In this homework, you will implement a Python script that finds the distance between two squares using CPLEX. In analytic geometry, a square is defined as the set of points $($x$,$y$)$ of the Cartesian plane that satisfy the following inequalities

a $12$ r $<=$ x $<=$ a $+12$ r b $12$ r $<=$ y $<=$ b $+12$ r

where $($a$,$ b$)$ is the center of the square and r is the side length. The Euclidean distance between two squares is the minimum distance between two points $($i$.$e$.,($x$1,$ y$1)$ and $($x$2,$ y$2))$ within these two squares $($i$.$e$.,($a$1,$ b$1,$ r$1)$ and $($a$2,$ b$2,$ r$2)),$ respectively.

The quadratic programming formulation of this problem becomes

z $=($x$1$x$2)2+($y$1$y$2)2$ x $1>=$ a $121$ r $1$

x $1<=$ a $1+12$ r $1$ y $1>=$ b $112$ r $1$ y $1<=$ b $1+12$ r $1$ x $2>=$ a $212$ r $2$ x $2<=$ a $2+12$ r $2$ y $2>=$ b $212$ r $2$ y $2<=$ b $2+12$ r $2.$

After solving the optimization problem, $$z will give the Euclidean distance between the two squares.

This problem will be represented using a $.$txt file, namely, squares.txt$.$ This file contains the parameters of two squares in two rows $($i$.$e$.,$ a$1,$ b$1$ and r$1$ in the first row and a$2,$ b$2$ and r$2$ in the second row$),$ and it is composed of the following two lines for an example problem:

squares.txt

$-------------$

$4\mathrm{.}06\mathrm{.}04\mathrm{.}0$

$11\mathrm{.}012\mathrm{.}02\mathrm{.}0$

minimize subject to:

$1$

The quadratic programming model for the example problem can be written as minimize z$=$x$21$x$1$x$2$x$2$x$1+$x$2+$y$12$y$1$y$2$y$2$y$1+$y$2$

subject to:

$+$ x$1+$x$1$

$+$y$1+$y$1$

$+$x$2+$x$2$

$+$ y$2+$y$2$

$>=2<=6>=4<=8>=10<=12>=11<=13.$

The optimum solution of the example problem is as follows:

x$1=5\mathrm{.}999999995573339,$ y$1=7\mathrm{.}999999994097785,$ x$2=10\mathrm{.}0,$

y$2=11\mathrm{.}0,$

distance$=5\mathrm{.}000000007082659.$

Implement your algorithm to find the distance between two squares in a single interactive Python notebook using Azure Lab Services. Your notebook should include at least the following function definition that takes the file path of the input file as parameter and returns the solution found.

def distance$\_$between$\_$squares$($squares$\_$file$)$:

# your implementation starts below

# your implementation ends above

return$($x$1\_$star, y$1\_$star, x$2\_$star, y$2\_$star, distance$\_$star$)$

What to submit: You are provided with a template file named as $0099999$ hw$04.$ipynb, where $99999$ should be replaced with your $5-$digit student number. You are allowed to change the template file between the following lines.

# your implementation starts below

# your implementation ends above

You need to submit your source code in a single file $(0099999$ hw$04.$py file that you will download from Azure Lab Services by following $$File$/$Save and Export Notebook As$.../$Executable Script$$ menu items$).$

How to submit: Submit the file you edited to Blackboard by following the exact style mentioned. Submissions that do not follow these guidelines will not be graded.