The Cross-Entropy Method for Continuous Multi-Extremal Optimization

Kroese, Dirk P.; Porotsky, Sergey; Rubinstein, Reuven Y.

doi:10.1007/s11009-006-9753-0

Cited by 223 publications

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“…Many other modifications can be found in [27,45,46] and in the list of references. When there are two or more optimal solutions the CE algorithm typically "fluctuates" between the solutions before focusing in on one of the solutions.…”

Section: The Cross-entropy Methods For Optimizationmentioning

confidence: 99%

“…Thus, the sampling distribution for X is characterized by a vector of means µ and a vector of standard deviations σ. At each iteration of the CE algorithm these parameter vectors are updated simply as the vectors of sample means and sample standard deviations of the elements in the elite set; see, for example, [27]. During the course of the algorithm, the sequence of mean vectors ideally tends to the maximizer x * , while the vector of standard deviations tend to the zero vector.…”

Section: Continuous Optimizationmentioning

confidence: 99%

“…Thus, by reducing the constrained problem ( (6) and (10)) to an unconstrained one ((6) with S instead of S), one can again apply Algorithm 2.1. Further details on constrained multi-extremal optimization with the CE method may be found in [27].…”

Section: Constrained Optimizationmentioning

confidence: 99%

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A Tutorial on the Cross-Entropy Method

et al. 2005

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Abstract:The cross-entropy method is a recent versatile Monte Carlo technique. This article provides a brief introduction to the cross-entropy method and discusses how it can be used for rare-event probability estimation and for solving combinatorial, continuous, constrained and noisy optimization problems. A comprehensive list of references on cross-entropy methods and applications is included.Keywords: cross-entropy, Kullback-Leibler divergence, rare events, importance sampling, stochastic search.The cross-entropy (CE) method is a recent generic Monte Carlo technique for solving complicated simulation and optimization problems. The approach was introduced by R.Y. Rubinstein in [41,42], extending his earlier work on variance minimization methods for rare-event probability estimation [40].The CE method can be applied to two types of problem:, where X is a random variable or vector taking values in some set X and H is function on X . An important special case is the estimation of a probability = P(S(X) γ), where S is another function on X .2. Optimization: Optimize (that is, maximize or minimize) S(x) over all x ∈ X , where S is some objective function on X . S can be either a known or a noisy function. In the latter case the objective function needs to be estimated, e.g., via simulation.In the estimation setting, the CE method can be viewed as an adaptive importance sampling procedure that uses the cross-entropy or Kullback-Leibler divergence as a measure of closeness between two sampling distributions, as is explained further in Section 1. In the optimization setting, the optimization problem is first translated into a rare-event estimation problem, and then the CE method for estimation is used as an adaptive algorithm to locate the optimum, as is explained further in Section 2.An easy tutorial on the CE method is given in [15]. A more comprehensive treatment can be found in [45]; see also [46, Chapter 8]. The CE method homepage can be found at www.cemethod.org .The CE method has been successfully applied to a diverse range of estimation and optimization problems, including buffer allocation [1], queueing models of telecommunication systems [14,16], optimal control of HIV/AIDS spread [48,49], signal detection [30], combinatorial auctions [9], DNA sequence alignment [24,38], scheduling and vehicle routing [3,8,11,20,23,53], neural and reinforcement learning [31,32,34,52,54], project management [12], rare-event simulation with light-and heavy-tail distributions [2,10,21,28], clustering analysis [4,5,29]. Applications to classical combinatorial optimization problems including the max-cut, traveling salesman, and Hamiltonian cycle 1

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Section: The Cross-entropy Methods For Optimizationmentioning

confidence: 99%

Section: Continuous Optimizationmentioning

confidence: 99%

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A Tutorial on the Cross-Entropy Method

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“…The CE method is employed to find the optimal control vector c * = {c i , and to update these parameters via CE to produce better performing control vectors in the next iteration [17]. We summarize the approach in the following algorithm.…”

Section: Main Proceduresmentioning

confidence: 99%

“…An alternative approach to deal with constraints on u is to add a penalty component to the objective function. However, it is often difficult to choose a good penalty function [17].…”

Section: Remark 32mentioning

confidence: 99%

Controlling the number of HIV infectives in a mobile population

Sani

Kroese

2008

Mathematical Biosciences

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Abstract. The spread of the human immunodeficiency virus (HIV) depends prominently on the migration of people between different regions. An important consequence of this population mobility is that HIV control strategies that are optimal in a regional sense may not be optimal in a national sense. We formulate various mathematical control problems for HIV spread in mobile heterosexual populations, and show how optimal regional control strategies can be obtained that minimize the national spread of HIV. We apply the cross-entropy method to solve these highly multi-modal and non-linear optimization problems. We demonstrate the effectiveness of the method via a range of experiments and illustrate how the form of the optimal control function depends on the mathematical model used for the HIV spread.

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Cross‐Entropy Method

Kroese

2011

Wiley Encyclopedia of Operations Research and Management Science

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The Cross-Entropy Method for Continuous Multi-Extremal Optimization

Cited by 223 publications

References 17 publications

A Tutorial on the Cross-Entropy Method

A Tutorial on the Cross-Entropy Method

Controlling the number of HIV infectives in a mobile population

Cross‐Entropy Method

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