## Definition **A cooling schedule** (also: annealing schedule) is the function that determines how the temperature parameter $T$ is lowered over the course of a [[Simulated Annealing]] run. It specifies three things: the initial temperature $T_0$, the rate at which $T$ decreases, and the stopping criterion. ## Standard Geometric Schedule The most common schedule multiplies $T$ by a constant factor $\alpha \in (0, 1)$ after each temperature stage: $T_{k+1} = \alpha \cdot T_k, \qquad \alpha \approx 0.95\text{–}0.99$ - **$T_0$ (initial temperature).** Set high enough that the [[Metropolis Acceptance Criterion]] accepts most proposed moves initially. A practical heuristic: sample random neighbours, compute their $\Delta$ values, and choose $T_0$ so that $\exp(-\bar\Delta / T_0) \approx 0.8$. - **$\alpha$ (cooling rate).** Slower cooling (larger $\alpha$) gives better solutions at the cost of more iterations. Typical values 0.95–0.99 are engineering compromises. - **Iterations per temperature level.** Usually tens to hundreds of neighbour evaluations before each temperature drop. - **Stopping criterion.** Minimum temperature $T_\text{min}$, budget of total iterations, or no improvement over $k$ consecutive temperature levels. ## Theoretical Guarantee Hajek (1988) proved that SA converges to the global optimum *with probability 1* provided the cooling schedule is no faster than logarithmic: $T_k \geq \frac{C}{\ln(k+1)}$ where $C$ depends on the depth of the deepest local optimum. The logarithmic schedule is far too slow for practice; geometric schedules sacrifice the guarantee but are computationally tractable. ## Adaptive Cooling Adaptive schedules adjust $\alpha$ dynamically based on observed acceptance rates rather than fixing it in advance: - If the acceptance rate is too high (too much random walking), cool faster. - If too low (stuck in a basin), slow down or reheat temporarily. This can outperform a fixed geometric schedule on problems where the energy landscape varies across the search. ## Related - [[Simulated Annealing]] — the algorithm the cooling schedule governs - [[Metropolis Acceptance Criterion]] — the probability formula whose behaviour the schedule controls through $T$ - [[Parallel Tempering]] — avoids the sensitivity to a single cooling schedule by running multiple replicas at fixed temperatures - [[Local vs Global Optimum]] — the schedule's slowness is the price paid to guarantee escape from local optima