Simulated annealing (dual_annealing) in SciPy - Mini Project: Build & Apply

Optimize a Simple Function Using Simulated Annealing (dual_annealing)

📖 Scenario: You are working as a data scientist and want to find the minimum value of a mathematical function. This is like trying to find the lowest point in a hilly landscape by exploring smartly.

🎯 Goal: Use the dual_annealing method from scipy.optimize to find the minimum of a given function within a specific range.

📋 What You'll Learn

Create a function to optimize

Set the bounds for the function inputs

Use dual_annealing to find the minimum

Print the minimum value found

💡 Why This Matters

🌍 Real World

Simulated annealing helps find good solutions in complex problems like scheduling, routing, or tuning machine learning models.

💼 Career

Understanding optimization techniques like dual_annealing is useful for data scientists and analysts who need to improve models or find best parameters.

1

Define the function to optimize

Write a function called func that takes one input x and returns the value of (x[0] - 3) ** 2 + (x[0] + 1) ** 2. This function represents the landscape you want to find the lowest point of.

SciPy

# Define the function func that takes x and returns the sum of squares expression
# Your code here

Hint

Remember, x will be a list or array, so access the first element with x[0].

2

Set the bounds for the input variable

Create a variable called bounds and set it to a list with one tuple: (-5, 5). This limits the search for the minimum to values between -5 and 5.

SciPy

def func(x):
    return (x[0] - 3) ** 2 + (x[0] + 1) ** 2

# Set bounds for x between -5 and 5
# Your code here

Hint

Bounds must be a list of tuples, even if there is only one variable.

3

Use dual_annealing to find the minimum

Import dual_annealing from scipy.optimize. Then create a variable called result by calling dual_annealing with func and bounds as arguments.

SciPy

def func(x):
    return (x[0] - 3) ** 2 + (x[0] + 1) ** 2

bounds = [(-5, 5)]

# Import dual_annealing and run optimization
# Your code here

Hint

Use from scipy.optimize import dual_annealing to import the function.

4

Print the minimum value found

Write a print statement to display the minimum value found by dual_annealing. Use result.fun to get the minimum function value.

SciPy

from scipy.optimize import dual_annealing

def func(x):
    return (x[0] - 3) ** 2 + (x[0] + 1) ** 2

bounds = [(-5, 5)]

result = dual_annealing(func, bounds)

# Print the minimum function value found
# Your code here

Hint

The minimum value should be 8.0 because the function has its lowest point at x = 1.

Practice

1. What is the main purpose of using dual_annealing in scipy.optimize?

easy

A. To find the minimum value of a function within given bounds

B. To sort a list of numbers in ascending order

C. To calculate the mean of a dataset

D. To generate random numbers following a normal distribution

Solution

1. Step 1: Understand the purpose of dual_annealing

   dual_annealing is an optimization method used to find the minimum of a function, especially when the function is complex and has many local minima.

2. Step 2: Identify the correct use case

   Among the options, only finding the minimum value of a function within bounds matches the purpose of dual_annealing.

3. Final Answer:

   To find the minimum value of a function within given bounds -> Option A

4. Quick Check:

   Optimization = Find minimum [OK]

Hint: dual_annealing is for minimizing functions with bounds [OK]

Common Mistakes:

• Confusing optimization with sorting or statistics
• Thinking dual_annealing generates random numbers
• Assuming it calculates averages

2. Which of the following is the correct way to import dual_annealing from scipy.optimize?

easy

A. from scipy.optimize import dual_annealing

B. import dual_annealing from scipy.optimize

C. from scipy import dual_annealing.optimize

D. import scipy.optimize.dual_annealing

Solution

1. Step 1: Recall Python import syntax

   The correct syntax to import a function from a module is from module import function.

2. Step 2: Match syntax to options

   from scipy.optimize import dual_annealing matches the correct syntax: from scipy.optimize import dual_annealing. Other options have incorrect syntax.

