Semi-Iterative Minimum Cross-Entropy Algorithms for Rare-Events,Counting, Combinatorial and Integer Programming

We present a new generic minimum cross-entropy method, called the semi-iterative MinxEnt, or simply SME, for rare-event probability estimation, counting, and approximation of the optimal solutions of a broad class of NP-hard linear integer and combinatorial optimization problems (COP’s). The main idea of our approach is to associate with each original problem an auxiliary single-constrained convex MinxEnt program of a special type, which has a closed-form solution. We prove that the optimal pdf obtained from the solution of such a specially designed MinxEnt program is a zero variance pdf, provided the “temperature” parameter is set to minus infinity. In addition we prove that the parametric pdf based on the product of marginals obtained from the optimal zero variance pdf coincides with the parametric pdf of the standard cross-entropy (CE) method. Thus, originally designed at the end of 1990s as a heuristics for estimation of rare-events and COP’s, CE has strong connection with MinxEnt, and thus, strong mathematical foundation. The crucial difference between the proposed SME method and the standard CE counterparts lies in their simulation-based versions: in the latter we always require to generate (via Monte Carlo) a sequence of tuples including the temperature parameter and the parameter vector in the optimal marginal pdf’s, while in the former we can fix in advance the temperature parameter (to be set to a large negative number) and then generate (via Monte Carlo) a sequence of parameter vectors of the optimal marginal pdf’s alone. In addition, in contrast to CE, neither the elite sample no the rarity parameter is needed in SME. As result, the proposed SME algorithm becomes simpler, faster and at least as accurate as the standard CE. Motivated by the SME method we introduce a new updating rule for the parameter vector in the parametric pdf of the CE program. We show that the CE algorithm based on the new updating rule, called the combined CE (CCE), is at least as fast and accurate as its standard CE and SME counterparts. We also found numerically that the variance minimization (VM)-based algorithms are the most robust ones. We, finally, demonstrate numerically that the proposed algorithms, and in particular the CCE one, allows accurate estimation of counting quantities up to the order of hundred of decision variables and hundreds of constraints. This research was supported by the Israel Science Foundation (grant No 191-565).     

Background risk,bivariate risk attitudes,and optimal prevention

This paper extends Eeckhoudt et al.’s (2012) results for precautionary effort to bivariate utility function framework. We establish an equivalence between the agent’s precautionary effort motive and the signs of successive cross-derivatives of the bivariate utility function. We show that the introduction (or deterioration) of an independent background risk induces more prevention to protect against wealth loss provided the individual exhibits correlation aversion of some given order. The conditions on the individual’s risk preferences are given to generate some specific prevention behaviors in the univariate framework with multiplicative risks. Our conclusion also indicates that an increase in the correlation between wealth risk and background risk leads to a reduction in optimal prevention.     

Managing Stochastic,Finite Capacity,Multi-Project Systems through the Cross-Entropy Methodology

This paper addresses the problem of loading a finite capacity, stochastic (random) and dynamic multi-project system. The system is controlled by keeping a constant number of projects concurrently in the system. A new approach, based on the Cross-Entropy (CE) method, is proposed to determine optimal loading of the system. Through numerical experiments, we demonstrate the CE method performance and show new insights into its behavior in a noisy system. Particularly, we suggest a trade-off between the convergence time, the number of iterations and the noise level. This research was partially supported by the Inga and Hal Marcus Research Fund     

基于可变模糊模型的非毗邻地区经济协调发展研究——以中俄豫伊两地为例

区域经济协调性是区域经济协同发展和可持续发展的基本保障,也是区域间经济合作和贸易的基础.以区域经济系统为研究对象,利用可变模糊识别模型对中俄豫伊两地2013-2019年的经济协调度发展进行了定量分析,探求两地经济协调发展的静态及动态机制.研究表明,近7年两地区经济经历了动态平衡到协调的调整和转变过程,向良好协调的状况发...     

