# Lecture 7-3: Convex Functions: Part 3

Download the original slides: [CMSE382-Lec7_3.pdf](CMSE382-Lec7_3.pdf)

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---

## This Lecture

**Topics:**

- Continuity and differentiability of convex functions
- Extended real-valued functions
- Maxima of a convex function

**Announcements:**

- Homework 3 due Friday!
- The homework uses CVXPY. If you missed last class, make sure you get your CVXPY installation working ASAP!

---

## Continuity and differentiability of convex functions

### Are convex functions always continuous?

*Nope.*

**Definition:**
A function $f: C \to \mathbb{R}$ defined over a convex set $C \subseteq \mathbb{R}^n$ is *convex* if for any $\mathbf{x}, \mathbf{y} \in C$ and $\lambda \in [0,1]$, we have
$$
f(\lambda \mathbf{x} + (1-\lambda) \mathbf{y}) \leq \lambda f(\mathbf{x}) + (1-\lambda) f(\mathbf{y}).
$$

![](../../../figures/Beck_Fig7_6.png)

---

### Lipschitz continuity

**Definition (Lipschitz Continuity):**
A function $f\colon S \to \mathbb{R}$ where $S \subseteq \mathbb{R}^n$ is *Lipschitz continuous* if there exists an $L > 0$ such that
$$
\|f(\mathbf{x}) - f(\mathbf{y})\| \leq L\|\mathbf{x}-\mathbf{y}\|, \quad \forall \mathbf{x}, \mathbf{y} \in S.
$$

- Note that $f$ isn't assumed to be continuous.
- Lipschitz continuity implies continuity, but the converse is not necessarily true.

---

### Local Lipschitz Continuity

**Definition (Local Lipschitz Continuity):**
A function $f\colon S \to \mathbb{R}$ where $S \subseteq \mathbb{R}^n$ is *locally Lipschitz continuous* if for every $\mathbf{x}_0 \in S$ there exist $\varepsilon > 0$ and $L > 0$ such that $B(\mathbf{x}_0,\varepsilon) \subseteq S$ and
$$
|f(\mathbf{x}) - f(\mathbf{x}_0)| \le L\|\mathbf{x} - \mathbf{x}_0\|, \quad \forall \mathbf{x} \in B(\mathbf{x}_0,\varepsilon).
$$

---

### Continuity of convex functions

**Are convex functions always continuous?**

**Theorem (Local Lipschitz continuity of convex functions):**
Let $f \colon C \to \mathbb{R}$ be a convex function defined over a convex set $C \subseteq \mathbb{R}^n$. Then $f$ is *locally Lipschitz continuous* at every $\mathbf{x}_0 \in \text{int}(C)$.

- Convex functions on $C \subseteq \mathbb{R}^n$ are continuous on $\text{int}(C)$.
- Convex functions on $\mathbb{R}^n$ are continous on $\mathbb{R}^n$.

![](../../../figures/Beck_Fig7_6.png)

---

### Directional derivatives for convex functions

**Definition (Recall):**
The **directional derivative** of $f:\mathbb{R}^n\to \mathbb{R}$ at $\mathbf{x}$ along the direction $\mathbf{d}$ is defined as
$$
f'(\mathbf{x};\mathbf{d})=\nabla f(\mathbf{x})^\top\mathbf{d}.
$$

**Theorem (Existence of directional derivatives for convex functions):**
Let $f: C \to \mathbb{R}$ be a convex function defined over the convex set $C\subseteq \mathbb{R}^n$.
Let $\mathbf{x} \in \text{int}(C)$. Then for any $\mathbf{d} \neq \mathbf{0}$, the directional derivative $f'(\mathbf{x};\mathbf{d})$ exists.

---

## Extended real-valued functions

### Extended real-valued functions

**Definition (Extended real-valued functions):**

- Real-valued functions take their values in $\mathbb{R} = (-\infty, \infty)$.
- Extended real-valued functions take their values in $\mathbb{R} \cup \{\infty\} = (-\infty, \infty]$

- **Warning:** Some sources define the extended real numbers as $\mathbb{R} \cup \{\pm\infty\} = [-\infty, \infty]$.

**Example: Indicator function**
$$
\delta_S(\mathbf{x}) = \begin{cases}
0, & \text{ if } \mathbf{x} \in S\\
\infty, & \text{ if } \mathbf{x} \notin S
\end{cases}
$$

**Arithmetic with $\infty$:**

- $a + \infty = \infty$ for any $a\in \mathbb{R}$.
- $a \cdot \infty = \infty$ for any $a \in \mathbb{R}_{++}$.
- $0 \cdot \infty = 0$.

