This course provides a rigorous treatment of linear optimization. It is designed for students who want to build large-scale optimization models and need an understanding of the underlying theory. The course is theoretical in nature and the vast majority of the assignments (as well as the exams) are theorem-proof oriented. It is designed for Ph.D. students only. The course will cover polyhedral theory, theory of systems of linear inequalities, simplex method, network simplex algorithm, polynomial time solvability of linear optimization, conic programming, and interior point methods in linear and semidefinite optimization.