Ising图模型概率推理的计算复杂性

图模型概率推理的主要任务是通过对联合概率分布进行变量求和来计算配分函数、变量边缘概率分布、条件 概率分布等。图模型概率推理计算复杂性及近似概率推理的计算复杂性是一重要的理论问题,也是设计概率推理算 法和近似概率推理算法的理论基础。研究了Ising图模型概率推理的计算复杂性,包括概率推理的难解性及不可近似 性。具体地,通过构建#2 SA"I'问题到Icing图模型概率推理问题的多项式时间计数归约,证明在一般 Ising图模型上 计算配分函数、变量边缘概率分布、条件概率分布的概率推理问题是#P难的,同时证明Icing图模型近似概率推理问 题是NP难的,即一般Icing图模型上的概率推理问题是难解且不可近似的。     

Ising图模型概率推理的参数化复杂性

Ising图模型概率推理的主要工作是通过变量求和来计算配分函数和边缘概率分布。传统计算复杂性理论证明Ising图模型精确概率推理是NP难的,并且Ising图模型近似概率推理是NP难的。研究了Ising图模型精确概率推理和Ising均值场近似概率推理的参数化复杂性。首先证明了不同参数的Ising图模型概率推理的参数化复杂性定理,指出基于变量个数或图模型树宽的参数化概率推理问题是固定参数可处理的。然后证明了Ising均值场的参数化复杂性定理,指出基于自由分布树宽、迭代次数和变量个数的参数化Icing均值场是固定参数可处理的;进一步,当Ising图模型参数满足Ising均值场迭代式压缩条件时,基于自由分布树宽和迭代次数的参数化Ising均值场是固定参数可处理的。     

A comparative study on reliability-index and target-performance-based probabilistic structural design optimization

Probabilistic structural design optimization enables designers and engineers to quantitatively take into account the uncertainties observed in the structural and environmental properties. In this paper, two approaches to determine the satisfaction of probabilistic constraints are discussed. One is the conventional reliability-index-based approach and the other is a more recently proposed target-performance-based approach. An algorithm, which detects and eliminates the excessive zigzagging iterations during the searches for the most probable failure point and the minimum performance target point, was incorporated.The number of iterations required by the two approaches was investigated in three examples: a cantilever beam, a three-bar truss and a ten-bar truss structure. Based on the results, the target-performance-based approach was found to be superior to the reliability-index-based one in view of both computational efficiency and numerical stability.     

An efficient method of solving design optimization problems with arbitrary random distributions in gradient-based design spaces

Reliability-Based Design Optimization (RBDO) algorithms, such as Reliability Index Approach (RIA) and Performance Measure Approach (PMA), have been developed to solve engineering optimization problems under design uncertainties. In some existing methods, the random design space is transformed to standard normal design space and the reliability assessment, such as reliability index from RIA or performance measure from PMA, is estimated in order to evaluate the failure probability. When the random variable is arbitrarily distributed and cannot be properly fitted to any known form of probability density function, the existing RBDO methods cannot perform reliability analysis in the original design space. This paper proposes a novel Ensemble of Gradient-based Transformed Reliability Analyses (EGTRA) to evaluate the failure probability of any arbitrarily distributed random variables in the original design space. The arbitrary distribution of the random variable is approximated by a merger of multiple Gaussian kernel functions in a single-variate coordinate that is directed toward the gradient of the constraint function. The failure probability is then estimated using the ensemble of each kernel reliability analysis. This paper further derives a linearly approximated probabilistic constraint at the design point with allowable reliability level in the original design space using the aforementioned fundamentals and techniques. Numerical examples with generated random distributions show that existing RBDO algorithms can improperly approximate the uncertainties as Gaussian distributions and provide solutions with poor assessments of reliabilities. On the other hand, the numerical results show EGTRA is capable of efficiently solving the RBDO problems with arbitrarily distributed uncertainties.     

Variational maximum A posteriori by annealed mean field analysis

This paper proposes a novel probabilistic variational method with deterministic annealing for the maximum a posteriori (MAP) estimation of complex stochastic systems. Since the MAP estimation involves global optimization, in general, it is very difficult to achieve. Therefore, most probabilistic inference algorithms are only able to achieve either the exact or the approximate posterior distributions. Our method constrains the mean field variational distribution to be multivariate Gaussian. Then, a deterministic annealing scheme is nicely incorporated into the mean field fix-point iterations to obtain the optimal MAP estimate. This is based on the observation that when the covariance of the variational Gaussian distribution approaches to zero, the infimum point of the Kullback-Leibler (KL) divergence between the variational Gaussian and the real posterior would be the same as the supreme point of the real posterior. Although global optimality may not be guaranteed, our extensive synthetic and real experiments demonstrate the effectiveness and efficiency of the proposed method.     

