This chapter introduces some basic concepts of discrete optimization. The emphasis is on providing mathematical description of some optimization algorithms. Convex sets and convex functions play an important role in many different types of optimization problems. We discuss hyperplane separation theorem and the fundamental Farkas lemma of polyhedral combinatorics, Petersen theorem on maximum matchings, and also K \(\ddot{\text {o}}\) nig-Egerváry theorem. We conclude the chapter with the classical Magner theorem and the max-flow min-cut theorem.