How to perform constrained optimization using limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm in Python

In Python, you can perform constrained optimization using the Limited-Memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm with the scipy.optimize module. The minimize function in SciPy provides an option for constrained optimization, and you can set the method parameter to ‘L-BFGS-B’ for L-BFGS optimization with bounds.

Here’s an example of how to perform constrained optimization using the L-BFGS algorithm in Python:

from scipy.optimize import minimize

# Define the objective function to minimize
def objective_function(x):
    return (x[0] - 2)**2 + (x[1] - 3)**2

# Define the bounds for the variables
bounds = [(0, None), (0, None)]  # The bounds are set for x[0] and x[1]

# Define the equality constraint function
def equality_constraint(x):
    return x[0] + x[1] - 5

# Define the inequality constraint function
def inequality_constraint(x):
    return x[0] - 2*x[1]

# Set initial guess
initial_guess = [0, 0]

# Perform constrained optimization using L-BFGS-B
result = minimize(objective_function, initial_guess, method='L-BFGS-B', bounds=bounds, constraints=[{'type': 'eq', 'fun': equality_constraint}, {'type': 'ineq', 'fun': inequality_constraint}])

# Display the result
print("Optimal values:", result.x)
print("Optimal objective function value:", result.fun)
print("Constraint results:", equality_constraint(result.x), inequality_constraint(result.x))

In this example:

• The objective_function is the function to minimize.
• The bounds variable defines the bounds for each variable. In this case, both x[0] and x[1] are non-negative ((0, None)).
• The equality_constraint and inequality_constraint functions represent the constraints. In this example, there is an equality constraint (x[0] + x[1] - 5) and an inequality constraint (x[0] - 2*x[1]).
• The method='L-BFGS-B' parameter specifies the L-BFGS-B optimization algorithm.

Make sure to adapt the objective function and constraints according to your specific problem. Additionally, provide appropriate bounds for the variables in the bounds parameter. The constraints parameter is a list containing dictionaries specifying the type and function of each constraint.

Always refer to the SciPy documentation for the most up-to-date information on optimization methods and their parameters.