Metropolis-Type Annealing Algorithms for Global Optimization in $\mathbb{R}^d $

Abstract

The convergence of a class of Metropolis-type Markov-chain annealing algorithms for global optimization of a smooth function $U( \cdot )$ on $\mathbb{R}^d $ is established. No prior information is assumed as to what bounded region contains a global minimum. The analysis contained herein is based on writing the Metropolis-type algorithm in the form of a recursive stochastic algorithm $X_{k + 1} = X_k - a_k (\nabla U(X_k ) + \xi _k ) + b_k W_k $, where $\{ {W_k } \}$ is a standard white Gaussian sequence, $\{ {\xi _k } \}$ are random variables, and $a_k = {A / k}$, $b_k = {{\sqrt B } / {\sqrt {k\log \log k} }}$ for k large. Convergence results for $\{ {X_k } \}$ are then applied from our previous work [SIAM Journal on Control and Optimization, 29 (1991), pp. 999–1018]. Since the analysis of $\{ {X_k } \}$ is based on the asymptotic behavior of the related Langevin-type Markov diffusion annealing algorithm $dY(t) = - \nabla U(Y(t))dt + c(t)dW(t)$, where $W( \cdot )$ is a standard Wiener process and $c(t) = {...