Value semigroups,value quantales,and positivity domains

In 1981 and 1997 Kopperman and Flagg, respectively, proved that every topological space is metrisable, provided the symmetry and separation axioms are removed from the requirements on the metric, and the metric is allowed to take values in, respectively, a value semigroup or a value quantale. Seeking to construct a value quantale from a value semigroup we focus on a small portion of the structure present in a value semigroup, comprising what we call a positivity domain, and we construct its enveloping value quantale, forming part of a detailed comparison between value semigroups and value quantales. We obtain a representation theorem for value quantales in terms of positivity domains, and we outline how products of positivity domains can be used in the theory of continuity spaces instead of (the non-existent) products of value quantales.     

A vanishing viscosity approach to the evolution of microstructures in finite plasticity

Material microstructures in finite single-slip crystal plasticity occur and evolve due to deformation. Their formation is not arbitrary, they tend to form structured spatial patterns. This hints at a universal underlying process. As in the approach of D. Kochmann and K. Hackl, we use a variational framework, focusing on the Lagrange functional to describe the evolving mircrostructure. We modify this approach by introducing a small smooth transition zone between the domains in order to improve the numerical treatment. We present explicit time-evolution equations for the volume fractions and the internal variables. We outline a numerical scheme. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The quaternary complex Hadamard matrices of orders 10, 12, and 14

A complete classification of quaternary complex Hadamard matrices of orders 10, 12 and 14 is given, and a new parametrization scheme for obtaining new examples of affine parametric families of complex Hadamard matrices is provided. On the one hand, it is proven that all 10×10 and 12×12 quaternary complex Hadamard matrices belong to some parametric family, but on the other hand, it is shown by exhibiting an isolated 14×14 matrix that there cannot be a general method for introducing parameters into these types of matrices.     

Nonintersecting Paths, Pfaffians, and ?-Matroids

In this paper, we study -matroids induced by nonintersecting paths in a directed graph. The association between nonintersecting paths and -matroids is derived from Pfaffians. On the one hand, certain numbers ofk-tuples of nonintersecting paths may (often) be expressed as a Pfaffian, while, on the other hand, representability problems for -matroids may be studied in terms of Pfaffians. It is shown that -matroids induced by nonintersecting paths are representable over fields of any characteristic and that weightings defined on the edge set and with values in some linearly ordered abelian group give rise to valuated -matroids.     

Connectivity modeling and optimization of linear consecutively connected systems with repairable connecting elements

Linear consecutively connected systems (LCCSs) are systems containing a linear sequence of ordered nodes. Connection elements (CE) characterized by diverse connection ranges, time-to-failure and time-to-repair distributions are allocated to different nodes to provide the system connectivity, i.e., a connection between the source and sink nodes of the LCCS. Examples of LCCSs abound in practical applications such as flow transmission systems and radio communication systems. Considerable research efforts have been expended in modeling and optimizing LCCSs. However, most of the existing works have assumed that CEs either are non-repairable or undergo a restrictive minimal repair policy with constant repair time. This paper makes new technical contributions by modeling and optimizing LCCSs with CEs under corrective maintenance with random repair time and different repair policies (minimal, perfect, and imperfect). The characteristics of CEs can depend on their location because the distance between adjacent nodes and conditions of CE operation and maintenance at different nodes can be different, which further complicates the problem. We first propose a discrete numerical algorithm to evaluate the instantaneous availability of each CE. A universal generating function based method is then implemented for assessing instantaneous and expected system connectivity for a specific CE allocation. As the CE allocation can have significant impacts on the system connectivity, we further define and solve the optimal CE allocation problem, whose objective is to find the CE allocation among LCCS nodes maximizing the expected system connectivity over a given mission time. Effects of different parameters including repair efficiency, mission time and repair time are investigated. As illustrated through examples, optimization results can facilitate optimal decisions on robust design and effective operation and maintenance managements of LCCSs.     

On Malliavin measures,SLE, and CFT

This paper is dedicated to the 70th birthday of V. Arnol’d. The paper is motivated by emerging connections between the conformal field theory (CFT) on the one hand and stochastic Löwner evolution (SLE) processes and measures that play the role of the Haar measures for the diffeomorphism group of a circle, on the other hand. We attempt to build a framework for widely spread beliefs that SLE processes would provide a picture of phase separation in a small massive perturbation of the CFT.     

