The Simplex Method (Dantzig, 1947) solves linear programs by traversing the vertices of the feasible polytope — the convex region defined by linear constraints. At each step, it moves to an adjacent vertex that improves the objective, following improving edges. The method terminates when no improving neighbor exists, guaranteeing a global optimum (linear programs have no local optima). Although worst-case exponential, simplex is remarkably fast in practice due to the shadow-vertex rule and degeneracy-breaking pivoting rules like Bland's rule.