# Lecture 8-3: Convex Optimization: Part 3

Download the original slides: [CMSE382-Lec8_3.pdf](CMSE382-Lec8_3.pdf)

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## This Lecture

**Topics:**

- Convex optimization with CVXPY
- Chebyshev center for a set of points

**Announcements:**

- Homework 4 due Friday.

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## Convex Optimization Example: Chebyshev Center of a set of points

### Chebyshev Center of a Set of Points

Given $m$ points $\mathbf{a}_1,\mathbf{a}_2, \ldots, \mathbf{a}_m$ in $\mathbb{R}^n$, find the center of the minimum-radius closed ball containing all the points.

$$
\min_{\mathbf{x},r} \quad r
$$

subject to

$$
\mathbf{a}_i \in B[\mathbf{x},r], \quad i=1,2,\ldots,m.
$$

Recall:

$$
B[\mathbf{x},r]=\{\mathbf{y}: \|\mathbf{y}-\mathbf{x}\| \leq r\}.
$$

So the problem can be rewritten as

$$
\begin{aligned}
\min_{\mathbf{x},r} \quad & r \\
\text{s.t.} \quad & \|\mathbf{x}-\mathbf{a}_i\| \leq r, \quad i=1,2,\ldots,m.
\end{aligned}
$$

![The Chebyshev center of a distribution of points](../../../figures/Chebyshev_center.png)