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The Cross-Entropy Method for Combinatorial and Continuous Optimization
Authors:Reuven Rubinstein
Institution:(1) Department of Mathematics, University of Queensland, Brisbane, QLD 4072, Australia;(2) School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia;(3) School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD 4001, Australia;(4) School of Mathematical Sciences, Monash University, Clayton, VIC, 3800, Australia
Abstract:We present a new and fast method, called the cross-entropy method, for finding the optimal solution of combinatorial and continuous nonconvex optimization problems with convex bounded domains. To find the optimal solution we solve a sequence of simple auxiliary smooth optimization problems based on Kullback-Leibler cross-entropy, importance sampling, Markov chain and Boltzmann distribution. We use importance sampling as an important ingredient for adaptive adjustment of the temperature in the Boltzmann distribution and use Kullback-Leibler cross-entropy to find the optimal solution. In fact, we use the mode of a unimodal importance sampling distribution, like the mode of beta distribution, as an estimate of the optimal solution for continuous optimization and Markov chains approach for combinatorial optimization. In the later case we show almost surely convergence of our algorithm to the optimal solution. Supporting numerical results for both continuous and combinatorial optimization problems are given as well. Our empirical studies suggest that the cross-entropy method has polynomial in the size of the problem running time complexity.
Keywords:combinatorial optimization  global optimization  importance sampling  markov chain monte carlo  simulated annealing  simulation
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