Generalized semi-infinite optimization : theory and applications in optimal control and discrete optimization

Abstract

In this survey article we are concerned with a very general class of nonlinear programming problems, which became of increasing interest in the last years. Several problems from science and engineering can be interpreted and treated as a differentiable generalized semi-infinite optímization problem While I is finite, being defined by finitely many equality and inequality constraints and , has infinitely many elements usually. These index sets of inequity constraints are locally assumed to be lìnearizable. Fìrstly, we look at simplifying problem representations, 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 tìme-mìnìmal control of heating processes. Interrelations between the different kinds of problem are regarded, and the importance of discrete structures and methods is pointed out. Thirdly, we mention two further examples from discrete mathematics which are related to our generalized semi-infinite optimization: random graphs and graphs based on Newton flows.