Simulated annealing (dual_annealing) in SciPy - Time & Space Complexity

Time Complexity: Simulated annealing (dual_annealing)

O(n * k)

Understanding Time Complexity

We want to understand how the time needed to find a solution using simulated annealing grows as the problem size increases.

Specifically, how does the number of steps in dual_annealing change with input size?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

from scipy.optimize import dual_annealing

n = 10

def objective(x):
    return sum((x - 2) ** 2)

bounds = [(-5, 5)] * n
result = dual_annealing(objective, bounds)
    

This code tries to find the minimum of a simple function using simulated annealing over n variables.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Evaluating the objective function many times during the annealing process.
• How many times: The number of function evaluations depends on the number of iterations and temperature schedule, which grows with problem size.

How Execution Grows With Input

As the number of variables n increases, each function evaluation takes longer because it sums over more elements.

Input Size (n)	Approx. Operations
10	Thousands of function evaluations x 10 operations each
100	Thousands of function evaluations x 100 operations each
1000	Thousands of function evaluations x 1000 operations each

Pattern observation: The total work grows roughly linearly with the number of variables times the number of evaluations.

Final Time Complexity

Time Complexity: O(n * k)

This means the time grows with the number of variables n and the number of function evaluations k during annealing.

Common Mistake

[X] Wrong: "The time only depends on the number of variables n."

[OK] Correct: The algorithm runs many steps, each evaluating the function. So total time depends on both n and how many steps k it takes.

Interview Connect

Understanding how optimization time grows helps you explain trade-offs in algorithms and shows you can think about performance beyond just code correctness.

Self-Check

"What if the objective function was more complex and took longer per evaluation? How would that affect the time complexity?"

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