1. Part 3. Linear Programming
3.2 Algorithm

2. General Formulation
Convex function
Convex
region

3. Graphical Solution

4. Degenerate Problems
Non-unique solutions
Unbounded minimum

5. Degenerate Problems – No Feasible Region

6. Remarks
The solution obtained from a cannonical system by setting the non-basic variables to zero is called a basic solution.
A basic feasible solution is a basic solution in which the values of the basi variables are nonnegative.
Every corner point of the feasible region corresponds to a basic feasible solution of the constraint equations. Thus, the optimum solution is a basic feasible solution.

7. Full Rank Assumption

9. Fundamental Theorem of Linear Programming
Given a linear program in standard form where A is an mxn matrix of rank m.
If there is a feasible solution, there is a basic feasible solution;
If there is an optimal solution, there is an optimal basic feasible solution.

10. Implication of Fundamental Theorem

11. Extreme Point

12. Theorem (Equivalence of extreme points and basic solutions)

13. Corollary
If there is a finite optimal solution to a linear programming problem, there is a finite optimal solution which is an extreme point of the constraint set.

14. Step 2
x1 and x2 are selected as
non-basic variables

15. Step 3: select new basic and non-basic variables
new basic variable

16. Which one of x3, x4, x5 should be selected as the new non-basic variables?