Convex Optimization - Cones

A non empty set C in $mathbb{R}^n$ is said to be cone with vertex 0 if $x in CRightarrow lambda x in C forall lambda geq 0$.

A set C is a convex cone if it convex as well as cone.

For example, $y=left | x ight |$ is not a convex cone because it is not convex.

But, $y geq left | x ight |$ 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 $lambda x + left ( 1-lambda ight )y in C : forall :lambda in left ( 0, 1 ight )$

Since C is cone, $lambda x in C$ and $left ( 1-lambda ight )y in C Leftrightarrow x,y in C$

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 pnear function is a convex cone.

Since a hyperplane is pnear, 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.