Characterizing Sets Defined By Inequalities

Abstract

Most engineering problems (design of products meeting specifications, off-line quality control, robust control…) can be formulated as constrained optimization problems, but characterization of the feasible set may turn out to be more important than optimization within this set, especially when some ad hoc optimality criterion is used and the constraints are the most realistic part of the problem definition. On the other hand, estimating or predicting parameters or state variables from noisy data leads to the idea of confidence regions. Feasible sets for design or control and confidence regions for estimation can be defined by inequalities. Methods are presented for characterizing such sets in a guaranteed way. Exact description and inner and outer approximations are considered. The techniques differ depending on whether the inequalities are linear or not, whether the description of the set is to be obtained off-line or on-line, and depending on the intended use of the result.