- Convex Optimization Tutorial
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- Convex Cones
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- Jensen's Inequality
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- 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
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- Convex Optimization - Discussion
Convex Optimization - Cones
A non empty set C in $\mathbb{R}^n$ is said to be cone with vertex 0 if $x \in C\Rightarrow \lambda x \in C \forall \lambda \geq 0$.
A set C is a convex cone if it convex as well as cone.
For example, $y=\left | x \right |$ is not a convex cone because it is not convex.
But, $y \geq \left | x \right |$ is a convex cone because it is convex as well as cone.
Note − A cone C is convex if and only if for any $x,y \in C, x+y \in C$.
Proof
Since C is cone, for $x,y \in C \Rightarrow \lambda x \in C$ and $\mu y \in C \:\forall \:\lambda, \mu \geq 0$
C is convex if $\lambda x + \left ( 1-\lambda \right )y \in C \: \forall \:\lambda \in \left ( 0, 1 \right )$
Since C is cone, $\lambda x \in C$ and $\left ( 1-\lambda \right )y \in C \Leftrightarrow x,y \in C$
Thus C is convex if $x+y \in C$
In general, if $x_1,x_2 \in C$, then, $\lambda_1x_1+\lambda_2x_2 \in C, \forall \lambda_1,\lambda_2 \geq 0$
Examples
The conic combination of infinite set of vectors in $\mathbb{R}^n$ is a convex cone.
Any empty set is a convex cone.
Any linear function is a convex cone.
Since a hyperplane is linear, it is also a convex cone.
Closed half spaces are also convex cones.
Note − The intersection of two convex cones is a convex cone but their union may or may not be a convex cone.