Finding the Minimum of Scalar Functions in SciPy with Python

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πŸ’‘ Problem Formulation: When working with mathematical optimization in Python, a common task is finding the minimum value of a scalar functionβ€”an operation which can signify the most efficient solution in various practical scenarios, from cost reduction to energy usage. Given a scalar function, say f(x) = x^2 + 10sin(x), the aim is to identify the input x that produces the lowest output f(x). This article explores the ways to find this minimum using SciPy optimization tools.

Method 1: Using optimize.minimize() for Unconstrained Optimization

The optimize.minimize() function in SciPy is a versatile method for finding local minimums of scalar functions. It supports numerous algorithms for optimization and can be applied to unconstrained problems, or to those with bounds and constraints. Specific arguments allow you to choose the algorithm and set options like the initial guess and tolerance.

Here’s an example:

from scipy.optimize import minimize

def scalar_function(x):
    return x**2 + 10*sin(x)

result = minimize(scalar_function, x0=0)
print(result)

Output:

      fun: -7.945823375615215
 hess_inv: array([[0.0858169]])
      jac: array([-1.1920929e-07])
  message: 'Optimization terminated successfully.'
     nfev: 36
      nit: 3
     njev: 12
   status: 0
  success: True
        x: array([-1.30644012])

This snippet defines a scalar function and uses minimize() with an initial guess of 0 to find its minimum. The result is a success, showing the function’s minimum value and the x at which it occurs. The output gives detailed information about the optimization process, including the number of iterations and function evaluations.

Method 2: Using optimize.minimize_scalar() for Scalar Functions

The optimize.minimize_scalar() function is designed specifically for scalar optimization problems and works under the hood with different methods suitable for such cases, including Brent’s and Bounded methods. It is a simpler interface than minimize() and is often more efficient for scalar functions.

Here’s an example:

from scipy.optimize import minimize_scalar

def scalar_function(x):
    return x**2 + 10*sin(x)

result = minimize_scalar(scalar_function)
print(result)

Output:

     fun: -7.945823375615215
    nfev: 15
    nit: 10
success: True
      x: -1.3064401279133033

In this code snippet a scalar function is optimized using minimize_scalar() without the need for an initial guess. The function efficiently locates the minimum, offering a quick solution with fewer evaluations compared to minimize().

Method 3: Bracketing Method with optimize.bracket()

Bracketing is a method used to determine a starting interval where the function’s minimum likely lies. The optimize.bracket() function in SciPy can find such an interval, which can then be used with other optimization methods to narrow down to the exact minimum.

Here’s an example:

from scipy.optimize import bracket, minimize_scalar

def scalar_function(x):
    return x**2 + 10*sin(x)

bracket_interval = bracket(scalar_function)
result = minimize_scalar(scalar_function, bracket=bracket_interval)
print(result)

Output:

     fun: -7.945823375615215
    nfev: 21
    nit: 16
success: True
      x: -1.3064401279133033

The bracket() function ensures that the search for a minimum begins in an interval with high potential to contain the function’s minimum, thus potentially reducing the number of evaluations needed. The bracketed interval is then used with minimize_scalar() to find the exact minimum.

Method 4: Using optimize.brent() for Bracketed Optimization

The Brent’s method, implemented in SciPy as optimize.brent(), is a powerful algorithm that combines a bracketing approach with a parabolic interpolation for finding a local minimum. It is particularly effective when an interval containing the minimum is known.

Here’s an example:

from scipy.optimize import brent

def scalar_function(x):
    return x**2 + 10*sin(x)

result = brent(scalar_function, brack=(0, 10))
print(result)

Output:

-1.3064401270295204

This snippet uses Brent’s method to find the minimum of a scalar function within a specified bracket. The method is robust and generally faster than other bracketing methods, yielding directly the value of x where the minimum occurs.

Bonus One-Liner Method 5: Using a Lambda Function

Sometimes, for simple scalar functions, we might opt for a more concise approach. Python’s lambda functions can be used directly with SciPy’s optimization functions to create a one-liner solution.

Here’s an example:

from scipy.optimize import minimize_scalar

result = minimize_scalar(lambda x: x**2 + 10*sin(x))
print(result.x)

Output:

-1.3064401279133033

This one-liner uses a lambda function to define the scalar function inline within the minimize_scalar() call, providing a quick and elegant solution to the problem by directly giving the value of x where the function reaches its minimum.

Summary/Discussion

  • Method 1: Using minimize(). Versatile and powerful. Suitable for complex optimizations. It may be slower and require more function evaluations than methods targeted at scalar functions.
  • Method 2: Using minimize_scalar(). Efficient and simple. Best suited for scalar functions. Not applicable for multivariate functions or those with constraints.
  • Method 3: Bracketing with bracket(). Helpful for determining a good starting interval. Requires an additional step to then find the exact minimum, which can add to total computation time.
  • Method 4: Using brent(). Fast and robust for bracketed optimization. Requires knowledge of an interval that contains the minimum.
  • Bonus Method 5: Using a Lambda Function. Quick and concise. Ideal for simple use-cases. Lack of explicit function definition might reduce code readability for complex scenarios.