# Fatih Pense's Blog

# Linear Programming, Affine Scaling with Julia

Wednesday, December 14th, 2022

My student friends passed me a homework question for Operations Research. Since it involves Matrices, I was attracted to try Julia. At the end I was very happy with the experience. Julia is a great language to do matrix operations.

## Question

Solve the following LP by using the Affine Scaling Approach of Interior Point Methods (IPM) with starting

## Solution

Wikipedia explains the algorithm in a very mathematical way: https://en.wikipedia.org/wiki/Affine_scaling. I guess you need to be very familiar with linear programming to understand that. (I’m not)

I found this great resource about using Julia for Operations Research by Changhyun Kwon: https://juliabook.chkwon.net/book/interior

I understood the problem and the method (except for the math magic that somehow gives you the direction to move for each variable). I wasn’t sure how to get `c`

and `A`

matrices below. Luckily, I found the exact example with these matrices in a PDF(page 41) by chance. It became a perfect puzzle, with all the parts in front of me.

So, here is the solution:

```
using LinearAlgebra
beta=0.8
x0 = [10, 2, 7, 13]
x=x0
c = [-2, 1, 0, 0]
A = [1 -1 1 0 ;
0 1 0 1 ]
b = [15, 15]
for i in 1:3
local X = Diagonal(x)
local p = inv(A*X^2*A')*A*X^2*c
local r = c - A'*p
global x = x - beta * X^2 * r / norm(X*r)
println((round.(x; digits=2)))
end
```