CSC2411 - Linear Programming and Combinatorial Optimization Lecture 3: Geometric Aspects of Linear Programming and an Introduction to the Simplex Algorithm

In this class, we discuss some geometrical interpretations of linear programs and a high-level description of the simplex algorithm. We also introduce methods for implementing the two phases of the simplex algorithm. These notes also include the tutorial presented on January 19. 1 Review In our previous class, we took some important steps toward defining an efficient algorithm for solving linear programs (LP). ¡ We defined basic feasible solutions (BFS). ¡ We proved that BFS and extreme points are equivalent. ¡ We showed that if an optimum exists for a LP, then there is a BFS which is optimum. – This gave us an exponential algorithm in which we check all sets of ¢ columns. – Candidate solutions arise when the columns are linearly independent and there is a nonnegative combination of the columns that equals £. – An easy corollary is that if there is a solution (¤ ¦ ¥ § © ¨), then there exists a solution which is a BFS. ¡ Finally, we proved the claim that if is a BFS, then there is a such that