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There exists many applications with so-called costly problems, which means that the objective function you want to maximize or minimize cannot be described using standard functions and expressions. Instead one considers these objective functions as ``black box'' where the parameter values are sent in and a function value is returned. This implies in particular that no derivative information is available.The reason for describing these problems as expensive is that it may take a long time to calculate a single function value. The black box could, for example, solve a large system of differential equations or carrying out a heavy simulation, which can take anywhere from several minutes to several hours!These very special conditions therefore requires customized algorithms. Common optimization algorithms are based on calculating function values every now and then, which usually can be done instantly. But with an expensive problem, it may take several hours to compute a single function value. Our main objective is therefore to create algorithms that exploit all available information to the limit before a new function value is calculated. Or in other words, we want to find the optimal solution using as few function evaluations as possible.A good example of real life applications comes from the automotive industry, where on the development of new engines utilize advanced models that are governed by a dozen key parameters. The goal is to optimize the model by changing the parameters in such a way that the engine becomes as energy efficient as possible, but still meets all sorts of demands on strength and external constraints.