Gradient descent gets trapped in the first valley it finds. Annealing borrows from metallurgy: start HOT — willing to jump UPHILL to escape a local dip — then cool down, growing pickier, until you settle in the deepest valley you found. Heat buys exploration; cooling buys refinement. Slide the cooling and watch it escape the trap.
Simulated annealing (Kirkpatrick 1983) explores by proposing random moves and accepting a WORSE move with probability e^(−Δ/T) — the Metropolis criterion — where T is a ‘temperature’ lowered on a cooling schedule. When T is high, uphill moves are often accepted (escaping local minima); as T→0 it becomes greedy and settles. With slow enough cooling it converges to the global optimum in probability. It solves hard combinatorial problems (TSP, layout, scheduling) where gradients don’t exist. A fail-loud self-check throws unless annealing from a bad start reaches a lower value than pure greedy descent. ◆ real optimization, node-verified.
The Metropolis accept rule and the escape-local-minima behaviour are exact; the global-optimum guarantee needs an impractically slow schedule, so real runs are heuristics. The landscape here is illustrative.