MINIMIZE_SCALAR

Overview

The minimize_scalar function finds the minimum value of a scalar mathematical function. This is useful for business users in Excel who need to optimize costs, maximize efficiency, or find the best value for a given scenario. The function leverages scipy.optimize.minimize_scalar and allows users to specify the function as a string, with optional bounds and method.

Arguments Table

Return Value Table

Detailed Examples

Example 1: Minimize a Quadratic Cost Function

Business Context: A manager wants to find the production level (x) that minimizes the cost function C(x) = x^2 + 3x + 2.

Excel Setup:

• Cell A1: ‘x**2 + 3*x + 2’ (Function expression)
• Cell B1: (Leave blank for unbounded)
• Cell C1: (Leave blank for default method)

Formula in Excel: =minimize_scalar(A1)

Expected Outcome: Returns a 2D list: [[x, minimum cost]].

Example 2: Minimize a Function with Bounds

Business Context: A logistics analyst wants to minimize the function f(x) = (x-5)^2 + 10, but only for x between 0 and 10.

Excel Setup:

• Cell A2: ‘(x-5)**2 + 10’ (Function expression)
• Cell B2: 0 (Lower bound)
• Cell C2: 10 (Upper bound)
• Cell D2: ‘bounded’ (Method)

Formula in Excel: =minimize_scalar(A2, (B2, C2), D2)

Expected Outcome: Returns a 2D list: [[x, minimum value]] within [0, 10].

Parameter and Output Types

• Inputs: func_expr (string), bounds (tuple of two floats or None), method (string or None)
• Outputs: 2D list: [[x, fun]] (both floats)

Edge Cases and Limitations

• The function expression must be a valid Python expression in terms of x.
• If bounds are provided, method should be ‘bounded’.
• If the function is not well-behaved (e.g., not continuous), results may be unreliable.
• Only scalar (single-variable) functions are supported.

Comparison to Regular Excel Functions

How Could This Be Done in Regular Excel?

In standard Excel, finding the minimum of a mathematical function typically requires one of the following approaches:

• Manual Calculation: Entering the function formula in a column for a range of x-values, then using the MIN function to find the minimum value and INDEX/MATCH to find the corresponding x.
• Solver Add-in: Using Excel’s built-in Solver add-in to set up an optimization problem, specifying the target cell (the function output), the variable cell (x), and constraints (bounds).

Example (Manual Table):

1. In column A, list possible x-values (e.g., from -10 to 10).
2. In column B, enter the formula for the function (e.g., =A2^2 + 3*A2 + 2).
3. Use =MIN(B2:B22) to find the minimum value.
4. Use =INDEX(A2:A22, MATCH(MIN(B2:B22), B2:B22, 0)) to find the x-value at the minimum.

Example (Solver):

1. Enter the function formula in a cell, referencing a variable cell for x.
2. Open Solver, set the objective to minimize the function cell by changing the x cell, and set bounds if needed.
3. Run Solver to find the minimum.

Advantages of This Approach

• Automation: No need to manually set up tables or configure Solver; the function can be called directly in a cell.
• Flexibility: Works for any valid mathematical expression in terms of x, with optional bounds and method selection.
• Integration: Can be used as a custom function in Excel formulas, making it easy to incorporate into larger models or automate repeated calculations.
• Works in Excel for the Web: Unlike the Solver add-in, which is not available in Excel Online, this function can be used in both desktop and web versions of Excel (when deployed as a custom function).
• Precision: Uses advanced optimization algorithms from scipy.optimize, which are more robust and accurate than grid search or manual methods.

This makes the minimize_scalar function especially useful for business users who need to perform optimization tasks frequently or programmatically within Excel.

Source Code

# Minimize Scalar Function
 
from scipy.optimize import minimize_scalar
import math
 
 
def minimize_scalar_function(func_expr, bounds=None, method=None):
    """
    Minimizes a scalar function using scipy.optimize.minimize_scalar.
 
    Args:
        func_expr (str): A string representing the function to minimize, e.g., 'x**2 + 3*x + 2'.
        bounds (list, optional): A 2D list [[min, max]] specifying the bounds for bounded methods.
        method (str, optional): Optimization method: 'brent', 'bounded', or 'golden'.
 
    Returns:
        list: [[x, fun]] where x is the location of minimum and fun is the minimum value
    """
    # Define the function from the string expression
    def func(x):
        return eval(func_expr, {"x": x, "math": math})
 
    # Check if 'x' is present in the function expression
    if 'x' not in func_expr:
        raise ValueError("Function expression must contain the variable 'x'.")
 
    kwargs = {}
    # Accept bounds as a 2D list [[min, max]] or as a scalar (not tuple)
    if bounds is not None:
        # If bounds is a 2D list (e.g., [[0, 10]]), extract min and max
        if isinstance(bounds, list) and len(bounds) == 1 and isinstance(bounds[0], list) and len(bounds[0]) == 2:
            min_val, max_val = bounds[0][0], bounds[0][1]
            kwargs['bounds'] = [min_val, max_val]
        else:
            kwargs['bounds'] = bounds
    if method is not None:
        kwargs['method'] = method
 
    result = minimize_scalar(func, **kwargs)
    # Return as a 2D list: [[x, fun]]
    return [[float(result.x), float(result.fun)]]