.
In mathematical programming a huge variety of objects to study
may be defined as the common
solution set of finite systems consisting of both equalities and inequalities
in .
Provided the defining functions to be smooth and in general position, the Thom isotopy lemma states that, locally, the considered solution set has the structure of a product of a so-called fiber with some Euclidean space. Here the fiber is a stratified set, i.e. it can be partitioned into differentiable manifolds in a certain, regular, way. Thus, the entire (local) complexity of the treated set is already contained in the fiber.
This leeds to various types of applications. A rather direct application of the lemma yields structural stability results. Here solution sets are compared, defined by the same system (of equalities and inequalities), however with slightly changed defining functions.
However, even sets defined by totally different defining systems can be (locally) compared; as long as one can be sure of the property that the sets of possible fibers coincide. These ideas, for example, provide a rather short proof of the well known manifold property of the Karush-Kuhn-Tucker set in parametric optimization.
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