Constraint Minimization using Python

The function `f(x,y) = 5 - (x - 2)^2 - 2*(y - 1)^2` is to be minimised subject to contraint `g(x,y) = x + 4y - 3`

Using Lagrange Multipliers to find maximum/minimum of `f(x,y)` subject to constraint `g(x,y)`

`L = f - lambdag`

`(delL)/(delx) = -2(x-2) - lambda = 0`

`(delL)/(dely) = -4(y-1) - 4lambda = 0`

`(delL)/(dellambda) = -x-4y+3 = 0`

Solving above 3 equations for 3 unknows `x, y and lambda`

`x = 5/3`

`y = 1/3`

`lambda = 2/3`

At `x = 5/3` and `y = 1/3`, `f(x,y) = 4`. This represents the maximum of the function under constraint `g(x,y)`.

Therefore this function does not have a minimum for the said constraint or the minimum of this function is at infinity or undefined.

Below is a Python program to plot the function on z-axis for corresponding range of values for `x` and `y` with constraint `x+4y -3 = 0`

""" To plot a given function under specified constraint"""


import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D


a = 1000


def z_value(l,c):
	
	t1 = pow((l - 2),2)
	t2 = pow((c - 1),2)

	z_i = 5 - t1 - 2*t2

	return z_i

x_i = []
y_i = []
for i in range(a):
	b = -i
	x_i.append(b)

for j in range(a):
	x_i.append(j)	


x = []
y = []
z = []


for l in x_i:

	x.append(l)
	c = (3 - l)/4
	y.append(c)
	z.append(z_value(l,c))





fig = plt.figure()
ax = Axes3D(fig)

ax.plot(x, y, z)
plt.xlabel('X-axis')
plt.ylabel('Y-axis')

plt.show()

Plots for the above program are as follows