Simulated annealing (dual_annealing) in SciPy - Step-by-Step Execution

Concept Flow - Simulated annealing (dual_annealing)

Start with initial guess

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Evaluate objective function

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Generate new candidate solution

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Calculate energy difference

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Accept new solution?

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Update current

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Lower temperature

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Temperature > min?

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Stop

Yes↓

Repeat steps

The algorithm starts with a guess, evaluates it, tries a new guess, decides to accept it or not based on energy difference and temperature, then lowers temperature and repeats until stopping.

Execution Sample

SciPy

from scipy.optimize import dual_annealing

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

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

This code finds the minimum of a simple function using dual_annealing from scipy.

Execution Table

Step	Current Solution	Candidate Solution	Energy Current	Energy Candidate	Energy Diff	Temperature	Accept Candidate	Reason
1	[0, 0]	[0.5, -0.3]	10	8.34	-1.66	2500	Yes	Candidate better (energy lower)
2	[0.5, -0.3]	[1.2, -0.8]	8.34	4.13	-4.21	2400	Yes	Candidate better
3	[1.2, -0.8]	[2.5, -1.5]	4.13	0.5	-3.63	2300	Yes	Candidate better
4	[2.5, -1.5]	[3.5, -0.5]	0.5	1.25	0.75	2200	No	Candidate worse, rejected
5	[2.5, -1.5]	[3.1, -1.1]	0.5	0.02	-0.48	2100	Yes	Candidate better
6	[3.1, -1.1]	[2.9, -0.9]	0.02	0.08	0.06	2000	Yes	Candidate worse but accepted by probability
...	...	...	...	...	...	...	...	...
N	[3.0, -1.0]	[3.0, -1.0]	0.0	0.0	0.0	1	Yes	Reached minimum, temperature low
Exit	Final solution found		0.0			0	Stop	Temperature reached minimum threshold

💡 Temperature reached minimum threshold, algorithm stops with best found solution.

Variable Tracker

Variable	Start	After 1	After 2	After 3	After 4	After 5	After 6	...	Final
Current Solution	[0, 0]	[0.5, -0.3]	[1.2, -0.8]	[2.5, -1.5]	[2.5, -1.5]	[3.1, -1.1]	[2.9, -0.9]	...	[3.0, -1.0]
Temperature	2500	2400	2300	2200	2100	2000	...	...	0

Key Moments - 3 Insights

Why does the algorithm sometimes accept a worse candidate solution?

Because of the temperature and energy difference, the algorithm accepts worse solutions with some probability to avoid local minima, as shown in step 6 where energy difference is positive but candidate is accepted.

What causes the algorithm to stop?

The algorithm stops when the temperature reaches a minimum threshold, as shown in the exit row where temperature is 0 and no further steps occur.

Why does the current solution sometimes stay the same after a candidate is rejected?

When the candidate solution has higher energy and is not accepted, the current solution remains unchanged, as in step 4 where candidate is worse and rejected.

Visual Quiz - 3 Questions

Look at the execution table at step 4. What is the reason the candidate solution was rejected?

A Candidate solution had lower energy

B Candidate solution had higher energy and was rejected

C Temperature was too high

D Algorithm reached stopping condition

💡 Hint

Check the 'Energy Diff' and 'Accept Candidate' columns at step 4 in the execution table.

According to the variable tracker, what is the temperature after step 3?

A 2100

B 2200

C 2300

D 2400

💡 Hint

Look at the 'Temperature' row and the 'After 3' column in the variable tracker.

If the temperature never decreased, how would the acceptance of worse candidates change?

A Worse candidates would be accepted more often

B Worse candidates would never be accepted

C Acceptance would be random

D Algorithm would stop immediately

💡 Hint

Recall that higher temperature increases probability to accept worse solutions, see step 6 in execution table.

Concept Snapshot

Simulated annealing (dual_annealing):
- Starts with initial guess and temperature
- Generates new candidate solutions
- Accepts better or sometimes worse solutions based on temperature
- Gradually lowers temperature
- Stops when temperature is low
- Helps find global minimum by escaping local minima

Full Transcript

Simulated annealing with dual_annealing starts by picking a guess and temperature. It checks the function value (energy). Then it tries a new guess nearby. If the new guess is better, it accepts it. If worse, it might still accept it depending on temperature. Temperature lowers step by step. This repeats until temperature is very low. The final guess is the best found minimum. This method helps avoid getting stuck in bad local minimum by sometimes accepting worse guesses early on.