Simulated Annealing: Escaping Local Minima

Search Controls

Starting temperature T₀ 4.0

Cooling rate α 0.970

Step size σ 0.55

Simulation speed 6 steps/frame

Search State

Press Start to begin the search

Temperature T

4.00

Iteration

Current f(x,y)

—

Best f(x,y) found

—

Accept rate (last 50)

—

Global minimum

—

How it works

Simulated annealing searches a rugged objective surface f(x,y) by proposing a random nearby point and deciding whether to move there using the Metropolis criterion:

Δf = f(x') − f(x)
if Δf ≤ 0: always accept (downhill)
else:      accept with probability P = exp(−Δf / T)

Cooling schedule:  T ← α · T   (α close to but below 1)

When T is high, uphill moves are frequently accepted, letting the search jump out of local minima that would trap plain gradient descent. As T cools toward zero, the acceptance probability for uphill moves collapses and the search behaves like greedy descent, settling into whichever basin it currently occupies.

T₀ — starting temperature. Higher starts explore more of the landscape before committing.
α (cooling rate) — how fast T shrinks each iteration; closer to 1 cools slower and explores longer but takes more iterations to converge.
σ (step size) — the standard deviation of the random proposal offset in x and y.
The gold marker tracks the current probe position; the small green marker marks the best point found so far — they can differ, since the probe may currently be exploring away from its best discovery.

Real-world relevance: this exact mechanism (borrowed from the physics of slow-cooling metal annealing into a low-energy crystal lattice) is used for scheduling, circuit-board layout, protein-structure search, and any objective landscape too bumpy for gradient methods to trust.

Current probe Best found

⚠ Couldn't load the 3D engineThree.js failed to load from the CDN. Check your connection and reload.