In this chapter we can finally introduce the most basic concepts in optimisation, the notion of maximum and minimum, and we shall define what an optimisation problem is. The rest of this chapter will then look at the more general and elementary necessary and/or sufficient conditions for optimality for convex functions and we shall introduce a special class of functions, the ‘coercive’ functions, that are particularly useful in optimisation problems, because they ensure, sometimes with minimal extra hypotheses, the existence of optimal solutions. First of all, however, we shall delve into the definition of the various types of maxima and minima, and define what linear and non-linear programming problems are.