We are concerned with a very general class of nonlinear programming problems, which became of increasing interest in the last years. Several problems from engineering can be described and treated as a differentiable generalized semi-infinite optimization problem. Here, we have a possibly infinite set of inequality constraints which depends on the state. This set is implicitly defined and assumed to be locally linearizable.
Firstly, we look at a simplifying problem representation, optimality conditions, the topological structure and stability of the problem under data perturbation, preparing convergence results for concepts of iteration procedures.
Secondly, we consider two classes of optimal control problems which can partially be interpreted and analyzed by generalized semi-infinite optimization. These special applications are optimal control of ordinary differential equations, and time-minimal control of heating processes.
Interrelations between the different kinds of problem and structural frontiers are regarded, and the importance of discrete structures is pointed out.
Thirdly, we reflect several continuous-discrete applications of cell decomposition, random graph evolution and interior point methods.
Key words: Generalized semi-infinite optimization, optimality condition, structural stability, iteration procedure, optimal control, discrete optimization, cells, random graphs, interior points.