(y)}{t} - \frac{1}{i}\\ &\leq \frac{f\left(y_{i}+t_{i} v_{i}\right)-f\left(y_{i}\right)}{t_{i}} \\ \end{align}

I would appreciate a lot if someone could explain to me what the thoughts behind this are.

mathsstudentTUD
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1 Answers1

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This is tricky! For me, the appearance of $1/i$ strongly suggested an approach that ultimately leads nowhere. Here's one way to work it out:

Define a family of functions, one for each $i \in \mathbb{N}$, by

$$ h_{i}\left(w,s\right) := \frac{f\left(w+sv_{i}\right) - f\left(w\right)} {s} $$

and consider the limit superior,

$$ f^{\circ}\left(x_{i},v_{i}\right) := \limsup\limits_{w \to x_{i}, s \to 0} \frac{f\left(w+sv_{i}\right) - f\left(w\right)} {s} = \limsup\limits_{w \to x_{i}, s \to 0} h_{i}\left(w,s\right). $$

Since the $\limsup$ is the largest of the limit points of the sequences that have the form $h_{i}\left(w_{j},s_{j}\right)$ and where $\left(w_{j},s_{j}\right)$ approaches $\left(x_{i},0\right)$, there must be at least one such sequence $\left(w,s\right)_{j} = \left(w_{j},s_{j}\right)$ making $h_{i}\left(w_{j},s_{j}\right)$ approach $f^{\circ}\left(x_{i},v_{i}\right)$. For that particular sequence, we have

$$ \lim_{j \to \infty} h_{i}\left(w_{j},s_{j}\right) = f^{\circ}\left(x_{i},v_{i}\right). $$

First note that we can go far enough down the sequence $\left(w,s\right)_{j}$ so that $\left\lVert w_{j} - x_{i}\right\rVert + s_{j}$ is as close to zero as we'd like; this works because $\left(w,s\right)_{j}$ was a sequence approaching $\left(x_{i},0\right)$. We can also go far enough down the sequence to make $h_{i}\left(w_{j},s_{j}\right)$ as close to its limit $f^{\circ}\left(x_{i},v_{i}\right)$ as we'd like.

In particular, for each $k \in \mathbb{N}$, we can let $j_{k}$ be the first index such that

\begin{gather*} \left\lVert w_{j_{k}} - x_{i}\right\rVert+ s_{j_{k}} < \frac{1}{k}\\ \text{ and }\\ f^{\circ}\left(x_{i},v_{i}\right) - \frac{1}{k} \le h_{i}\left(w_{j_{k}},s_{j_{k}}\right). \end{gather*}

By taking $k=i$, this gives

\begin{gather*} \left\lVert w_{j_{i}} - x_{i}\right\rVert+ s_{j_{i}} < \frac{1}{i}\\ \text{ and }\\ f^{\circ}\left(x_{i},v_{i}\right) - \frac{1}{i} \le h_{i}\left(w_{j_{i}},s_{j_{i}}\right). \end{gather*}

Now all we have to do is define new sequences by $y_{i} := w_{j_{i}}$ and $t_{i} := s_{j_{i}}$. Then

\begin{gather*} \left\lVert y_{i} - x_{i}\right\rVert + t_{i} < \frac{1}{i}\\ \text{ and }\\ f^{\circ}\left(x_{i},v_{i}\right) - \frac{1}{i} \le h_{i}\left(y_{i},t_{i}\right) = \frac{f\left(y_{i}+t_{i}v_{i}\right) - f\left(y_{i}\right)} {t_{i}}. \end{gather*}

Michael Orlitzky
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