Quadratically constrained minimum cross-entropy analysis

Quadratically constrained minimum cross-entropy problem has recently been studied by Zhang and Brockett through an elaborately constructed dual. In this paper, we take a geometric programming approach to analyze this problem. Unlike Zhang and Brockett, we separate the probability constraint from general quadratic constraints and use two simple geometric inequalities to derive its dual problem. Furthermore, by using the dual perturbation method, we directly prove the strong duality theorem and derive a dual-to-primal conversion formula. As a by-product, the perturbation proof gives us insights to develop a computation procedure that avoids dual non-differentiability and allows us to use a general purpose optimizer to find an-optimal solution for the quadratically constrained minimum cross-entropy analysis.     

Stochastic Order and Generalized Weighted Mean Invariance

In this paper, we present order invariance theoretical results for weighted quasi-arithmetic means of a monotonic series of numbers. The quasi-arithmetic mean, or Kolmogorov–Nagumo mean, generalizes the classical mean and appears in many disciplines, from information theory to physics, from economics to traffic flow. Stochastic orders are defined on weights (or equivalently, discrete probability distributions). They were introduced to study risk in economics and decision theory, and recently have found utility in Monte Carlo techniques and in image processing. We show in this paper that, if two distributions of weights are ordered under first stochastic order, then for any monotonic series of numbers their weighted quasi-arithmetic means share the same order. This means for instance that arithmetic and harmonic mean for two different distributions of weights always have to be aligned if the weights are stochastically ordered, this is, either both means increase or both decrease. We explore the invariance properties when convex (concave) functions define both the quasi-arithmetic mean and the series of numbers, we show its relationship with increasing concave order and increasing convex order, and we observe the important role played by a new defined mirror property of stochastic orders. We also give some applications to entropy and cross-entropy and present an example of multiple importance sampling Monte Carlo technique that illustrates the usefulness and transversality of our approach. Invariance theorems are useful when a system is represented by a set of quasi-arithmetic means and we want to change the distribution of weights so that all means evolve in the same direction.     

求解全局最优问题的多重点样本水平值估计的相对熵算法

研究有界闭箱约束下的全局最优化问题,利用相对熵及广义方差函数方程的最大根与全局最小值之间的等价关系,设计求解全局最优值的积分型水平值估计算法.对采用重点样本采样技巧产生的函数值按一定规则进行聚类,从而在各聚类中产生的若干新重点样本,结合相对熵算法,构造出多重点样本进行全局搜索的新算法.该算法的优点在于每次迭代选用当前较好的函数值信息,以达到随机搜索到更好的函数值信息.同时多重点样本可有利挖掘出更好的全局信息.一系列的数值实验表明该算法是非常有效的.     

Application of the Cross-Entropy Method to the Buffer Allocation Problem in a Simulation-Based Environment

The buffer allocation problem (BAP) is a well-known difficult problem in the design of production lines. We present a stochastic algorithm for solving the BAP, based on the cross-entropy method, a new paradigm for stochastic optimization. The algorithm involves the following iterative steps: (a) the generation of buffer allocations according to a certain random mechanism, followed by (b) the modification of this mechanism on the basis of cross-entropy minimization. Through various numerical experiments we demonstrate the efficiency of the proposed algorithm and show that the method can quickly generate (near-)optimal buffer allocations for fairly large production lines.     

The Role of Information in Managing Interactions from a Multifractal Perspective

In the framework of the multifractal hydrodynamic model, the correlations informational entropy–cross-entropy manages attractive and repulsive interactions through a multifractal specific potential. The classical dynamics associated with them imply Hubble-type effects, Galilei-type effects, and dependences of interaction constants with multifractal degrees at various scale resolutions, while the insertion of the relativistic amendments in the same dynamics imply multifractal transformations of a generalized Lorentz-type, multifractal metrics invariant to these transformations, and an estimation of the dimension of the multifractal Universe. In such a context, some correspondences with standard cosmologies are analyzed. Since the same types of interactions can also be obtained as harmonics mapping between the usual space and the hyperbolic plane, two measures with uniform and non-uniform temporal flows become functional, temporal measures analogous with Milne’s temporal measures in a more general manner. This work furthers the analysis published recently by our group in “Towards Interactions through Information in a Multifractal Paradigm”.     

