Applied nonlinear optimization

Modelling and simulation of complex processes in key technologies are the main issues of the DFG research center. Being able to efficiently deal with the complex models established and having the corresponding mathematical basis available, engineers and practitioners are interested in optimizing these models according to some objective. This is reflected by several scientific projects in the center. They cover various fields of optimization ranging from discrete, nonlinear, or stochastic optimization to optimal control of ordinary or partial differential equations. In some of the projects, different fields interact and grow together. Traditionally, Berlin is a good place to deal with optimization. All fields mentioned above are covered by well known mathematicians working at one of the scientific institutions in Berlin. In this paper, we first introduce some basic ideas of nonlinear optimization, aiming at guiding the reader from the calculus of variations started in 1696 to necessary optimality conditions for nonlinear optimization problems in Banach spaces. Finally, we arrive at specific applied optimization problems which shall be the subject of study in the DFG research center. A detailed presentation of all aspects of optimization that shall be treated in the center would go beyond the scope of this contribution. Therefore, we will only shed light on a few selected projects contributing to the fields of stochastic optimization and optimal control of partial differential equations, where the two authors are engaged. Nonlinear optimization developed as an independent field of applied mathematics in the early fifties of the 20th century. It grew out of linear programming, which had posed new questions in the field of extremal problems during World War II, and had invented the simplex method. Over a long time this method was the most successful numerical technique to solve optimization problems. Since a long period before, nonlinear optimization – in the sense of extremal problems – is part of calculus. The unconstrained minimization of the Rosenbrock function