Ants Searching for a Minimum

Abstract

We are solving optimization problems every day. For example, we look for the best investments for our money, the best path to get back home, the best restaurant according to our wishes/budget. Clearly, the notion of “optimality” is very subjective and varies a lot from one person to another (have you ever tried to organize a party with some friends?). In a more abstract framework, we can say that we are looking for the best solution within a set of admissible solutions (the constraints which limit our decision). If we denote by S the set of admissible solutions, two different subjects can judge “optimal” two completely different options x ^* and y ^* satisfying the constraint, i.e. x ^*, y ^* ∈ S. This is not surprizing since every one has his/her own criterium to optimize and it can be very difficult to define it properly (do you remember the motivations your friend gave to have pizza instead of burgers at your graduation party?). From the mathematical point of view we need to have a clear definition of the priorities and we assume that they are represented by an objective function f : S → ℝ. This allow us to determine if a solution x in better than y, because for every x ∈ S, the functions f gives a value f(x) that we can compare with other values.