Integer Polyhedra and Total Dual Integrality

Understanding the boundary between continuous linear programming and discrete integer optimization is fundamental to solving complex combinatorial problems. This text-based course introduces you to the geometric and algebraic principles behind integer polyhedra and total dual integrality. You will build a rigorous conceptual foundation, discovering how rational cones, discrete lattice points, and duality theorems guarantee integer solutions to linear programs. What you'll learn: - Understand the fundamental properties of convex cones, polyhedra, and integer lattice points - Learn the existence and construction of finite Hilbert bases in rational cones - Master total dual integrality (TDI) and its role in characterizing integral polyhedra - Apply duality theory to classical combinatorial results such as Dilworth's theorem on chains and antichains - Analyze weighted extensions and proof techniques for certifying TDI systems - Connect polyhedral theory with modern algorithmic formulations in discrete optimization The course begins with fundamental terminology and linear algebra concepts before methodically advancing through cone geometry, Hilbert bases, TDI systems, and combinatorial applications. Designed for beginners in advanced discrete mathematics and optimization, all concepts are explained from first principles without requiring specialized prior coursework. Start reading today to deepen your grasp of discrete optimization theory.