Improving Simulated Annealing through Derandomization

TL;DR

This paper introduces QMC-SA, a derandomized version of simulated annealing using low-discrepancy sequences, proving its almost sure convergence to the global optimum without objective-specific conditions, and demonstrating its superiority through numerical experiments.

Contribution

It develops a new QMC-based simulated annealing method with proven convergence properties, extending the theoretical understanding of derandomized optimization algorithms.

Findings

QMC-SA converges almost surely to the global optimum for any R in natural numbers.

The deterministic version of QMC-SA converges for univariate objective functions.

Numerical results show QMC-SA outperforms traditional simulated annealing algorithms.

Abstract

We propose and study a version of simulated annealing (SA) on continuous state spaces based on $(t,s)_R$-sequences. The parameter $R\in\bar{\mathbb{N}}$ regulates the degree of randomness of the input sequence, with the case $R=0$ corresponding to IID uniform random numbers and the limiting case $R=\infty$ to $(t,s)$-sequences. Our main result, obtained for rectangular domains, shows that the resulting optimization method, which we refer to as QMC-SA, converges almost surely to the global optimum of the objective function $\varphi$ for any $R\in\mathbb{N}$. When $\varphi$ is univariate, we are in addition able to show that the completely deterministic version of QMC-SA is convergent. A key property of these results is that they do not require objective-dependent conditions on the cooling schedule. As a corollary of our theoretical analysis, we provide a new almost sure convergence result for SA which shares this property under minimal assumptions on $\varphi$. We further explain how our results in fact apply to a broader class of optimization methods including for example threshold accepting, for which to our knowledge no convergence results currently exist. We finally illustrate the superiority of QMC-SA over SA algorithms in a numerical study.