Convex Optimization - Direction

Let S be a closed convex set in $\mathbb{R}^n$. A non zero vector $d \in \mathbb{R}^n$ is called a direction of S if for each $x \in S,x+\lambda d \in S, \forall \lambda \geq 0.$

• Two directions $d_1$ and $d_2$ of S are called distinct if $d \neq \alpha d_2$ for $ \alpha>0$.

• A direction $d$ of $S$ is said to be extreme direction if it cannot be written as a positive linear combination of two distinct directions, i.e., if $d=\lambda _1d_1+\lambda _2d_2$ for $\lambda _1, \lambda _2>0$, then $d_1= \alpha d_2$ for some $\alpha$.

• Any other direction can be expressed as a positive combination of extreme directions.

• For a convex set $S$, the direction d such that $x+\lambda d \in S$ for some $x \in S$ and all $\lambda \geq0$ is called recessive for $S$.

• Let E be the set of the points where a certain function $f:S \rightarrow$ over a non-empty convex set S in $\mathbb{R}^n$ attains its maximum, then $E$ is called exposed face of $S$. The directions of exposed faces are called exposed directions.

• A ray whose direction is an extreme direction is called an extreme ray.

Example

Consider the function $f\left ( x \right )=y=\left |x \right |$, where $x \in \mathbb{R}^n$. Let d be unit vector in $\mathbb{R}^n$

Then, d is the direction for the function f because for any $\lambda \geq 0, x+\lambda d \in f\left ( x \right )$.