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 $d eq alpha d_2$ for $ alpha>0$.

A direction $d$ of $S$ is said to be extreme direction if it cannot be written as a positive pnear 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 $f:S ightarrow$ 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 $fleft ( x ight )=y=left |x ight |$, 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 $lambda geq 0, x+lambda d in fleft ( x ight )$.