We certainly know how to maximize a real-valued function f. If f is regular, we solve the equation f'(x)=0 and, if a interior point of maximum exists, then it must be one of the solutions. But what happens when the function is vector-valued (i.e. its image is a vector of R^n)? We first have to introduce an order relation in R^n. This is never completely satisfactory (there is not a suitable total order relation on R^n). Suppose that we have chosen an order relation on R^n. What is the analogous of the Fermat rule f'=0? And what about the case of a set-valued function (i.e. its image is a set?). The problem of maximizing a vector-valued function (or even a set-valued function) with respect to some criterium (an order relation) is quite common in the real life. Think for example, to the situation where you have to choose the best candidate for a job position (you are a head-hunter). You evaluate the candidates by three independent parameters: (curriculum, adaptiveness, level of English). So, the output of each evaluation is a vector of R^3. How can you take the best one? Arguments and tools: linear algebra, differential calculus for functions of several variables, elements of Banach spaces, optimal control, game theory.