Global optimization (differential_evolution) in SciPy - Step-by-Step Execution

Concept Flow - Global optimization (differential_evolution)

Define objective function

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Set bounds for variables

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Initialize population randomly

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Mutation and recombination

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Selection: choose better solutions

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Check stopping criteria

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Return best

The differential evolution algorithm starts with a random population, then improves solutions by mutation, recombination, and selection until stopping criteria are met.

Execution Sample

SciPy

from scipy.optimize import differential_evolution

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

bounds = [(-5, 5), (-5, 5)]
result = differential_evolution(f, bounds)
print(result.x, result.fun)

This code finds the minimum of a simple function using differential evolution within given bounds.

Execution Table

Step	Population Sample	Mutation Vector	Trial Vector	Selection Result	Best Solution So Far
1	[[0.1, -4.5], [2.0, 0.0], [-3.0, 1.0], [4.5, -2.0]]	[2.0, 0.0]	[2.0, 0.0]	Selected (better)	[2.0, 0.0] (fun=2)
2	[[2.0, 0.0], [2.5, -1.0], [-3.0, 1.0], [4.5, -2.0]]	[2.5, -1.0]	[2.5, -1.0]	Selected (better)	[2.5, -1.0] (fun=0.25)
3	[[2.0, 0.0], [-3.0, 1.0], [3.0, -1.0], [4.5, -2.0]]	[3.0, -1.0]	[3.0, -1.0]	Selected (better)	[3.0, -1.0] (fun=0)
4	[[3.0, -1.0], [-3.0, 1.0], [3.0, -1.0], [4.5, -2.0]]	[4.5, -2.0]	[4.5, -2.0]	Not selected (worse)	[3.0, -1.0] (fun=0)
5	[[3.0, -1.0], [-3.0, 1.0], [3.0, -1.0], [4.5, -2.0]]	[2.9, -1.1]	[2.9, -1.1]	Selected (better)	[2.9, -1.1] (fun=0.02)
Exit	-	-	-	-	Converged to best solution [2.9, -1.1] with fun=0.02

💡 Algorithm stops when improvement is minimal or max iterations reached.

Variable Tracker

Variable	Start	After 1	After 2	After 3	After 4	After 5	Final
Population	[[0.1, -4.5], [2.0, 0.0], [-3.0, 1.0], [4.5, -2.0]]	[[2.0, 0.0], [2.5, -1.0], [-3.0, 1.0], [4.5, -2.0]]	[[2.0, 0.0], [-3.0, 1.0], [3.0, -1.0], [4.5, -2.0]]	[[3.0, -1.0], [-3.0, 1.0], [3.0, -1.0], [4.5, -2.0]]	[[3.0, -1.0], [-3.0, 1.0], [3.0, -1.0], [4.5, -2.0]]	[[3.0, -1.0], [2.9, -1.1], [3.0, -1.0], [4.5, -2.0]]	[[3.0, -1.0], [2.9, -1.1], [3.0, -1.0], [4.5, -2.0]]
Best Solution	None	[2.0, 0.0]	[2.5, -1.0]	[3.0, -1.0]	[3.0, -1.0]	[2.9, -1.1]	[2.9, -1.1]
Best Fun Value	None	2	0.25	0	0	0.02	0.02

Key Moments - 3 Insights

Why does the algorithm sometimes keep a worse solution instead of the trial vector?

Because selection only replaces a population member if the trial vector has a better (lower) function value, as shown in execution_table step 4 where the trial vector was worse and not selected.

How does the population change over iterations?

The population updates by replacing individuals with better trial vectors, as seen in variable_tracker where population members change after each iteration.

What stops the differential evolution algorithm?

The algorithm stops when improvements become very small or after a max number of iterations, as noted in the exit_note and the last row of execution_table.

Visual Quiz - 3 Questions

Look at the execution_table at step 3. What is the best solution and its function value?

A [2.0, 0.0] with fun=2

B [-3.0, 1.0] with fun=40

C [3.0, -1.0] with fun=0

D [4.5, -2.0] with fun=unknown

💡 Hint

Check the 'Best Solution So Far' column at step 3 in execution_table.

At which step does the algorithm reject a trial vector because it is worse?

A Step 4

B Step 2

C Step 5

D Step 1

💡 Hint

Look for 'Not selected (worse)' in the Selection Result column in execution_table.

If the bounds were changed to narrower ranges, how would the population likely change?

A Population would become larger

B Population values would be limited within new bounds

C Population would not change

D Algorithm would stop immediately

💡 Hint

Variable_tracker shows population values; bounds restrict possible values.

Concept Snapshot

differential_evolution(func, bounds)
- Starts with random population within bounds
- Uses mutation and recombination to create trial vectors
- Selects better solutions to form next generation
- Stops when convergence or max iterations reached
- Returns best solution found and its function value

Full Transcript

Differential evolution is a global optimization method. It starts by defining the function to minimize and setting variable bounds. Then it creates a random population of candidate solutions. Each iteration, it mutates and recombines population members to create trial vectors. If a trial vector improves the function value, it replaces the original member. This process repeats until the solution converges or a maximum number of iterations is reached. The best solution and its function value are returned.