Lecture 6-1: Convex Sets: Part 1

Lecture 6-1: Convex Sets: Part 1#

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Convex Set Definitions#

Topics Covered#

Topics:

  • Definition of convex sets

  • Algebraic operations on Convex Sets

  • Convex Hull

Convex Sets#

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 \(C\).

Examples: Lines#

A line in \(\mathbb{R}^n\) is a set of points in the form

\[L=\{\mathbf{z}+t\mathbf{d}:t\in\mathbb{R}\},\]

where \(\mathbf{z},\mathbf{d}\in\mathbb{R}^n\) and \(\mathbf{d}\neq 0\). The point \(\mathbf{z}\) is on the line, and the line is parallel to the direction \(\mathbf{d}\).

Closed and open line segments between two different points \(\mathbf{x}, \mathbf{y}\in\mathbb{R}^n\):

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

  • \((\mathbf{x},\mathbf{y}):=\{\lambda\mathbf{x}+(1-\lambda)\mathbf{y}:0<\lambda< 1\}\)

  • \([\mathbf{x},\mathbf{y}):=\{\lambda\mathbf{x}+(1-\lambda)\mathbf{y}:0<\lambda\leq 1\}\)

More Examples of Convex Sets#

  • empty set \(\emptyset\) and the whole space \(\mathbb{R}^n\).

  • hyperplane: \(\{\mathbf{x}\in\mathbb{R}^n:\mathbf{a}^\top\mathbf{x}=b\}\) for \(\mathbf{0}\neq\mathbf{a}\in\mathbb{R}^n\) and \(b\in\mathbb{R}\).

  • closed and open half-spaces: Given \(\mathbf{0}\neq\mathbf{a}\in\mathbb{R}^n\) and \(b\in\mathbb{R}\)

    • \(\{\mathbf{x}\in\mathbb{R}^n:\mathbf{a}^\top\mathbf{x}\leq b\}\)

    • \(\{\mathbf{x}\in\mathbb{R}^n:\mathbf{a}^\top\mathbf{x}<b\}\)

  • balls: Given \(\mathbf{c}\in\mathbb{R}^n\) and \(r>0\)

    • \(B(\mathbf{c}, r) := \{\mathbf{x}\in\mathbb{R}^n:\|\mathbf{x}-\mathbf{c}\|<r\}\)

    • \(B[\mathbf{c}, r] := \{\mathbf{x}\in\mathbb{R}^n:\|\mathbf{x}-\mathbf{c}\|\leq r\}\)

  • ellipsoid: \(\{\mathbf{x}\in\mathbb{R}^n:(\mathbf{x}-\mathbf{b})^\top\mathbf{Q}(\mathbf{x}-\mathbf{b})\leq c\}\) with \(\mathbf{Q}\in\mathbb{R}^{n\times n}\) being positive semidefinite, \(\mathbf{b}\in\mathbb{R}^n\), and \(c>0\).

Algebraic Operations on Convex Sets#

Intersection of Convex Sets#

Property: The intersection of any convex sets is convex.

Example: Convex polytopes \(\{\mathbf{x}\in\mathbb{R}^n:\mathbf{A}_i{x}\leq\mathbf{b}_i\}\) with \(\mathbf{A}\in\mathbb{R}^{m\times n}\) and \(\mathbf{b}\in\mathbb{R}^m\).

Linear Combinations#

Property: The linear combination of a finite number of convex sets is convex.

  • Let \(C_1, C_2, \ldots, C_k \subseteq \mathbb{R}^n\) be convex sets

  • Let \(\mu_1, \mu_2, \ldots, \mu_k \in \mathbb{R}\). Then the set

\[\mu_1C_1+\mu_2C_2+\ldots+\mu_kC_k = \left\{\sum\limits_{i=1}^k \mu_i\textbf{x}_i \mid \textbf{x}_i \in C_i, i=1, 2, \ldots, k\right\}\]

is convex.

Products#

Property: The Cartesian product of a finite number of convex sets is convex.

  • Let \(C_i \subseteq \mathbb{R}^{k_i}\) be a convex set for any \(i = 1, 2, \ldots, m\).

  • Then

\[C_1\times C_2 \times \ldots \times C_m = \{(\textbf{x}_1, \textbf{x}_2, \ldots, \textbf{x}_m) \mid \textbf{x}_i \in C_i, i=1, 2, \ldots, m\}\]

is convex.

Linear Transforms#

Property: The image of a convex set under a linear transformation is convex.

Let \(M\subseteq \mathbb{R}^n\) be a convex set and let \(\textbf{A} \in \mathbb{R}^{m \times n}\). Then the set

\[\textbf{A}(M) = \{\textbf{A}\textbf{x} \mid \textbf{x} \in M\}\]

is convex.

Property: If a convex set is a linear transform of another set, then that set is convex.

