Convex Optimization - Polyhedral Set

A set in $\mathbb{R}^n$ is said to be polyhedral if it is the intersection of a finite number of closed half spaces, i.e.,

$S=\left \{ x \in \mathbb{R}^n:p_{i}^{T}x\leq \alpha_i, i=1,2,....,n \right \}$

For example,

• $\left \{ x \in \mathbb{R}^n:AX=b \right \}$

• $\left \{ x \in \mathbb{R}^n:AX\leq b \right \}$

• $\left \{ x \in \mathbb{R}^n:AX\geq b \right \}$

Polyhedral Cone

A set in $\mathbb{R}^n$ is said to be polyhedral cone if it is the intersection of a finite number of half spaces that contain the origin, i.e., $S=\left \{ x \in \mathbb{R}^n:p_{i}^{T}x\leq 0, i=1, 2,... \right \}$

Polytope

A polytope is a polyhedral set which is bounded.

Remarks

• A polytope is a convex hull of a finite set of points.
• A polyhedral cone is generated by a finite set of vectors.
• A polyhedral set is a closed set.
• A polyhedral set is a convex set.