Classification approach for reliability-based topology optimization using probabilistic neural networks

This research explores the usage of classification approaches in order to facilitate the accurate estimation of probabilistic constraints in optimization problems under uncertainty. The efficiency of the proposed framework is achieved with the combination of a conventional topology optimization method and a classification approach- namely, probabilistic neural networks (PNN). Specifically, the implemented framework using PNN is useful in the case of highly nonlinear or disjoint failure domain problems. The effectiveness of the proposed framework is demonstrated with three examples. The first example deals with the estimation of the limit state function in the case of disjoint failure domains. The second example shows the efficacy of the proposed method in the design of stiffest structure through the topology optimization process with the consideration of random field inputs and disjoint failure phenomenon, such as buckling. The third example demonstrates the applicability of the proposed method in a practical engineering problem.     

自组网Random Direction移动模型点空间概率分布的研究

在无线自组网的研究中，研究移动模型点的空间概率分布是研究自组网协议的仿真评价和分析移动自组网许多相关特性(例如网络连通性、最小生成树、平均路径长度、网络容量等)的重要的基础问题．针对自组网经常使用的Random Direction移动模型运动节点的空间分布进行了研究，给出了一维及二维情况下运动节点空间概率分布的精确公式．研究结果为基于Random Direction移动模型的证明、仿真与应用提供了严格的理论依据．     

Structural topology synthesis with dynamics and nonlinearities using equivalent linear systems

Topology optimization for nonlinear and dynamic problems is expensive because of the necessity to solve the equations of motion at every optimization iteration in order to evaluate the objective function and constraints. In this work, an iterative methodology is developed using the concept of an equivalent linear system for the topology synthesis of structures undergoing nonlinear and dynamic response, using minimal nonlinear response evaluations. The approach uses equivalent loads obtained from nonlinear dynamic analysis to perform optimization iterations, during the course of which the nonlinear and dynamic system is continuously approximated. In this process, the optimization is decoupled from the nonlinear dynamic analysis. Results are presented for various kinds of nonlinear and dynamic problems showing the effectiveness of the developed approach.     

Conservative reliability-based design optimization method with insufficient input data

Reliability analysis and reliability-based design optimization (RBDO) require an exact input probabilistic model to obtain accurate probability of failure (PoF) and RBDO optimum design. However, often only limited input data is available to generate the input probabilistic model in practical engineering problems. The insufficient input data induces uncertainty in the input probabilistic model, and this uncertainty forces the PoF to be uncertain. Therefore, it is necessary to consider the PoF to follow a probability distribution. In this paper, the probability of the PoF is obtained with consecutive conditional probabilities of input distribution types and parameters using the Bayesian approach. The approximate conditional probabilities are obtained under reasonable assumptions, and Monte Carlo simulation is applied to calculate the probability of the PoF. The probability of the PoF at a user-specified target PoF is defined as the conservativeness level of the PoF. The conservativeness level, in addition to the target PoF, will be used as a probabilistic constraint in an RBDO process to obtain a conservative optimum design, for limited input data. Thus, the design sensitivity of the conservativeness level is derived to support an efficient optimization process. Using numerical examples, it is demonstrated that the conservativeness level should be involved in RBDO when input data is limited. The accuracy and efficiency of the proposed design sensitivity method is verified. Finally, conservative RBDO optimum designs are obtained using the developed methods for limited input data problems.     

System reliability based vehicle design for crashworthiness and effects of various uncertainty reduction measures

Reliability-based design optimization of automobile structures for crashworthiness has been studied by many researchers by using either single component probabilistic constraints or single failure mode based probabilistic constraints, while system reliability considerations are mostly disregarded. In this paper, we perform system reliability based design optimization (SRBDO) of an automobile for crashworthiness and analyze the effect of reliability allocation in different failure modes. In addition, effects of various uncertainty reduction measures (e.g., reducing variability in material properties, reducing error of finite element analysis) are investigated and tradeoff plots of uncertainty reduction, system reliability and structural weight are generated. These types of tradeoff plots can be used by a company manager to decide whether to allocate the company resources for employing uncertainty reduction measures or allocating the resources for the excess weight to protect against the unreduced uncertainties. Furthermore, relative importance of automobile structural members in different crash scenarios is quantified. Submitted for publication in the Structural and Multidisciplinary Optimization (SMO).     

