Convex Optimization Tutorial
Convex Optimization Resources
Selected Reading
- Algorithms for Convex Problems
- Karush-Kuhn-Tucker Optimality Necessary Conditions
- Fritz-John Conditions
- Convex Programming Problem
- Pseudoconvex Function
- Strongly Quasiconvex Function
- Strictly Quasiconvex Function
- Differentiable Quasiconvex Function
- Quasiconvex & Quasiconcave functions
- Sufficient & Necessary Conditions for Global Optima
- Differentiable Convex Function
- Convex & Concave Function
- Direction
- Extreme point of a convex set
- Polyhedral Set
- Conic Combination
- Polar Cone
- Convex Cones
- Fundamental Separation Theorem
- Closest Point Theorem
- Weierstrass Theorem
- Caratheodory Theorem
- Convex Hull
- Affine Set
- Convex Set
- Minima and Maxima
- Inner Product
- Norm
- Linear Programming
- Introduction
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Convex Optimization Resources
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- HR Interview Questions
- Effective Resume Writing
- Questions and Answers
- UPSC IAS Exams Notes
Polyhedral Set
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.,
$S=left { x in mathbb{R}^n:p_{i}^{T}xleq alpha_i, i=1,2,....,n ight }$
For example,
$left { x in mathbb{R}^n:AX=b ight }$
$left { x in mathbb{R}^n:AXleq b ight }$
$left { x in mathbb{R}^n:AXgeq b ight }$
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., $S=left { x in mathbb{R}^n:p_{i}^{T}xleq 0, i=1, 2,... ight }$
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.