Let \(D \subseteq \mathbb{R}^m\) be a convex set, and let \(\textbf{A} \in \mathbb{R}^{m \times n}\). Then the set

\[\textbf{A}^{-1}(D) = \{\textbf{x} \in \mathbb{R}^n \mid \textbf{A}\textbf{x}\in D\}\]

is convex.

Nonexample#

Unions of convex sets are not necessarily convex.

Topological Properties of Convex Sets#

Recall: Interior Points#

Definition: Given a set \(U \subset \mathbb{R}^n\), a point \(\mathbf{c}\in U\) is an interior point of \(U\) if there exists \(r >0\) such that \(B(\mathbf{c}, r) \subset U\). The set of all interior points is called the interior and defined as

\[\text{int}(U)=\{\mathbf{x}\in U:B(\mathbf{c},r)\subset U\text{ for some }r>0\}.\]

Examples:

  • \(\text{int}(\mathbb{R}_+^n)=\mathbb{R}_{++}^n\)

  • \(\text{int}(B[\mathbf{c},r]) = B(\mathbf{c},r)\)

  • \(\text{int}([\mathbf{x},\mathbf{y}])=(\mathbf{x},\mathbf{y})\) if \(\mathbf{x}\neq\mathbf{y}\)

  • \(\text{int}([\mathbf{x},\mathbf{x}]) = \emptyset\)

Examples of Convex Sets with Empty Interior#

Topological Properties of Convex Sets#

Theorem: The closure of a convex set is convex.

Theorem: The interior of a convex set is convex.

Theorem: Let \(C\) be a convex set with a nonempty interior. Then:

  1. \(\text{cl}(\text{int}(C)) = \text{cl}(C)\)

  2. \(\text{int}(\text{cl}(C)) = \text{int}(C)\)

Theorem: The convex hull of a finite set is closed.

Convex Hulls#

Convex Combinations#

Definition: Given \(k\) points \(\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_k \in\mathbb{R}^n\), a convex combination of these \(k\) points is a point of the form

\[\lambda_1\mathbf{x}_1 +\lambda_2\mathbf{x}_2 +\cdots +\lambda_k \mathbf{x}_k\]

where \(\lambda_1,\lambda_2,\dots,\lambda_k\) are nonnegative numbers satisfying \(\lambda_1 +\lambda_2 +\dots +\lambda_k = 1\). We denote

\[\Delta_m=\{(\lambda_1,\lambda_2,\cdots,\lambda_m)\mid \lambda_i\geq 0,~\forall i,~\sum_{i=1}^m\lambda_i=1\}.\]

A convex set is defined by the property that any convex combination of two points from the set is also in the set.

Convex Combinations Theorem#

Theorem: Let \(C\subset \mathbb{R}^n\) be a convex set and \(\mathbf{x}_1,\mathbf{x}_2,\cdots,\mathbf{x}_m\in C\). Then for any \(\lambda\in\Delta_m\), we have \(\sum_{i=1}^m\lambda_i\mathbf{x}_i\in C\).

  • A convex combination of any number of points from a convex set is in the set.

Convex Hulls#

Let \(S\subset \mathbb{R}^n\). The convex hull of \(S\) is defined as

\[\text{conv}(S):=\left\{\sum_{i=1}^k\lambda_i\mathbf{x}_i\mid\mathbf{x}_1,\mathbf{x}_2,\cdots,\mathbf{x}_k\in S,\lambda\in\Delta_k, k\in \mathbb{N}\right\}\]

Lemma: Let \(S \subseteq \mathbb{R}^n\). If \(S \subseteq T\) for some convex set \(T\), then \(\text{conv}(S) \subseteq T\).

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

Theorem: If \(x\in \mathrm{Conv}(S)\subset \mathbb{R}^{d}\), then \(x\) is the convex sum of at most \(d+1\) points of \(S\).

Example: Convex Hull#

Let \(S\) contain four points:

\[\begin{split}\mathbf{x}_1=\begin{bmatrix}1\\1\end{bmatrix},\quad\mathbf{x}_2=\begin{bmatrix}1\\2\end{bmatrix},\quad\mathbf{x}_3=\begin{bmatrix}2\\1\end{bmatrix},\quad\mathbf{x}_4=\begin{bmatrix}2\\2\end{bmatrix}.\end{split}\]
\[\text{conv}(S)=\{(x_1,x_2):1\leq x_1\leq 2; 1\leq x_2\leq 2\}.\]

Given \(\mathbf{x}=\begin{bmatrix}1.5\\1.4\end{bmatrix}\), we can find three points from the four points such that \(\mathbf{x}\) is a linear combination of these three points.

  • \(\mathbf{x}\) is a linear combination of \(\mathbf{x}_1\), \(\mathbf{x}_2\), and \(\mathbf{x}_3\): \(\mathbf{x}=0.1\mathbf{x}_1+0.4\mathbf{x}_2+0.5\mathbf{x}_3\)

Convex Hull Examples#

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