Inverse propagation of uncertainties in finite element model updating through use of fuzzy arithmetic

A fuzzy finite element model updating (FFEMU) method is presented in this study for the damage detection problem. The uncertainty caused by the measurement noise in modal parameters is described by fuzzy numbers. Inverse analysis is formulated as a constrained optimization problem at each α-cut level. Membership functions of each updating parameter which correspond to reduction in bending stiffness of the finite elements is determined by minimizing an objective function using a hybrid version of genetic algorithms (GA) and particle swarm optimization method (PSO) which is very efficient in terms of accuracy and robustness. Practical evaluation of the approximate bounds of the interval modal parameters in FFEMU iterations is addressed. A probabilistic analysis is performed using Monte Carlo simulation (MCS) and the results are compared with presented FFEMU method. It is apparent from numerical simulations that the proposed method is well capable in finding the membership functions of the updating parameters within reasonable accuracy. It is also shown that the results obtained by FFEMU are in good agreement with the MCS results while FFEMU is not as computationally expensive as the MCS method. Nevertheless, the proposed FFEMU do not required derivatives of the objective function like existing methods except in the deterministic case.     

Optimization of forging processes using Finite Element simulations

During the last decades, simulation software based on the Finite Element Method (FEM) has significantly contributed to the design of feasible forming processes. Coupling FEM to mathematical optimization algorithms offers a promising opportunity to design optimal metal forming processes rather than just feasible ones. In this paper Sequential Approximate Optimization (SAO) for optimizing forging processes is discussed. The algorithm incorporates time-consuming nonlinear FEM simulations. Three variants of the SAO algorithm—which differ by their sequential improvement strategies—have been investigated and compared to other optimization algorithms by application to two forging processes. The other algorithms taken into account are two iterative algorithms (BFGS and SCPIP) and a Metamodel Assisted Evolutionary Strategy (MAES). It is essential for sequential approximate optimization algorithms to implement an improvement strategy that uses as much information obtained during previous iterations as possible. If such a sequential improvement strategy is used, SAO provides a very efficient algorithm to optimize forging processes using time-consuming FEM simulations.     

Simulation based optimization of stochastic systems with integer design variables by sequential multipoint linear approximation

Optimization problems are considered for which objective function and constraints are defined as expected values of stochastic functions that can only be evaluated at integer design variable levels via a computationally expensive computer simulation. Design sensitivities are assumed not to be available. An optimization approach is proposed based on a sequence of linear approximate optimization subproblems. Within each search subregion a linear approximate optimization subproblem is built using response surface model building. To this end, N simulation experiments are carried out in the search subregion according to a D-optimal experimental design. The linear approximate optimization problem is solved by integer linear programming using corrected constraint bounds to account for any uncertainty due to the stochasticity. Each approximate optimum is evaluated on the basis of M simulation replications with respect to objective function change and feasibility of the design. The performance of the optimization approach and the influence of parameters N and M is illustrated via two analytical test problems. A third example shows the application to a production flow line simulation model. Received April 28, 2000     

Component and system reliability-based topology optimization using a single-loop method

We perform reliability-based topology optimization by combining reliability analysis and material distribution topology design methods to design linear elastic structures subject to random inputs, such as random loadings. Both component reliability and system reliability are considered. In component reliability, we satisfy numerous probabilistic constraints which quantify the failure of different events. In system reliability, we satisfy a single probabilistic constraint which encompasses the component events. We adopt the first-order reliability method to approximate the component reliabilities and the inclusion-exclusion rule to approximate the system reliability. To solve the probabilistic optimization problem, we use a variant of the single loop method, which eliminates the need for an inner reliability analysis loop. The proposed method is amenable to implementation with existing deterministic topology optimization software, and hence suitable for practical applications. Designs obtained from component and system reliability-based topology optimization are compared to those obtained from traditional deterministic topology optimization and validated via Monte Carlo simulation.     

概率图模型研究进展综述

概率图模型作为一类有力的工具,能够简洁地表示复杂的概率分布,有效地(近似)计算边缘分布和条件分布,方便地学习概率模型中的参数和超参数.因此,它作为一种处理不确定性的形式化方法,被广泛应用于需要进行自动的概率推理的场合,例如计算机视觉、自然语言处理.回顾了有关概率图模型的表示、推理和学习的基本概念和主要结果,并详细介绍了这些方法在两种重要的概率模型中的应用.还回顾了在加速经典近似推理算法方面的新进展.最后讨论了相关方向的研究前景.     

