Dimensionality reducing by alpha-dense curves: application to global optimization, multiple integration and mathematical programming

Abstract

In Mathematics the word complexity, as synonym of difficulty, has not a clear meaning. However, it seems that it is very easy to understand that one mathematical problem involving functions of several variables is, in general, more complex than one problem with a univariable function. The Alpha-Dense Curves theory tries to reduce the complexity associated to the dimension and establishes an Approximation Theory on the domains, in the sense of the Hausdorff metric, instead of that of the functions. Some applications to optimize multivariable continuous functions, to approximate a multiple integral by a simple one, and to determine if a non-linear inequality has solution, will be the subjects which we shall deal with techniques based on Alpha-Dense Curves.