Approaching the best
While it is often possible to theoretically demonstrate the existence of an optimal solution to a given problem, an explicit and complete form of this solution is more difficult to obtain outside of a few textbook cases. It is therefore necessary to settle for calculating an estimate of it using algorithms. Examples are numerous, both in a "psychological" context (optimization of collective well-being using the Pareto model) and in physics. Thus, when the quantity to be optimized varies like a fluid, clever procedures can reach the precious minimum or maximum, provided care is taken not to remain stuck in a local pocket: it is the global optimum that is sought!
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The Monte Carlo method
In practice, finding an optimal value often involves computing demanding integrals. How can this be done? Physicists developed the Monte Carlo method, whose complexity does not increase with the dimension of the integrals involved.
Linear programming
Linear programming deals with problems that seem elementary in formulation: optimizing linear functions over a set defined by linear inequalities. Yet this theory has many highly practical applications.
Optimizing welfare?
Is the welfare of a population simply the sum of its members' satisfaction scores? This is far from certain, especially since measuring, comparing and aggregating individual utilities is no straightforward matter. Italian economist Vilfredo Pareto proposed a novel approach.
Gradient descent: skiing your way to a minimum
You are on a ski slope, surrounded by fog. Which route should you take to get all the way down? One approach is to follow the steepest slope—that is, the gradient. This idea yields both a numerical method and a way of finding optima.
Simulated annealing
During an approximate computation, how can we distinguish a local minimum from a global one? Simulated annealing, based on a common practice in metallurgy, offers a subtle yet effective heuristic method for many applications.