1. Prepared by Po-Chuan on 2016/05/24
The Simplex Method
Prepared by Po-Chuan on 2016/05/24

2. What is simplex method? A way to solve LP problems.
Then, how to solve them?
There are many forms of LP problems.
But there’s a standard form of all LP problems.
So we focus on the solution of standard LP problems.

3. Standard form of LP problems
All the RHS (right hand side) values are non-negative.
If not, multiply by -1.
All the variables are non-negative.
If 𝑥 𝑖 is non-positive, multiply by -1.
If 𝑥 𝑖 is free, replace 𝑥 𝑖 by 𝑥 𝑖 ′ − 𝑥 𝑖 ′′ , where 𝑥 𝑖 ′ , 𝑥 𝑖 ′′ >0.
𝑥 1 + 𝑥 2 ≤4→ 𝑥 1 ′ − 𝑥 1 ′′ + 𝑥 2 ≤4, 𝑥 1 ′ , 𝑥 1 ′′ ≥0
All the constraints are equalities.
If not, add and extra variable.
𝑥 1 + 𝑥 2 ≤4→ 𝑥 1 + 𝑥 2 + 𝑥 3 =4, 𝑥 3 ≥0
No limit on the object function.

4. Standard form in matrices
The standard form is expressed as min 𝑐 𝑇 𝑥 s.t. 𝐴𝑥=𝑏 𝑥≥0
For example, min 2 𝑥 1 − 𝑥 2 s.t. 𝑥 𝑥 2 + 𝑥 = 𝑥 1 −6 𝑥 𝑥 4 =4

5. The matrices
𝐴= − , 𝑏= , 𝑐= 2 −1 0 0
Where 𝐴∈ 𝑅 𝑚×𝑛 is the coefficient matrix 𝑏∈ 𝑅 𝑚 is the RHS vector c∈ 𝑅 𝑛 is the objective vector.
The objective function can be min or max.

6. Basic solutions - bases
Consider a standard form LP with 𝑚 constraints and 𝑛 variables.
We can assume rank 𝐴=𝑚, which means all rows are independent.
We can also assume that 𝑚<𝑛, because when 𝑚≥𝑛, the problem is trivial.
We then choose 𝑚 columns from 𝐴, called 𝐴 𝐵 , as a basis of 𝐴 in order to get a unique solution for 𝐴𝑥=𝑏, which also implies 𝐴 𝐵 must be invertible.
Each basis leads to a basic solution, the columns (variables) not selected as filled by 0.

7. Basic solutions – an example
min 6 𝑥 𝑥 2 s.t. 𝑥 𝑥 2 + 𝑥 = 𝑥 𝑥 𝑥 4 =6
𝑚=2, 𝑛=4, so we choose 2 columns (variables) to form a basis.
If we choose 𝑥 1 and 𝑥 2 … (we can also choose 𝑥 2 and 𝑥 4 ) 𝑥 1 +2 𝑥 2 =6 2 𝑥 1 + 𝑥 2 =6
The basic solution for this basis is 𝑥 1 , 𝑥 2 , 𝑥 3 , 𝑥 4 = 2,2,0,0

8. Basic solutions – all solutions
In this example, we can get at most =6 bases.
Calculate all these possible combinations, and we’ll get all possible basic solutions.
Different bases may lead to a same basic solution.
# of distinct basic solutions ≤ # of distinct bases ≤ 𝑛 𝑚 .
When multiple bases correspond to one single basic solution, the LP is degenerated; otherwise, it is non- degenerated.

9. Basic feasible solutions
We just found many basic solutions, but are they feasible?
A basic solution is feasible if all variables are non-negative.
A feasible basic solution is called basic feasible solution (BF solution).
In the previous example, basis 𝑥 1 , 𝑥 4 is not feasible because its basic solution 𝑥 1 , 𝑥 2 , 𝑥 3 , 𝑥 4 = 6,0,0,−6 does not satisfy the requirement.
So why do we find BF solutions?
They’re extreme points!

10. Basic feasible solutions – an example

11. Solving LP Now we get the feasible solutions. What’s next?
Iterate through all BF solutions and find the best one.
Too slow, since there are 𝑛 𝑚 ones.
So we introduce the concept of “adjacent BF solution”.
2 solutions are called adjacent if the bases are differed only by one column (variable).
𝑥 1 , 𝑥 3 and 𝑥 3 , 𝑥 4 are adjacent.
𝑥 1 , 𝑥 2 and 𝑥 3 , 𝑥 4 are not adjacent.

12. Adjacent BF solutions by graph

13. So… what’s the optimal solution?
At each BF solution, move to an adjacent solution that is better.
There can be many adjacent solutions at a certain solution, any one is acceptable.
If there’s no better adjacent solution, then the current solution is optimal.
(水往低處流，人往高處爬)

14. Simplex method 的幾何及代數意義
欲知後事為何，且待下回分解

15. Thanks!
Prepared by Po-Chuan