Lecture 8-3: Convex Optimization: Part 3#
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This Lecture#
Topics:
Convex optimization with CVXPY
Chebyshev center for a set of points
Announcements:
Homework 4 due Friday.
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
\[\begin{split}
\begin{aligned}
\min_{\mathbf{x},r} \quad & r \\
\text{s.t.} \quad & \|\mathbf{x}-\mathbf{a}_i\| \leq r, \quad i=1,2,\ldots,m.
\end{aligned}
\end{split}\]
