Lecture 6-3: Convex Sets: Part 3

Lecture 6-3: Convex Sets: Part 3#

Download the original slides: CMSE382-Lec6_3.pdf

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Feasible Region#

Topics Covered#

Topics:

  • Convex polytope

  • Feasible region

  • Basic feasible solutions

  • Extreme points

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 \(C\).

Definition: A set \(C \subseteq \mathbb{R}^n\) is a cone if for any \(\textbf{x} \in C\) and \(\lambda \geq 0\), we have \(\lambda \textbf{x} \in C\).

Theorem: A set \(C\) is a convex cone if and only if for any \(\textbf{x}, \textbf{y} \in C\) we have \(\textbf{x} + \textbf{y} \in C\).

Motivation: Linear Optimization (Linear Programming)#

Polytopes#

Definition: A Polytope is a geometric object with flat faces.

  • It generalizes polyhedra to higher dimensions.

    • Polyhedron is a \(3\)-polytope.

    • Polygon is a \(2\)-polytope.

Definition: A polyhedron is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices.

Convex Polytopes#

Definition: We define a convex polytope as the set

\[P = \{\textbf{x} \in \mathbb{R}^n \mid \textbf{A}\textbf{x}\boldsymbol{\leq}\textbf{b}\}, \text{ where } A \in \mathbb{R}^{m\times n}, b \in \mathbb{R}^m.\]

Feasible Region#

Definition: The feasible region of an optimization problem is the set of all possible points that satisfy the problem’s constraints.

  • It represents all the possible candidates for the optimization solution.

Example Feasible Region - 2D#

What is the feasible region for the linear system

\[\begin{split}\begin{aligned} 3y + 2x &\leq 40 \\ 4y + x & \leq 30\\ x, y &\geq 0 \end{aligned}\end{split}\]

Example Feasible Region - 3D#

What is the feasible region for the linear system

\[\begin{split}\begin{aligned} x_1+x_2+x_3 &=6 \\ x_2+x_3 &=3\\ x_1, x_2, x_3 &\geq 0 \end{aligned}\end{split}\]

Basic Feasible Solutions#

Basic Feasible Solutions#

Definition: Let

\[P = \{\textbf{x} \in \mathbb{R}^n \mid \textbf{A}\textbf{x} = \textbf{b},\, x \geq 0\},\]

where \(A \in \mathbb{R}^{m\times n}\), \(b \in \mathbb{R}^m\), and \(A\)’s rows are linearly independent.

\(\overline{\textbf{x}}\) is a basic feasible solution (bfs) of \(P\) if the columns of \(\textbf{A}\) corresponding to the indices of the positive values of \(\overline{\textbf{x}}\) are linearly independent.

  • If \(P\) is non-empty, then it contains at least one bfs.

    • This is shown by using the conic representation theorem from last class.

  • A bfs has at most \(m\) non-zero elements.

Example: Basic Feasible Solution#

\[\begin{split}\begin{aligned} x_1+x_2+x_3 &=6 \\ x_2+x_3 &=3\\ x_1, x_2, x_3 &\geq 0 \end{aligned}\end{split}\]

  • Is \((x_1, x_2, x_3) = (3,3,0)\) a solution? Yes because \(3 + 3 + 0 = 6\); \(3 + 0 = 3\); \(3,3,0 \geq 0\).

  • Is it a bfs? Yes because the columns of \(A\) corresponding to the indices of the positive values of \(\overline{\textbf{x}}\) (which are \(x_1\) and \(x_2\)) are linearly independent.

Extreme Points#

Definition: Let \(S\) be a convex set. A point \(\textbf{x} \in S\) is an extreme point of \(S\) if there do not exist two distinct points \(\textbf{x}_1, \textbf{x}_2 \in S\) and \(\lambda \in (0,1)\) such that \(\textbf{x} = \lambda\textbf{x}_1+(1-\lambda)\textbf{x}_2\).

  • It is a point in \(S\) that cannot be represented as a nontrivial convex combination of two different points in \(S\).

  • The set of all extreme points is denoted \(\text{ext}(S)\).

Equivalence of Extreme Points and BFS#

Theorem: Let \(P = \{\textbf{x} \in \mathbb{R}^n \mid \textbf{A}\textbf{x} = \textbf{b}, \textbf{x}\boldsymbol{\geq}\textbf{0}\}\), where \(\textbf{A} \in \mathbb{R}^{m \times n}\) has linearly independent rows and \(\textbf{b} \in \mathbb{R}^m\). Then \(\overline{\textbf{x}}\) is a basic feasible solution of \(P\) if and only if it is an extreme point of \(P\).

Extreme Points and Convex Hull#

Theorem: Let \(S \subseteq \mathbb{R}^n\) be a closed and bounded convex set. Then

\[S = \text{conv}(\text{ext}(S))\]

  • A compact convex set is the convex hull of its extreme points.

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