Generalized Semi-Infinite Optimization and Anticipatory Systems

This article is a small survey and pioneering as a starting point for a longer research project : to utilize generalized semi-infinite optimization for purposes of prediction. Firstly, it reflects tbe analytical and inverse (intrinsic) behaviour of generalized semi-infinite optimization problems P(f,h,g,u,v) and presents interpretations of them from the viewpoint of anticipatory systems. These differentiable problems admit an infinite set Y(x) of inequality constraints y, which depends on the state x. Under suitable assumptions, we present global stability properties of the feasible set and corresponding structural stability properties of the entire optimization problem (Weber, 2002 ; Weber, 2003). The achieved results are a basis of algorithm design.

In the course of explanation, the perturbational approach gives rise to reconstructions. By studying three applications of generalized semi-infinite optimization, secondly, we interpret these aspects of inverse problems in the sense of prediction. The three anticipatory systems are : (i) Reverse Chebycchev approximation, where we describe a given system by a neighbouring easier one as long as possible under some error tolerance. We begin by a motivating problem from chemical engineering and turn then to time-dependent systems. (ii) Time-minimal or -maximal optimization problems, where we want to pull or push the time-horizon of some process to present time or into the future. We mention global warming and turn to further kinds of biosystems. (iii) Computational biology, where we are concerned with prediction and stability of DNA microarray gene-expression patterns.