Single-loop system reliability-based topology optimization considering statistical dependence between limit-states

This paper presents a single-loop algorithm for system reliability-based topology optimization (SRBTO) that can account for statistical dependence between multiple limit-states, and its applications to computationally demanding topology optimization (TO) problems. A single-loop reliability-based design optimization (RBDO) algorithm replaces the inner-loop iterations to evaluate probabilistic constraints by a non-iterative approximation. The proposed single-loop SRBTO algorithm accounts for the statistical dependence between the limit-states by using the matrix-based system reliability (MSR) method to compute the system failure probability and its parameter sensitivities. The SRBTO/MSR approach is applicable to general system events including series, parallel, cut-set and link-set systems and provides the gradients of the system failure probability to facilitate gradient-based optimization. In most RBTO applications, probabilistic constraints are evaluated by use of the first-order reliability method for efficiency. In order to improve the accuracy of the reliability calculations for RBDO or RBTO problems with high nonlinearity, we introduce a new single-loop RBDO scheme utilizing the second-order reliability method and implement it to the proposed SRBTO algorithm. Moreover, in order to overcome challenges in applying the proposed algorithm to computationally demanding topology optimization problems, we utilize the multiresolution topology optimization (MTOP) method, which achieves computational efficiency in topology optimization by assigning different levels of resolutions to three meshes representing finite element analysis, design variables and material density distribution respectively. The paper provides numerical examples of two- and three-dimensional topology optimization problems to demonstrate the proposed SRBTO algorithm and its applications. The optimal topologies from deterministic, component and system RBTOs are compared with one another to investigate the impact of optimization schemes on final topologies. Monte Carlo simulations are also performed to verify the accuracy of the failure probabilities computed by the proposed approach.     

Hybrid ant colony optimization algorithm applied to the multi-depot vehicle routing problem

The article deals with the hybrid Ant Colony Optimization algorithm and its application to the Multi-Depot Vehicle Routing Problem (MDVRP). The algorithm combines both probabilistic and exact techniques. The former implements the bio-inspired approach based on the behaviour of ants in the nature when searching for food together with simulated annealing principles. The latter complements the former. The algorithm explores the search space in a finite number of iterations. In each iteration, the deterministic local optimization process may be used to improve the current solution. Firstly, the key parts and features of the algorithm are presented, especially in connection with the exact optimization process. Next, the article deals with the results of experiments on MDVRP problems conducted to verify the quality of the algorithm; moreover, these results are compared to other state-of-the-art methods. As experiments, Cordreau’s benchmark instances were used. The experiments showed that the proposed algorithm overcomes the other methods as it has the smallest average error (the difference between the found solution and the best known solution) on the entire set of benchmark instances.

     

Sequential probabilistic analytical target cascading method for hierarchical multilevel optimization under uncertainty

Probabilistic Analytical Target Cascading (PATC) is a methodology for hierarchical multilevel optimization under uncertainty. In PATC, the statisticalmoments of the stochastic interrelated responses are matched between neighbouring levels to ensure the consistency of the solution. When the interrelated response is far from normal distribution, high order moments may need to be matched in the PATC formulation, which results in great computational difficulty. To overcome this disadvantage, a sequential PATC (SPATC) approach is proposed in this paper. SPATC firstly decouples the original probabilistic design problem into deterministic optimization problem and probabilistic analysis, and then hierarchically decomposes them into subproblems. The statistical information matching between neighbouring levels in the existing PATC framework is eliminated in SPATC. All in one probabilistic analysis and hierarchical probabilistic analysis are established to calculate the probabilistic characteristic of the interrelated responses and linking variables. Three examples are used to demonstrate the effectiveness and efficiency of the proposed SPATC approach. The results show that the SPATC approach is more efficient and accurate than PATC, especially for the multilevel design problem with non-normal interrelated responses.     

一种实数编码多目标贝叶斯优化算法

提出了一种采用基于决策树概率模型表示各变量之间条件相关性的分布估算算法:实数编码多目标贝叶斯优化算法(RCMBOA)。通过构建这样的概率模型,继而对模型进行抽样以产生新个体。再对生成的新个体进行变异操作,以提高算法的搜索能力,增加种群的多样性。这种生成新个体的方法结合非劣分层与截断选择机制,可以很好地逼近多目标问题的Pareto前沿。同时,在进行截断选择时,每次只删除一个排挤距离小的个体,之后重新估算个体的排挤距离,以获得分布均匀的非劣解集。对于约束多目标优化问题,算法采用带约束支配关系判别个体的优劣。用该算法对8个较难的测试问题进行了优化计算,获得的非劣解集与NSGA-II算法得到的相比,非劣解集的质量更高,分布更为均匀。计算结果说明RCMBOA是一种有效、鲁棒的多目标优化算法。