- Convex Optimization Tutorial
- Home
- Introduction
- Linear Programming
- Norm
- Inner Product
- Minima and Maxima
- Convex Set
- Affine Set
- Convex Hull
- Caratheodory Theorem
- Weierstrass Theorem
- Closest Point Theorem
- Fundamental Separation Theorem
- Convex Cones
- Polar Cone
- Conic Combination
- Polyhedral Set
- Extreme point of a convex set
- Direction
- Convex & Concave Function
- Jensen's Inequality
- Differentiable Convex Function
- Sufficient & Necessary Conditions for Global Optima
- Quasiconvex & Quasiconcave functions
- Differentiable Quasiconvex Function
- Strictly Quasiconvex Function
- Strongly Quasiconvex Function
- Pseudoconvex Function
- Convex Programming Problem
- Fritz-John Conditions
- Karush-Kuhn-Tucker Optimality Necessary Conditions
- Algorithms for Convex Problems
- Convex Optimization Resources
- Convex Optimization - Quick Guide
- Convex Optimization - Resources
- Convex Optimization - Discussion
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.
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