# Worksheet 7-3: Convex Functions

Download: [CMSE382-WS7_3.pdf](CMSE382-WS7_3.pdf)

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This is an AI-generated transcript of the worksheet and may contain errors or inaccuracies. Please refer to the original course materials for authoritative content.
```

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## Worksheet 7-3: Q1

Consider $\max{\{\mathbf{x}^{\top} Q \mathbf{x}\mid \|\mathbf{x}\| \leq 1}\}$ where $\mathbf{x} \in \mathbb{R}^n$ and $Q \succeq 0$.

(a) Does a maximizer exist? Justify by checking all the conditions of the relevant theorem.

(b) Let $\mathbf{x} \in \mathbb{R}^2$ (that is, assume $n=2$ above) and answer the following questions.

(i) Using $L_1$-norm for $\mathbf{x}$: Sketch the feasible region and point out where a maximizer, if it exists, can be found.

(ii) Using $L_2$-norm for $\mathbf{x}$: Sketch the feasible region and point out where a maximizer, if it exists, can be found.

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## Worksheet 7-3: Q2

Consider the function

$$
\begin{array}{rccc}
f: & [0,\infty) & \rightarrow & \mathbb{R} \cup \{\infty\} \\
& x & \mapsto &
\begin{cases}
\infty & \text{if } x = 0 \\
1/x & \text{if } x > 0
\end{cases}
\end{array}
$$

- What is the effective domain of $f$, $\text{dom}(f)$?

- Sketch the epigraph of $f$.

- Is $f$ convex? Justify your answer.

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## Worksheet 7-3: Q3

![Epigraph problem](../../../figures/epigraph_problem.png)

For each of the functions shown in the figure, answer the following:

(i) Sketch the sublevel sets at level $\alpha$.

(ii) Sketch the epigraph.

(iii) **Based only on your sketches**, is the function convex? Justify your answer.

(iv) **Based only on your sketches**, is the function quasi-convex?