Enhanced Slime Mould Algorithm for Multilevel Thresholding Image Segmentation Using Entropy Measures

Image segmentation is a fundamental but essential step in image processing because it dramatically influences posterior image analysis. Multilevel thresholding image segmentation is one of the most popular image segmentation techniques, and many researchers have used meta-heuristic optimization algorithms (MAs) to determine the threshold values. However, MAs have some defects; for example, they are prone to stagnate in local optimal and slow convergence speed. This paper proposes an enhanced slime mould algorithm for global optimization and multilevel thresholding image segmentation, namely ESMA. First, the Levy flight method is used to improve the exploration ability of SMA. Second, quasi opposition-based learning is introduced to enhance the exploitation ability and balance the exploration and exploitation. Then, the superiority of the proposed work ESMA is confirmed concerning the 23 benchmark functions. Afterward, the ESMA is applied in multilevel thresholding image segmentation using minimum cross-entropy as the fitness function. We select eight greyscale images as the benchmark images for testing and compare them with the other classical and state-of-the-art algorithms. Meanwhile, the experimental metrics include the average fitness (mean), standard deviation (Std), peak signal to noise ratio (PSNR), structure similarity index (SSIM), feature similarity index (FSIM), and Wilcoxon rank-sum test, which is utilized to evaluate the quality of segmentation. Experimental results demonstrated that ESMA is superior to other algorithms and can provide higher segmentation accuracy.     

基于混合权重和VIKOR的黑启动方案评估

黑启动作为电力体系安全防御和事故后快速恢复措施之一,对其路径的评估是黑启动辅助决策的一个重要组成部分。本文将一种利用最小叉熵准则集成组合权重的思想运用到黑启动方案评估上。首先利用可变熵模型,在克服了经典熵权法权重分配差别过大、权重无法体现评价矩阵微小变化等问题的基础上,依据各指标客观数据信息的差异得到可变熵权重;接着又尝试性引入决策者效用函数,将原始属性评价矩阵转换为带有决策者主观偏好的判断矩阵,结合Kullback-Leibler距离模型得出带有决策者意愿的偏好权重,再根据最小叉熵准则,对可变熵权重以及偏好权重进行集成,求得指标综合权重。最后利用VIKOR法对方案间关系进行细致分析,得到最优方案。采用天津电网黑启动数据进行了验证,验证结果表明了所提方法的有效性。     

一种新的求总极值的水平值估计算法

提出了一种求解总极值问题的新水平值估计算法. 为此, 引入一类变差函数并研究它的性质; 给出基于变差函数的全局最优性条件, 并构造出一种求总极值的水平值估计算法. 为了实现这种算法, 采用了基于重点样本技术的Monte-Carlo方法来计算变差,并利用相对熵算法的主要思想更新取样密度.初步的数值实验说明了算法的有效性.     

Stochastic level-value approximation for quadratic integer convex programming

We propose a stochastic level value approximation method for a quadratic integer convex minimizing problem in this paper. This method applies an importance sampling technique, and make use of the cross-entropy method to update the sample density functions. We also prove the asymptotic convergence of this algorithm, and report some numerical results to illuminate its effectiveness.     

一种无约束全局优化的水平值下降算法

本文研究无约束全局优化问题,建立了一种新的水平值下降算法(Level-value Descent Method,LDM).讨论并建立了概率意义下取全局最小值的一个充分必要条件,证明了算法LDM是依概率测度收敛的.这种LDM算法是基于重点度取样(Improtance Sampling)和Markov链Monte-Carlo随机模拟实现的,并利用相对熵方法(TheCross-Entropy Method)自动更新取样密度,算例表明LDM算法具有较高的数值精度和较好的全局收敛性.