Lipschitz Conditions#

Definition: Local Lipschitz Condition

Let \(f: \mathcal{D}_f \subseteq \mathbb{R}^n \to \mathbb{R}\) be a real scalar field and let \(S \subseteq \mathcal{D}_f\).

We say that \(f\) satisfies a local Lipschitz condition on \(S\) with respect to the \(i_1, \dotsc, i_m\)-th variables if each \(\boldsymbol{p} \in S\) has some neighborhood \(N\) of \(\boldsymbol{p}\) and some \(L \in \mathbb{R}_{\ge 0}\) such that for all \(\boldsymbol{x}, \boldsymbol{y} \in N \cap S\) with \(x_k = y_k\) for all indices \(k \notin \{i_1, \dotsc, i_m\}\), we have:

\[|f(\boldsymbol{x}) - f(\boldsymbol{y})| \le L \max_{j \in \{i_1, \dotsc, i_m\}}|x_j - y_j|\]

Example: \(f(t, y) = 2t + y^2\)

Consider the following real scalar field:

\[f: \mathbb{R}^2 \to \mathbb{R} \qquad f(t, y) = 2t + y^2\]

It satisfies a local Lipschitz condition on \(\mathbb{R}^2\) because for each \((t, y) \in \mathbb{R}^2\), if we choose some \(R \gt 0\), then for each \((t, y_1)\) and \((t, y_2)\) with \(|y_1| \lt R\) and \(|y_2| \lt R\), we get

\[\begin{aligned}|f(t, y_1) - f(t, y_2)| & = |2t + y_1^2 - 2t - y_2^2| \\ & = |y_1^2 - y_2^2| \\ & = |y_1 - y_2|\cdot|y_1 + y_2| \\ & \le |y_1 - y_2|(|y_1| + |y_2|) \\ & \le 2R |y_1 - y_2| \le L |y_1 - y_2|\end{aligned}\]

with \(L = 2R\).

Example: \(f(t, y) = \sqrt{|y|}\)

Consider the following real scalar field:

\[f: \mathbb{R}^2 \to \mathbb{R} \qquad f(t, y) = \sqrt{|y|}\]

It cannot satisfy local Lipschitz condition around any neighborhood of \((t, y)\) whenever \(y = 0\). Specifically:

\[|f(t, y_1) - f(t, y_2)| = |\sqrt{y_1} - \sqrt{y_2}| = \frac{|y_1 - y_2|}{\sqrt{y_1} + \sqrt{y_2}}\]

For \(y_1, y_2 \to 0\), this approaches \(\infty\) and so no such \(L\) can exist.

Definition: Global Lipschitz Condition

Let \(f: \mathcal{D}_f \subseteq \mathbb{R}^n \to \mathbb{R}\) be a real scalar field and let \(S \subseteq \mathcal{D}_f\).

We say that \(f\) satisfies a global Lipschitz condition on \(S\) with respect to the \(i_1, \dotsc, i_m\)-th variables if there exists some \(L \in \mathbb{R}_{\ge 0}\) such that for all \(\boldsymbol{x}, \boldsymbol{y} \in S\) with \(x_k = y_k\) for all indices \(k \notin \{i_1, \dotsc, i_m\}\), we have:

\[|f(\boldsymbol{x}) - f(\boldsymbol{y})| \le L \max_{j \in \{i_1, \dotsc, i_m\}}|x_j - y_j|\]

Example: \(f(t, y) = t + 2y\)

Consider the following real scalar field:

\[f: \mathbb{R}^2 \to \mathbb{R} \qquad f(t, y) = t + 2y\]

It satisfies a global Lipschitz conditionon \(\mathbb{R}^2\) with \(L = 2\):

\[|f(t, y_1) - f(t, y_2)| = |t + 2y_1 - t - 2y_2| = 2|y_1 - y_2|\]

Theorem: Continuous Partial Differentiability \(\implies\) Local Lipschitz Condition

Let \(f: \mathcal{D}_f \subseteq \mathbb{R}^n \to \mathbb{R}\) be a real scalar field and let \(S \subseteq \mathcal{D}_f\) be locally convex.

If \(f\) is continuously partially differentiable on \(S\) w.r.t. the \(i_1, \dotsc, i_m\)-th variables, then \(f\) satisfies a local Lipschitz condition on \(S\) w.r.t. the \(i_1, \dotsc, i_m\)-th variables.

Proof

TODO