An optimization problem's objective is to find an optimal solution among a large set of candidates. The shape of your favorite soda can was derived from an optimization problem, trying to minimize the amount of material to use (the surface) for a given volume (33 cl). In this case, the surface, the quantity to minimize, is also called the objective function.
Optimization problems often come with some constraints on the variables. The fact that a length has to be positive is already a constraint, mathematically speaking. But constraints can be expressed in many different forms.
The simpler form of an optimization problem is so-called linear optimization, where both the objective function and the constraints are linear.
Graph optimization problems are also part of mathematical optimization problems. The most famous of them is the traveling-salesman problem (TSP). We are going to talk a bit more about this particular problem in the following section...