Diffusion for Global Optimization in Rⁿ

We seek a global minimum of U:Rⁿ → R. The solution to ( * )(d / dt)X(t) = - ∇ U(X(t)) will find local minima. Using the idea of simulated annealing, we consider the diffusion process, dX(t) = - ∇ U(X(t))dt + σ (t)dW(t), $X(0) = x$, where W( · ) is the n-dimensional standard Brownian motion and 1/2σ ² (t) is the annealing rate which decreases to zero as t goes to ∞. Under suitable condition on $U(x)$, we prove that $X(t)$ converges weakly to a probability measure π if for large t, σ ² (t) = c / log t with c > c₀, where c₀ has a simple expression involving the action function of the dynamical system $( * )$, π concentrates on the global minima of U and is the weak limit of the Gibbs densities π ₜ (x) ∝ exp ( - 2U(x) / σ ² (t)). The above result can also be formulated as follows: consider the Fokker–Planck equation (forward equation) \[{∂ }{{∂ t}}V(t,y) = 1/2σ ^2 (t)Δ V(t,y) + ∇ · (V(t,y)∇ U(y))\] with V(0,y) = δ ₓ (y). If σ ² (t) = c / log t for large t and c > c₀, then V(t,y) → π weakly.