Infeasible linear inequality systems arise in many different contexts. For instance, linear programs might turn out to be infeasible because of modeling errors or data inaccuracies. To resolve the infeasibility one often looks for the smallest number of inequalities whose removal renders the program feasible. An interesting application is to find a solution to a linear equation system with smallest support. Irreducible infeasible subsystems are other basic building blocks of infeasible systems. In many applications one searches for such systems of smallest size. The aim of project is to investigate basic structures of infeasible linear inequality systems and to develop solution methods for the related optimization problems.