The optimization problem

Optimization theory regards methods of solving problems that are of the general following form:

Definition: Optimization problem, feasible region, cost function

An optimization problem is a problem of the following form: where is a set called the feasible region, and where is a real-valued function called the cost function or objective function.

Remark

Unless otherwise stated, is assumed to be a subset of .

Definition: Optimal value, feasible solution, optimal solution

The optimal value of an optimization problem is .

A vector is called a feasible solution whenever .

A vector is called a (global) optimal solution whenever for all .

Remark

If an optimal solution exists, then is the optimal value. This is because is a minimum of and is thus the infimum.

Definition: Local optimal solution, global optimal solution

A vector is called a local optimal solution if there exists a such that whenever such that , .