The linear optimization problem

An easy kind of optimization problem is where everything is linear.

Definition: Inequalities between vectors of

The expression where is componentwise, that is,

Definition: Linear optimization problem

An optimization problem is called linear whenever it is of the form for some matrices , , some vectors , and some vector .

Remark

Equivalently, the problem is linear whenever the feasible region is and the cost function is

Remark

Each equality constraint can be captured by two inequalities, so each linear optimization problem is equivalent to one of the form