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Minimzing_With_condition_using_python.qmd
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Minimzing_With_condition_using_python.qmd
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---
title: Minimizing a function with a condition
format: gfm
---
## Problem:
$$
\begin{aligned}
& f_X(x) = 12x^2 (1- x), \quad 0 < x < 1, \\
& A = \underset{b - a}{\min}\{(a, b)~|~\mathcal{P}(a < X < b) = 0.5\}\\
& \implies A = ?
\end{aligned}
$$
#### Soloution:
$$
\begin{aligned}
& X \sim \text{Beta}(3, 2), \\
& \mathcal{P}(a < X < b) = \int_a^b 12x^2 (1 - x) dx = \\
& 3(a^4 - b^4) + 4(b^3 - a^3) = 0.5 \implies \\
& \text{Using Lagrange Method:}\quad \gamma(a, b, \lambda) = b - a + \lambda \left[\int_a^b f(x)dx - 0.5\right] \implies \\
& (\hat{a}, \hat{b}): ~~ \begin{cases} \frac{\partial \gamma}{\partial \lambda} = 0 \\
\frac{\partial \gamma}{\partial a} = 0 \\
\frac{\partial \gamma}{\partial b} = 0\end{cases} \implies \\
& \begin{cases}3(a^4 - b^4) + 4(b^3 - a^3) = 0.5\\
b^2 - b^3 = a^2 - a^3 \end{cases}\quad \overset{\text{Using Numerical Method with python}}{\to}
\end{aligned}
$$
***
#### Using python
```{python}
import numpy as np
import sympy
from sympy import symbols, simplify, integrate, diff
x, a, b, l = symbols(['x', 'a', 'b', 'lambda'])
fx = 12 * x**2 * (1 - x)
ress = integrate(fx, (x, a, b))
ress
```
```{python}
gam = b - a + l * (ress - 0.5)
eq1 = diff(gam, l)
eq2 = diff(gam, a)
eq3 = diff(gam, b)
eq1
```
```{python}
eq2
```
```{python}
eq3
```
#### soloution
```{python}
from scipy.optimize import minimize
# Define the function to minimize
def func(x):
return x[1] - x[0]
# Define the constraint
cons = ({'type': 'eq', 'fun': lambda x: 3 * (x[0]**4 - x[1]**4) +
4 * (x[1]**3 - x[0]**3) - 0.5},
{'type': 'ineq', 'fun': lambda x: x[0] - 0},
{'type': 'ineq', 'fun': lambda x: x[1] - 0},
{'type': 'ineq', 'fun': lambda x: x[1] - x[0] - 0},
{'type': 'ineq', 'fun': lambda x: 1 - x[0] - 0},
{'type': 'ineq', 'fun': lambda x: 1 - x[1] - 0})
x0 = [0.75, 0.4]
# Call the minimize function with the method 'SLSQP' (Sequential Least Squares Programming)
res = minimize(func, x0, method = 'SLSQP', constraints = cons)
result = res.x
print(f"""
The solution is: \n\n
a: {result[0]}, \n
b: {result[1]}""")
```
## check soloution:
```{python}
round(integrate(fx, (x, result[0], result[1])), 6)
```
```{python}
a = result[0]; b = result[1]
print(f"""
a**2 - a**3: { round(a**2 - a**3, 6)}, \n
b**2 - b**3: { round(b**2 - b**3, 6)}
""")
```