On posets and independence spaces

By constructing a correspondence relationship between independence spaces and posets, under isomorphism, this paper characterizes loopless independence spaces and applies this characterization to reformulate certain results on independence spaces in poset frameworks. These state that the idea provided in this paper is a new approach for the study of independence spaces. We outline our future work finally.     

Individual irrationality,network structure,and collective intelligence: An agent‐based simulation approach

An agent‐based model is constructed to study the effects of individual irrationality and network structure on collective performance. We find that individual irrationality and network density are the most influential factors. Moderate degree of irrationality results in moderate knowledge unity and superior quality of knowledge; increasing network connections or decreasing average path length (APL) can promote quality of knowledge integration and knowledge unity. Furthermore, APL is more influential than clustering coefficient and centralization (CE). Less clustering may contribute to higher quality of knowledge integration, while higher CE may contribute to higher knowledge unity. © 2015 Wiley Periodicals, Inc. Complexity 21: 44–54, 2016     

Mpi-空间与偏序集

通过建立mpi-空间和偏序集之间的对应关系,在同构意义下,得到无环mpi-空间的特征.利用这种持征,mpi-空间的一些结果可以转化到偏序集框架结构下进行研究.这些成果清楚地表明这里所提供的思想是研究mpi-空间的一个新方法.最后,概述了我们未来的一些工作.     

A Phase Field Model for Ductile to Brittle Failure Mode Transition

The computational modeling of failure mechanisms in solids due to fracture based on sharp crack discontinuities suffers in dynamic problems with complex crack topologies. This can be overcome by a diffusive crack modeling based on the introduction of a crack phase field. We outline a conceptual framework for phase field models of crack propagation in brittle elastic and ductile elastic-plastic solids under dynamic loading and investigate the ductile to brittle failure mode transition observed in the experiment performed by Kalthoff and Winkeler [3]. We develop incremental variational principles and consider their numerical implementations by multi-field finite element methods. To this end, we define energy storage and dissipation functions for the plastic flow including the fracture phase field. The introduction of local history fields that drive the evolution of the crack phase field inspires the construction of robust operator split schemes. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis and modeling of the hydro-mechanics of deformable fractures

In the present contribution, we investigate and model the flow of a viscous fluid into a single deformable fracture and the associated elastic deformation of the surrounding porous rock. Due to the coupled nature of the problem, conceptually and technically different strategies can be applied to model and solve the resulting hydro-mechanical system. Therefore, two different modeling approaches are analyzed. On the one hand, we derive the governing equations from the conservation of mass avoiding any of the frequently used approximations or simplifications. On the other hand, we describe the physics based on Biot's quasi-static poroelastic equations. We compare and critically discuss the results obtained with the presented different solution approaches. The results show that both models are able to capture the same physical effects, also those caused by the fracture deformation. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

CE equivalence and shape equivalence of 1-dimensional compacta

In this paper the relationship between CE equivalence and shape equivalence for locally connected, 1-dimensional compacta is investigated. Two theorems are proved. The first asserts that every path connected planar continuum is CE equivalent either to a bouquet of circles or to the Hawaiian earring. The second asserts that for every locally connected, 1-dimensional continuum X there is a cell-like map of X onto a planar continuum. It follows that CE equivalence and shape equivalence are the same for the class of all locally connected, 1-dimensional compacta. In addition, an example of Ferry is generalized to show that for every n⩾1 there exists an n-dimensional, LCⁿ⁻² continuum Y such that Sh(Y)=Sh(S¹) but Y is not CE equivalent to S¹.     

Extremal indices, geometric ergodicity of Markov chains, and MCMC

We investigate the connections between extremal indices on the one hand and stability of Markov chains on the other hand. Both theories relate to the tail behaviour of stochastic processes, and we find a close link between the extremal index and geometric ergodicity. Our results are illustrated throughout with examples from simple MCMC chains.