Least squares solutions of linear inequality systems: a pedestrian approach

International audienceWith the help of elementary results and techniques from Real Analysis and Optimization at the undergraduate level, we study least squares solutions of linear inequality systems. We prove existence of solutions in various ways, provide a characterization of solutions in terms of nonlinear systems, and illustrate the applicability of results as a mathematical tool for checking the consistency of a system of linear inequalities and for proving "theorems of alternative" like the one by Gordan. Since a linear equality is the conjunction of two linear inequalities, the proposed results cover and extend what is known in the classical context of least squares solutions of linear equality systems."De tous les principes qu'on peut proposer pour cet objet, je pense qu'il n'en est pas de plus général, de plus exact, ni d'une application plus facile que celui qui consistè a rendre minimum la somme des carrés des erreurs" "Of all the principles that can be proposed, I think there is none more general, more exact, and more easy of application, than that which consists of minimizing the sum of the squares of the errors" A.-M.Legendre, Nouvelles méthodes pour la détermination des orbites des comètes, Paris (1805)