Linear Optimization Solutions
Why
Do solutions exist to a linear optimization problem which is feasible and bounded? Yes.
Result
Suppose $A \in \R ^{m \times n}$, $b \in
\R ^n$, and $c \in \R ^n$ so that
\[ P = \Set{x \in \R ^n}{Ax \leq b} \neq \varnothing \]
and\[ \delta = \inf \Set{c^\top x}{x \in P} > -\infty \]
Then there exists $x^\star \in \R ^n$ with $c^\top x^\star = \delta $.For this reason, a linear program is sometimes abbreviated $\min\Set{cx}{Ax \leq b}$ instead of $\inf\set{c^\top x}{Ax \leq b}$.