# Worksheet 7-1: Convex Functions

Download: [CMSE382-WS7_1.pdf](CMSE382-WS7_1.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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**Atom functions (assume convex unless you are proving directly from the definition):**

| | | |
|---|---|---|
| $f(t) = mt + b$, $m,b \in \mathbb{R}$ | $f(t) = t^2$ | $f(t) = e^t$ |
| | $f(t) = t^4$ | $f(t) = -\ln(t)$, $t > 0$ |

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## Worksheet 7-1: Q1

Justify each step (i)--(vi) in the following proof that $f(t) = t^2$ is convex.
Pick any $\lambda \in [0,1]$ and $t_1, t_2 \in \mathbb{R}$. Then:

$$
\begin{aligned}
f(\lambda t_1+(1-\lambda)t_2)
&= (\lambda t_1+(1-\lambda)t_2)^2 \qquad\qquad\textit{(i)}\\
&= \lambda^2 t_1^2 + 2\lambda(1-\lambda)t_1 t_2 + (1-\lambda)^2 t_2^2 \qquad\qquad\textit{(ii)}\\
&= \lambda t_1^2 - \lambda(1-\lambda)t_1^2 + 2\lambda(1-\lambda)t_1 t_2
    + (1-\lambda)t_2^2 - \lambda(1-\lambda)t_2^2 \qquad\qquad\textit{(iii)}\\
&= \lambda t_1^2+(1-\lambda)t_2^2-\lambda(1-\lambda)(t_1-t_2)^2 \qquad\qquad\textit{(iv)}\\
&\le \lambda t_1^2+(1-\lambda)t_2^2 \qquad\qquad\textit{(v)}\\
&= \lambda f(t_1)+(1-\lambda)f(t_2). \qquad\qquad\textit{(vi)}
\end{aligned}
$$

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## Worksheet 7-1: Q2

Show that each of the following functions is convex. Use the theorems from lecture to justify your answer.

(a) $f(x,y) = x^2 + 3y^2$

(b) $h(x,y) = \exp(x + y)$

(c) $f(x,y) = x^2 + 2xy + 3y^2 + 2x - 3y$

(d) $f(x,y,z) = \exp(x - y + z) + \exp(2y) + x$

(e) $f(x,y) = -\ln(xy)$, for $x > 0$, $y > 0$

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## Worksheet 7-1: Q3

Are the following functions convex? Justify your answer.

(a) $f(x,y) = xy$

(b) $\|\mathbf{x}\|$ for any norm on $\mathbb{R}^n$. Use the definition of convex function.

(c) $h(\mathbf{x}) = \exp(\|\mathbf{x}\|)$

(d) $\|\mathbf{x}\|^2$ for any norm on $\mathbb{R}^n$