In these years, we note a growing interest in theoretical foundations and methods which disrete optimization and continuous optimization have in common. There is hope that both fields may learn from each other, and that there will be such a unifying theory that tradition, character and importance of each field remain respected.
This survey article bases on research about the large class of generalized semi-infinite (continuous) optimization problems, including optimal control theory. We indicate discrete-combinatorial aspects of an algorithms based on local linearization, and present an algorithm for solving a time-minimal problem of heating.
Furthermore, we explain combinatorial relations between graphs and nonlinear optimization problems, topological properties of graphs, topological properties and optimal control applications of networks, symmetric multi-processing systems, random graphs and their Morse theoretical aspects, Newton flows and their discrete features, discrete tomography and related inverse problems, exact and approximate experimental designs.
Throughout the article we pay attention to structural frontiers, and we motivate future research.
The results have been joint work with Erik Kropat (TU Darmstadt) and Stefan W. Pickl (Univ. Köln).