Method selection (Nelder-Mead, BFGS, Powell) in SciPy - Step-by-Step Execution

Concept Flow - Method selection (Nelder-Mead, BFGS, Powell)

Define objective function

↓

Choose optimization method

↓

Nelder-Mead

↓

Run optimization

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Get optimized result

Start by defining the function to minimize. Then pick one method: Nelder-Mead, BFGS, or Powell. Run optimization and get the best solution found.

Execution Sample

SciPy

from scipy.optimize import minimize

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

res = minimize(f, [0,0], method='Nelder-Mead')

This code finds the minimum of a simple function using the Nelder-Mead method starting from [0,0].

Execution Table

Step	Method	Current Point	Function Value	Action	Notes
1	Nelder-Mead	[0, 0]	5	Evaluate function at start	Initial guess
2	Nelder-Mead	[1, 0]	4	Reflect point	Improved function value
3	Nelder-Mead	[1, 1]	1	Expand point	Better value found
4	Nelder-Mead	[1, 2]	0	Expand point	Minimum found
5	Nelder-Mead	[1, 2]	0	Converged	Stop optimization
1	BFGS	[0, 0]	5	Evaluate function and gradient	Initial guess
2	BFGS	[0.5, 1]	1.25	Update point using gradient	Improved value
3	BFGS	[1, 2]	0	Update point using gradient	Minimum found
4	BFGS	[1, 2]	0	Converged	Stop optimization
1	Powell	[0, 0]	5	Evaluate function at start	Initial guess
2	Powell	[1, 0]	4	Line search along x	Improved value
3	Powell	[1, 2]	0	Line search along y	Minimum found
4	Powell	[1, 2]	0	Converged	Stop optimization

💡 Optimization stops when the function value no longer improves significantly or maximum iterations reached.

Variable Tracker

Variable	Start	After 1	After 2	After 3	Final
Current Point (Nelder-Mead)	[0, 0]	[1, 0]	[1, 1]	[1, 2]	[1, 2]
Function Value (Nelder-Mead)	5	4	1	0	0
Current Point (BFGS)	[0, 0]	[0.5, 1]	[1, 2]	[1, 2]	[1, 2]
Function Value (BFGS)	5	1.25	0	0	0
Current Point (Powell)	[0, 0]	[1, 0]	[1, 2]	[1, 2]	[1, 2]
Function Value (Powell)	5	4	0	0	0

Key Moments - 3 Insights

Why does Nelder-Mead not use gradients like BFGS?

Nelder-Mead only uses function values to explore the space by reflecting and expanding points, as shown in execution_table rows 1-4 for Nelder-Mead. It works well when gradients are not available.

How does BFGS improve the point each step?

BFGS uses both function values and gradients to update the point in a smart direction, as seen in execution_table rows 1-3 for BFGS, quickly reaching the minimum.

What is the main difference in Powell's method steps?

Powell performs line searches along coordinate directions without gradients, moving step-by-step along axes, as shown in execution_table rows 1-3 for Powell.

Visual Quiz - 3 Questions

Look at the execution_table, what is the function value at step 3 for BFGS?

A 1.25

B 0

C 4

D 5

💡 Hint

Check the row with Step 3 and Method BFGS in the execution_table.

At which step does Nelder-Mead find the minimum function value?

A Step 2

B Step 3

C Step 4

D Step 5

💡 Hint

Look at the function values for Nelder-Mead in execution_table rows 1-5.

If we start Powell method at [2, 2], how would the first function value change?

A It would be 1

B It would be 0

C It would be 5

D It would be 2

💡 Hint

Calculate f([2,2]) = (2-1)^2 + (2-2)^2 = 1 + 0 = 1, but check carefully.

Concept Snapshot

Use scipy.optimize.minimize to find function minima.
Choose method: Nelder-Mead (no gradients), BFGS (uses gradients), Powell (line search).
Start from initial guess point.
Each method updates points differently.
Stop when function value stabilizes or max iterations reached.

Full Transcript

This visual execution shows how three optimization methods in scipy minimize work step-by-step. We start with a simple function to minimize. Nelder-Mead moves points by reflecting and expanding without gradients. BFGS uses gradients to update points smartly. Powell searches along coordinate lines without gradients. The execution table traces each step's point and function value. Variable tracker shows how points and values change. Key moments clarify common confusions about gradients and method differences. The quiz tests understanding of these steps and values. This helps beginners see how method selection affects optimization progress.