3. Final Answer:

   from scipy.optimize import dual_annealing -> Option A

4. Quick Check:

   Correct import syntax = from scipy.optimize import dual_annealing [OK]

Hint: Use 'from module import function' to import dual_annealing [OK]

Common Mistakes:

• Using 'import function from module' which is invalid
• Trying to import submodules incorrectly
• Using dot notation in import statements wrongly

3. What will be the output of the following code snippet?

from scipy.optimize import dual_annealing

def f(x):
    return (x[0] - 3)**2 + (x[1] + 1)**2

bounds = [(-5, 5), (-5, 5)]
result = dual_annealing(f, bounds)
print(round(result.fun, 2))

medium

A. 10.00

B. 0.00

C. 4.00

D. Error

Solution

1. Step 1: Understand the function and bounds

   The function f(x) calculates the sum of squares of (x[0]-3) and (x[1]+1). The minimum is at x[0]=3 and x[1]=-1, where the function value is 0.

2. Step 2: dual_annealing finds the minimum within bounds

   The bounds allow x[0]=3 and x[1]=-1. So the optimizer should find the minimum function value close to 0. The print statement rounds the result to 2 decimals.

3. Final Answer:

   0.00 -> Option B

4. Quick Check:

   Minimum value = 0.00 [OK]

Hint: Minimum of squared distance function is zero at target point [OK]

Common Mistakes:

• Assuming the minimum is outside bounds
• Confusing function value with input values
• Expecting an error due to function shape

4. Identify the error in the following code using dual_annealing:

from scipy.optimize import dual_annealing

def f(x):
    return x**2

bounds = [(-2, 2)]
result = dual_annealing(f, bounds)
print(result.x)

medium

A. dual_annealing requires no bounds argument

B. Bounds should be a tuple, not a list

C. Function f expects a scalar but dual_annealing passes an array

D. Missing import for numpy

Solution

1. Step 1: Check function input type

   dual_annealing passes an array (even if one variable), but f(x) expects a scalar x. This mismatch causes an error.

2. Step 2: Verify bounds and imports

   Bounds as a list of tuples is correct. dual_annealing requires bounds. No numpy import needed here.

3. Final Answer:

   Function f expects a scalar but dual_annealing passes an array -> Option C

4. Quick Check:

   Function input type mismatch = Function f expects a scalar but dual_annealing passes an array [OK]

Hint: dual_annealing passes array input; function must accept array [OK]

Common Mistakes:

• Assuming bounds format is wrong
• Thinking numpy import is mandatory here
• Ignoring input type mismatch

5. You want to minimize the function f(x) = (x[0]-2)^2 + (x[1]-3)^2 but only allow x[0] between 0 and 1, and x[1] between 2 and 4. Which code correctly uses dual_annealing to find the minimum within these bounds?

hard

A. bounds = [(0, 1), (2, 4)]
result = dual_annealing(f)

B. bounds = [(2, 3), (3, 4)]
result = dual_annealing(f, bounds)

C. bounds = [(0, 2), (2, 3)]
result = dual_annealing(f, bounds)

D. bounds = [(0, 1), (2, 4)]
result = dual_annealing(f, bounds)

Solution

1. Step 1: Understand the function and bounds

   The function minimum is at x[0]=2, x[1]=3. But bounds restrict x[0] to [0,1] and x[1] to [2,4]. So the optimizer must search within these bounds.

2. Step 2: Check code options for correct bounds and usage

   The code bounds = [(0, 1), (2, 4)] result = dual_annealing(f, bounds) correctly sets bounds as [(0,1), (2,4)] and passes them to dual_annealing. The code bounds = [(2, 3), (3, 4)] result = dual_annealing(f, bounds) has wrong bounds. The code bounds = [(0, 2), (2, 3)] result = dual_annealing(f, bounds) has wrong bounds. The code bounds = [(0, 1), (2, 4)] result = dual_annealing(f) misses bounds argument.

3. Final Answer:

   bounds = [(0, 1), (2, 4)] result = dual_annealing(f, bounds) -> Option D

4. Quick Check:

   Correct bounds and function call = bounds = [(0, 1), (2, 4)] result = dual_annealing(f, bounds) [OK]

Hint: Bounds must match variable limits and be passed to dual_annealing [OK]

Common Mistakes:

• Using wrong bounds that exclude minimum
• Not passing bounds argument to dual_annealing
• Confusing variable order in bounds