Optimierungsprobleme : Verschwundene Minima in der nichtlinearen Optimierung

Bei vielen Dingen des täglichen Lebens, bei technologischen Prozessen oder auch in der Wirtschaft will man ein Optimum finden. Das betrifft etwa den minimalen Energieverbrauch bei der Herstellung eines Produkts, die kürzeste Zeit, um von einem Punkt zu einem anderen zu gelangen oder die Gewinnmaximierung in einem Unternehmen. Bei vielen dieser Optimierungsaufgaben spielt Mathematik eine große Rolle.
Optimization is part of our daily lives. We aim to minimize the time it takes to reach a desired location or maximize a company’s profit. The number of optimization variables is quite high in many applications – it may range from several hundreds to billions. However, here we only discuss settings with two variables. Optimality conditions are used to compute critical points, some of which may be saddle points or have no meaning at all. This is not a problem – we can check additional conditions to reject these points. However, if we lose a minimizer, the optimal point is lost forever. Two simple examples illustrate the loss of solutions of two standard optimization methods (elimination and Lagrange). This knowledge is important for applications, as there are several classes of optimization problems (including Stackelberg games, optimization problems governed by variational inequalities) that generically possess such undesired properties.
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