Lecture 6-2: Convex Sets: Part 2

Lecture 6-2: Convex Sets: Part 2#

Download the original slides: CMSE382-Lec6_2.pdf

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

Cones#

Topics Covered#

Topics:

  • Convex cones

  • Conic combinations

Review: Last Time#

Definition: A set \(C \subseteq \mathbb{R}^n\) is convex if for any \(\textbf{x}, \textbf{y} \in C\), the line segment \([\textbf{x}, \textbf{y}]\) is also in \(C\).

\[[\textbf{x}, \textbf{y}] = \{\lambda\textbf{x}+(1-\lambda)\textbf{y}:0\leq\lambda\leq 1\}\]

Definition: The convex hull of a set \(S \subseteq \mathbb{R}^n\) is the smallest convex set containing \(S\).

Definition: A convex combination of points \(\textbf{x}_1, \textbf{x}_2, \ldots, \textbf{x}_k\) is a point of the form \(\lambda_1\textbf{x}_1 + \lambda_2\textbf{x}_2 + \ldots + \lambda_k\textbf{x}_k\), where \(\lambda_i \geq 0\) and \(\sum_{i=1}^k \lambda_i = 1\).

Cones#

Definition: A set \(S\) is a cone if for any \(\textbf{x} \in S\) and \(\lambda \geq 0\), \(\lambda\textbf{x} \in S\).

  • Union of cones is a cone.

  • Intersection of cones is a cone.

  • (Remember that the union of convex sets is not necessarily convex, but the intersection of convex sets is convex)

Convex Cones#

Theorem: A cone \(S\) is convex if and only if \(\textbf{x}, \textbf{y} \in S \implies \textbf{x}+\textbf{y} \in S\).

Conic Combinations#

Conic Combinations#

Definition: Given \(k\) points \(\textbf{x}_1, \textbf{x}_2, \ldots, \textbf{x}_k \in \mathbb{R}^n\), a conic combination of these \(k\) points is a vector of the form \(\lambda_1\textbf{x}_1+\lambda_2\textbf{x}_2+\ldots+\lambda_k\textbf{x}_k\), where \(\lambda_i \geq 0\).

  • This is different than a convex combination because the \(\lambda_i\) do not need to sum to 1.

Conic Combination Examples#

Conic Hull#

Definition: Let \(S \subseteq \mathbb{R}^n\). Then the conic hull of \(S\), denoted by \(\mathrm{cone}(S)\), is the set of all conic combinations of vectors from \(S\):

\[\mathrm{cone}(S)= \left\{\sum\limits_{i=1}^k \lambda_i\textbf{x}_i \mid \textbf{x}_1, \textbf{x}_2, \ldots, \textbf{x}_k \in S, \lambda_i \geq 0, k \in \mathbb{N}\right\}\]

  • The conic hull of a set \(S\) is the smallest convex cone containing \(S\).

  • For \(S \subset \mathbb{R}^n\), \(\mathrm{conv}(S) \subseteq \mathrm{cone}(S)\).

Conic Hull Examples#

Conic Representation Theorem#

Theorem: Let \(S \subseteq \mathbb{R}^n\) and let \(\textbf{x} \in \mathrm{cone}(S)\). Then there exist \(k\) linearly independent vectors \(\textbf{x}_1, \textbf{x}_2, \ldots, \textbf{x}_k \in S\) such that \(\textbf{x} \in \mathrm{cone}(\{\textbf{x}_1, \textbf{x}_2, \ldots, \textbf{x}_k\})\); that is, there exists \(\lambda_1, \ldots, \lambda_k \in \mathbb{R}\) such that

\[\textbf{x}=\sum\limits_{i=1}^k \lambda_i\textbf{x}_i\]

In addition, \(k \leq n\).

  • This says that each vector in the conic hull of a set \(S \subseteq \mathbb{R}^n\) can be represented as a convex combination of at most \(n\) vectors from \(S\).

Conic Representation Theorem Examples#

  • If \(\mathbf{x} \in \mathrm{cone}(S)\), then \(\mathbf{x}\) is the nonnegative sum of at most \(n\) points of \(S\).

  • If \(\mathbf{x}\in \mathrm{conv}(S)\), then \(\mathbf{x}\) is the convex sum of at most \(n+1\) points of \(S\).

Conic Combination Versus Convex Combination#

For \(S \subset \mathbb{R}^n\):

Conic

Convex

Set definition

cone: \(\forall \mathbf{x}\in S, \lambda\geq 0 \Rightarrow \lambda\mathbf{x}\in S\)

convex: \(\forall \mathbf{x},\mathbf{y}\in S\), segment \([\mathbf{x},\mathbf{y}]\subset S\)

Combination

\(\lambda_1 \mathbf{x}_1 +\ldots+ \lambda_n \mathbf{x}_n\), \(\lambda_i \geq 0\)

\(\lambda_1 \mathbf{x}_1 +\ldots+ \lambda_n \mathbf{x}_n\), \(\lambda_i \geq 0\) and \(\sum_{i=1}^n \lambda_i=1\)

Representation

at most \(n\) points

at most \(n+1\) points

Hull

smallest convex cone containing \(S\)

smallest convex set containing \(S\)
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