Simplex Algorithm: Solution on a Vertex

This lecture introduces the simplex algorithm, the most famous optimization algorithm designed by George Dantzig in 1949 to solve linear optimization problems. The algorithm is based on the concept that the solution to a problem lies on one of the vertices of the constraint polyhedron. It illustrates this idea using a one-dimensional problem and explains how the optimal solution can be found on a vertex. The lecture then generalizes this concept to multiple dimensions, highlighting that for a linear optimization problem with a feasible solution, the optimal solution can be found on a vertex of the constraint polyhedron. The lecture also discusses the practical challenges of enumerating all the base solutions of a polyhedron and emphasizes the importance of understanding feasible base solutions in the context of the simplex algorithm.