![](../../../figures/indicator_function.png)

---

### Effective domain of extended real-valued functions

**Definition:**
The **effective domain of an extended real-valued function** is the set of vectors for which the function takes a real value:
$$
 \text{dom}(f ) = \{\mathbf{x} \in \mathbb{R}^n \mid f(\mathbf{x}) < \infty \}.
$$

$$
\delta_S(x) = \begin{cases}
0, & \text{ if } \mathbf{x} \in S;\\
\infty, & \text{ if } \mathbf{x} \notin S.
\end{cases}
$$

$$
f(x) = \begin{cases}
\infty, & \text{ if } \mathbf{x} \in S;\\
0, & \text{ if } \mathbf{x} \notin S.
\end{cases}
$$

$$
f(x) = \begin{cases}
\infty, & \text{ if } x = 0;\\
x^2, & \text{ if } x \in (0,1].
\end{cases}
$$

![](../../../figures/extended_function_convex_nonconvex_example1.png)
![](../../../figures/extended_function_convex_nonconvex_example2.png)
![](../../../figures/extended_function_convex_nonconvex_example3.png)

---

### Convexity of extended real-valued functions

**Theorem:**
An extended real-valued function $f$ is convex if and only if $\text{dom}(f)$ is a convex set and $f$ is convex over $\text{dom}(f)$.

$$
\delta_S(x) = \begin{cases}
0, & \text{ if } \mathbf{x} \in S;\\
\infty, & \text{ if } \mathbf{x} \notin S.
\end{cases}
$$

$$
f(x) = \begin{cases}
\infty, & \text{ if } \mathbf{x} \in S;\\
0, & \text{ if } \mathbf{x} \notin S.
\end{cases}
$$

$$
f(x) = \begin{cases}
\infty, & \text{ if } x = 0;\\
x^2, & \text{ if } x \in (0,1].
\end{cases}
$$

![](../../../figures/extended_function_convex_nonconvex_example1.png)
![](../../../figures/extended_function_convex_nonconvex_example2.png)
![](../../../figures/extended_function_convex_nonconvex_example3.png)

---

### Epigraph of a function

**Definition (Epigraph of a function):**
Let $f:\mathbb{R}^n \to \mathbb{R} \cup \{\infty\}$. The epigraph set $\text{epi}(f ) \subseteq \mathbb{R}^{n+1}$ is defined by
$$
 ext{epi}(f)=\bigg\{\begin{pmatrix}\mathbf{x}\\t \end{pmatrix}:f(\mathbf{x}) \leq t \bigg\}.
$$

- It contains all the points $\begin{pmatrix}\mathbf{x}\\t \end{pmatrix}$ on or above the function graph.

**Theorem (Function and epigraph convexity):**
An extended real-valued (or a real-valued) function $f$ is convex if and only if its epigraph set $\text{epi}(f)$ is convex.

![](../../../figures/epigraph.png)

---

### Preservation of convexity under maximum for extended real-valued functions

**Theorem:**
Let $f_i:\mathbb{R}^n \to \mathbb{R} \cup \{\infty\}$ be an extended real-valued convex function for any $i \in I$ (I being an arbitrary index set).
Then the function
$$
f(\mathbf{x}) = \max_{i\in I} f_i(\mathbf{x})
$$
is an extended real-valued convex function.

![](../../../figures/extended_function_pointwise_max_convexity.png)

---

## Maxima of a convex function

### Recall: Extreme points

**Definition (Recall: Extreme points):**
Let $S$ be a convex set. A point $\textbf{x} \in S$ is an **extreme point** of $S$ if there do not exist two distinct points $\textbf{x}_1, \textbf{x}_2 \in S$ and $\lambda \in (0,1)$ such that $\textbf{x} = \lambda\textbf{x}_1+(1-\lambda)\textbf{x}_2$.

- It is a point in $S$ that cannot be represented as a nontrivial convex combination of two different points in $S$.
- The set of all extreme points is denoted $\mbox{ext}(S)$.

![](../../../figures/extreme_points_example.png)

---

### Maxima of convex functions

**Theorem:**
Let $f: C \to \mathbb{R}$ be a convex function which is not constant over the convex set $C$.
Then $f$ does not attain a maximum at a point in $\text{int}(C)$.

- Maximum of non-constant convex function defined on a convex set cannot occur at an interior point in the set.

![](../../../figures/convex_function_maxima.png)

---

### Maxima of convex functions

**Theorem:**
Let $f: C \to \mathbb{R}$ be a convex and continuous function over the convex and compact set $C \subseteq \mathbb{R}^n$. Then there exists at least one maximizer of $f$ over $C$ that is an extreme point of $C$.

![](../../../figures/maximizer_at_extreme_points.png)

---

### Groups - Round 3

**Group 1**
Lowell, Tianjuan, Lauryn, Atticus

**Group 2**
Alice, Aidan, Dev, Anthony

**Group 3**
Abigail, Michal, Breena, Andrew

**Group 4**
Kyle, Vinod, Dori, Joseph

**Group 5**
Yousif, Jamie, Jay, K.M Tausif

**Group 6**
Shanze, Saitej, Karen, Jack

**Group 7**
Arjun, Noah, Luis, Arya

**Group 8**
Morgan, Jonid, Sanskaar, Jake

**Group 9**
Quang Minh, Monirul Amin, Daniel, Ha

**Group 10**
Braedon, Dominic, Zheng, Lora

**Group 11**
Sai, Brandon, Purvi, Aaron

**Group 12**
Igor, Scott, Maye, Long