Optimization in cartesian closed categories, lagrange’s method of multipliers and applications

Abstract

A category is defined as an algebraic structure that has objects that are linked by morphisms. Categories were created as a foundation of mathematics and as a way of relating algebraic structures and systems of topological spaces. Any foundation of mathematics must include algebra, topology, and analysis. Algebra and topology have been studied extensively in category theory but not the analysis. This is partly due to the algebraic nature of category theory and the fact that the axiom of choice is not used in category theory. However, with the introduction of infinitesimals, it has been possible to study synthetic differentiation that is consistent with categories. It has been pointed out that, in order to treat mathematically the decisive abstract general relations of physics, it is necessary that the mathematical world picture involves a Cartesian closed category of smooth morphisms between smooth spaces. Algebra and topology have been studied in Cartesian closed categories but optimization has not so far been considered. This study aimed at defining a cone, derivative, extremal object and then used these definitions to obtain optimization results using the Lagrange method of multipliers in Cartesian closed categories. The study also provides a discussion of the various areas that the results can be applied such as building spreadsheet application, neuroscience, cognitive neural network architectures and program optimizations.