2001 , Volume 6, № 2, p.25-46

Generalized semi-infinite optimization problems M[h, g, u, v] are subject to a set of inequality constraints that possibly is of infinite cardinality and depending on the state x. This article summarizes investigations from Weber [57] based on various reseach of authors such as Guddat, Jongen, Rиuckmann and Twilt. We give a survey about manifold and continuity properties of the feasible M[h, g, u, v] and about the corresponding behaviour of (f, h, g, u, v) under slight perturbations. Here, suitable boundedness assumptions and constraint qualifications on the upper stage of x and on the lower stage of inequality constraints y are provided. We state Manifold Theorem, Continuity Theorem, Genericity Theorem, Stability Theorem and Structural Stability Theorem. Results of this kind play an important role for the development of iterative solution algorithms of P(f; h; g; u; v). Finally, we briefly describe extensions in cases of unboundedness and nondifferentiability, and we indicate under which structural frontiers our investigations extend to optimal control of ordinary differential equations. Here, directed graphs become a valuable mean.

*34H05 Control problems

90C31 Sensitivity, stability, parametric optimization

90C34 Semi-infinite programming