On the Topology of Generalized Semi-Infinite Optimization

This survey article reflects the topological and inverse behaviour of generalized semi-infinite optimization problems P(f, h, g, u, v) and presents the analytical methods. These differentiable problems admit an infinite set Y(x) of inequality constraints y which depends on the state x. We extend our previous investigations ["Generalized Semi-Infinite Optimization and Related Topics", Research and Expositions in Mathematics 29, Heldermann Verlag (2003)] based on research of Guddat, Jongen, Rueckmann, Twilt and others. Under suitable assumptions on boundedness and qualifying conditions on lower y-stage and upper x-stage, we present manifold, continuity and global stability properties of the feasible set M[h, g, u, v], and corresponding structural stability properties of P(f, h, g, u, v) referring to slight data perturbations. Hereby, the character of our investigation is essentially specialized by the linear independence constraint qualification locally imposed on Y(x). The achieved results are important for algorithm design and convergence. Two extensions refer to unboundedness and nondifferentiable max-min-type objective functions. In the course of explanation, the perturbational approach gives rise to study inverse problems of reconstruction. We trace them into optimal control of ordinary differential equations, and indicate related investigations in heating processes, continuum mechanics and discrete tomography. Throughout the article, we realize discrete-combinatorial aspects and methods.