A simple constrained minimization problem

P8.4.1A simple constrained minimization problem

A rectangular field with area $A = 10000\;\mathrm{m^3}$ is to be fenced-off beside a straight river (the boundary with the river does not need to be fenced). What dimensions $$a,b$$ minimize the amount of fencing required? Verify that a constrained minimization algorithm gives the same answer as the algebraic analysis.

Show solution

The amount of fencing required is $$L = a+2b$$ and the area is constrained to be $$A=ab=10000$$ . The analytical answer is easily obtained by calculus to be $a = \sqrt{2A}, b = \sqrt{A/2}$ . Using numerical minimization gives the same result:

constrainedminimization

from scipy.optimize import minimize

A = 10000
res = minimize(
    lambda X: X[0] + 2 * X[1],
    (100, 100),
    method="slsqp",
    constraints={"type": "eq", "fun": lambda X: A - X[0] * X[1]},
)
print(res)

Output:

 message: Optimization terminated successfully
 success: True
  status: 0
     fun: 282.84271247440637
       x: [ 1.414e+02  7.071e+01]
     nit: 11
     jac: [ 1.000e+00  2.000e+00]
    nfev: 33
    njev: 11

The result is reported to be $$a=141.4$$ m and $$b=70.71$$ m. The exact result is:

In [x]: import numpy as np
In [x]: A = 10000
In [x]: print("a =", np.sqrt(2 * A))
a = 141.4213562373095

In [x]: print("b =", np.sqrt(A / 2))
b = 70.71067811865476