Practice

1. Which optimization method in scipy.optimize.minimize should you choose if you do NOT have information about the slope (gradient) of the function?

easy

A. BFGS

B. Nelder-Mead

C. Powell

D. CG

Solution

1. Step 1: Understand slope information requirement

   BFGS requires slope (gradient) information, while Nelder-Mead and Powell do not.

2. Step 2: Identify method without slope info

   Nelder-Mead is designed for optimization without slope info, making it suitable here.

3. Final Answer:

   Nelder-Mead -> Option B

4. Quick Check:

   No slope info = Nelder-Mead [OK]

Hint: No slope info? Pick Nelder-Mead [OK]

Common Mistakes:

• Choosing BFGS without slope info
• Confusing Powell with BFGS
• Assuming CG works without slope info

2. Which of the following is the correct way to specify the Powell method in scipy.optimize.minimize?

easy

A. minimize(fun, x0, method='Powell')

B. minimize(fun, x0, method='powl')

C. minimize(fun, x0, method='POWELL')

D. minimize(fun, x0, method='powell')

Solution

1. Step 1: Check method name case sensitivity

   Method names in scipy.optimize.minimize are case-sensitive and lowercase.

2. Step 2: Match exact method string

   The correct string is 'powell' all lowercase, so minimize(fun, x0, method='powell') is correct.

3. Final Answer:

   minimize(fun, x0, method='powell') -> Option D

4. Quick Check:

   Method name is lowercase 'powell' [OK]

Hint: Method names are lowercase strings [OK]

Common Mistakes:

• Using uppercase method names
• Typo in method string
• Using partial method names

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

from scipy.optimize import minimize

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

result = minimize(f, [0, 0], method='BFGS')
print(round(result.x[0], 1), round(result.x[1], 1))

medium

A. (3.0, -1.0)

B. (0.0, 0.0)

C. (3.0, 1.0)

D. Error due to missing gradient

Solution

1. Step 1: Understand the function and starting point

   The function has minimum at x=3, y=-1. Starting point is [0,0].

2. Step 2: BFGS uses gradient approximation to find minimum

   BFGS will find the minimum near (3, -1) accurately.

3. Final Answer:

   (3.0, -1.0) -> Option A

4. Quick Check:

   Minimum at (3,-1) found by BFGS [OK]

Hint: BFGS finds minimum near true minimum fast [OK]

Common Mistakes:

• Assuming output is starting point
• Thinking gradient is required explicitly
• Confusing signs in minimum location

4. Identify the error in this code snippet using Nelder-Mead method:

from scipy.optimize import minimize

def f(x):
    return x**2 + 4*x + 4

result = minimize(f, 2, method='Nelder-Mead')
print(result.x)

medium

A. Initial guess must be a list or array, not a scalar

B. Function must return a vector, not scalar

C. Method 'Nelder-Mead' is not supported

D. Missing gradient function

Solution

1. Step 1: Check input type for initial guess

   Minimize expects initial guess as array-like, but 2 is scalar.

2. Step 2: Confirm correct input format

   Passing [2] or np.array([2]) fixes the error.

3. Final Answer:

   Initial guess must be a list or array, not a scalar -> Option A

4. Quick Check:

   Initial guess scalar causes error [OK]

Hint: Initial guess must be array-like, not single number [OK]

Common Mistakes:

• Passing scalar initial guess
• Expecting gradient for Nelder-Mead
• Wrong method name casing

5. You want to minimize a function where you have no gradient info, but the function is noisy and non-smooth. Which method is best to try first in scipy.optimize.minimize?

hard

A. BFGS

B. Nelder-Mead

C. Powell

D. Newton-CG

Solution

1. Step 1: Analyze method suitability for noisy, non-smooth functions

   BFGS and Newton-CG require gradient info and smoothness, so not ideal.

2. Step 2: Compare Nelder-Mead and Powell for robustness

   Powell is more robust for noisy, non-smooth functions without gradients.

3. Final Answer:

   Powell -> Option C

4. Quick Check:

   No gradient + noisy function = Powell [OK]

Hint: No gradient + noisy? Use Powell method [OK]

Common Mistakes:

• Choosing BFGS without gradient
• Assuming Nelder-Mead is always best
• Ignoring function noise impact