1) What does simulated annealing solve?
It searches for a low objective value when a function has many local minima. The method balances random exploration with gradual cooling to improve the chance of finding a near-global minimum.
Calculator inputs
Use benchmark functions, choose bounds, set thermal controls, and run a configurable simulated annealing search.
Example data table
Sample optimization history for a two-variable Rastrigin search. Values below are illustrative and help users understand output structure.
| Iteration | Temperature | Current Objective | Best Objective | Accepted | x1 | x2 |
|---|---|---|---|---|---|---|
| 1 | 100.0000 | 47.2861 | 47.2861 | Yes | 4.2000 | -3.5000 |
| 25 | 92.0000 | 21.6418 | 18.5924 | Yes | 2.1700 | -1.1900 |
| 70 | 77.8707 | 9.5532 | 7.1089 | No | 1.0050 | -0.8800 |
| 160 | 55.2212 | 2.7413 | 1.9986 | Yes | 0.3100 | -0.0700 |
| 310 | 28.3886 | 0.6930 | 0.4204 | Yes | 0.0580 | -0.0210 |
| 480 | 13.4124 | 0.1288 | 0.0261 | Yes | 0.0100 | -0.0040 |
Formula used
1) Neighbor generation
For each variable, the calculator proposes a nearby candidate using:
x′i = clamp(xi + ui, lower, upper)
where each random offset ui is sampled from [−step size, +step size].
2) Energy difference
The objective function value acts like energy in annealing:
Δ = f(x′) − f(x)
If Δ ≤ 0, the new point is always accepted because it improves or matches the current solution.
3) Metropolis acceptance probability
When the candidate is worse, the method may still accept it to escape local minima:
P(accept) = exp(−Δ / T)
Here, T is the current temperature. Higher temperatures allow more uphill moves.
4) Cooling schedule
Exponential: Tk+1 = αTk
Linear: Tk+1 = max(Tk − α, Tmin)
Logarithmic: Tk = T0 / ln(2 + αk)
How to use this calculator
- Choose a benchmark function that matches the landscape you want to study.
- Set the dimension, initial coordinates, and common lower and upper bounds.
- Enter the thermal settings: initial temperature, minimum temperature, cooling rule, and alpha value.
- Choose iterations per temperature, total temperature levels, neighborhood step size, and stall limit.
- Optionally add a random seed when you want reproducible optimization paths.
- Press Run Simulated Annealing to calculate the best point, best objective value, convergence history, and acceptance statistics.
- Use the CSV button for raw iteration export and the PDF button for a report-style snapshot.
- Review the charts to see whether cooling was too fast, too slow, or appropriately balanced.
Frequently asked questions
2) Why accept worse moves at all?
Accepting some worse moves prevents the search from getting trapped too early. At higher temperatures, uphill moves are more likely, which improves exploration across rough landscapes.
3) What is a good initial temperature?
A higher initial temperature increases exploration and acceptance of uphill moves. Start large enough to allow movement, then reduce gradually until the algorithm settles around strong candidates.
4) How does step size affect results?
Large steps explore widely but may overshoot good areas. Small steps refine local neighborhoods but can slow global exploration. Good performance usually needs a balanced step size.
5) When should I use exponential cooling?
Exponential cooling is a practical default because it is simple and stable. It usually works well when you want smooth, predictable temperature decay with limited tuning effort.
6) Why does the calculator include a stall limit?
The stall limit stops the run after many non-improving iterations. It saves time when the search has effectively converged or when the chosen settings no longer generate useful progress.
7) Can I reproduce the same run later?
Yes. Enter the same random seed and keep all settings unchanged. That makes the pseudo-random sequence repeat, so the optimization path becomes reproducible.
8) What does the contour chart show?
For two-variable problems, it plots the objective surface and overlays the search path. This helps you see where the algorithm wandered, improved, and finally settled.