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

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

Define objective function

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Choose optimization method

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Nelder-Mead

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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.