Kolmogorov-conditions for vectorial approximation problems

A general vectorial best approximation problem in linear and locally convex topological spaces, respectively, is considered. The approximation is based on socalled vectorial norms. For efficient, weakly efficient and strongly efficient solutions sufficient optimality conditions which can be interpreted as generalized Kolmogorov-conditions are obtained in vectorial as well as scalarized form.     

Propagation of vectorial Lorentz beam beyond the paraxial approximation

Based on the vectorial Rayleigh–Sommerfeld integrals, the analytical propagation expression of vectorial Lorentz beam beyond paraxial approximation is presented. The far field expression and the scalar paraxial result are obtained as special cases of the general formulae. According to the analytical representation, the light intensity distribution of vectorial Lorentz beam is depicted at the different reference planes. This research provides an approach to further investigate the propagation of Lorentz beam beyond the paraxial approximation.     

Stability results for approximately efficient solutions

We introduce a concept for approximately efficient solutions in vector optimization and compare it with another recent concept given in [8]. Further, we study relations between the set of approximately efficient solutions of a vector optimization problem and the approximate solutions of a corresponding parametric surrogate optimization problem. Finally, we prove stability properties for the scalar surrogate problem.     

Kolmogorov conditions for vectorial optimization problems

Necessary and sufficient conditions of Kolmogorov type are given for characterizing efficient, weakly efficient, and minimal solutions of vectorial optimization problems.     

一类非凸集值优化问题的对偶定理

对偶理论是数学规划研究领域的重点问题之一,通对偶模型可以实现一个最小化问题与一个最大化问题之间的相互转化.本文的目的是建立一类非凸约束集值优化问题的对偶理论,在逼近多值函数定义的不变凸性假设下,研究了原集值优化问题的Mond-Weir型和Wolfe型对偶问题.利用分析的方法,本文得到了两种对偶模型下关于弱极小元的弱对偶定理,强对偶定理和逆对偶定理.这些对偶定理揭示了原问题与所讨论的Mond-Weir型和Wolfe型对偶问题之间存在着明确的对偶关系.本文所得结果丰富和深化了集值优化理论及其应用的研究内容.     

锥-逼近多值函数和集值优化的近似解

最优性条件和对偶理论是集值向量优化研究领域的重点问题之一．本文的目的是建立一类广义凸集值优化的最优性条件和对偶定理,在锥-逼近多值函数概念的基础上,定义集值映射的一类新的广义不变凸性,称之为次不变凸集值映射,在这类广义凸性假设下,研究最优性条件和对偶定理．利用分析的方法,本文得到了集值优化问题关于弱近似极小元的一个最优性充分条件,以及Mond-Weir和Wolfe两种模型下的弱对偶定理、强对偶定理和逆对偶定理．本文所得结果丰富和深化了集值优化理论及其应用的研究内容．     

A quadratic approximation for structural topology optimization

In topology optimization, it is customary to use reciprocal‐like approximations, which result in monotonically decreasing approximate objective functions. In this paper, we demonstrate that efficient quadratic approximations for topology optimization can also be derived, if the approximate Hessian terms are chosen with care. To demonstrate this, we construct a dual SAO algorithm for topology optimization based on a strictly convex, diagonal quadratic approximation to the objective function. Although the approximation is purely quadratic, it does contain essential elements of reciprocal‐like approximations: for self‐adjoint problems, our approximation is identical to the quadratic or second‐order Taylor series approximation to the exponential approximation. We present both a single‐point and a two‐point variant of the new quadratic approximation. Copyright © 2009 John Wiley & Sons, Ltd.     

一类不可微广义凸多目标规划的最优性条件和对偶

本文研究带不等式约束的不可微多目标规划问题,引入了广义d-I型一致不变凸函数的概念,证明了Pareto有效解和Pareto弱有效解的Karush-Kuhn-Tucker型充分条件.构造出了混合对偶模型,并证明了相应的对偶定理.     

Local Pareto approximation for multi-objective optimization

The design process of complex systems often resorts to solving an optimization problem, which involves different disciplines and where all design criteria have to be optimized simultaneously. Mathematically, this problem can be reduced to a vector optimization problem. The solution of this problem is not unique and is represented by a Pareto surface in the objective function space. Once a Pareto solution is obtained, it may be very useful for the decision-maker to be able to perform a quick local approximation in the vicinity of this Pareto solution for sensitivity analysis. In this article, new linear and quadratic local approximations of the Pareto surface are derived and compared to existing formulas. The case of non-differentiable Pareto points (solutions) in the objective space is also analysed. The concept of a local quick Pareto analyser based on local sensitivity analysis is proposed. This Pareto analysis provides a quantitative insight into the relation between variations of the different objective functions under constraints. A few examples are considered to illustrate the concept and its advantages.     

A ‘best points’ interpolation method for efficient approximation of parametrized functions

We present an interpolation method for efficient approximation of parametrized functions. The method recognizes and exploits the low‐dimensional manifold structure of the parametrized functions to provide good approximation. Basic ingredients include a specific problem‐dependent basis set defining a low‐dimensional representation of the parametrized functions, and a set of ‘best interpolation points’ capturing the spatial‐parameter variation of the parametrized functions. The best interpolation points are defined as solution of a least‐squares minimization problem which can be solved efficiently using standard optimization algorithms. The approximation is then determined from the basis set and the best interpolation points through an inexpensive and stable interpolation procedure. In addition, an a posteriori error estimator is introduced to quantify the approximation error and requires little additional cost. Numerical results are presented to demonstrate the accuracy and efficiency of the method. Copyright © 2007 John Wiley & Sons, Ltd.     

An efficient response surface method using moving least squares approximation for structural reliability analysis

The response surface method (RSM) is widely adopted for structural reliability analysis because of its numerical efficiency. However, the RSM is time consuming for large-scale applications and sometimes shows large errors in the calculation of the sensitivity of the reliability index with respect to random variables. In order to overcome these problems, this study proposes an efficient RSM applying a moving least squares (MLS) approximation instead of the traditional least squares approximation generally used in the RSM. The MLS approximation gives higher weight to the experimental points closer to the most probable failure point (MPFP), which allows the response surface function (RSF) to be closer to the limit state function at the MPFP. In the proposed method, a linear RSF is constructed at first and a quadratic RSF is formed using the axial experimental points selected from the reduced region where the MPFP is likely to exist. The RSF is updated successively by adding one new experimental point to the previous set of experimental points. Numerical examples are presented to demonstrate the improved accuracy and computational efficiency of the proposed method compared to the conventional RSM.     

基于鞍点逼近的机械结构可靠性稳健优化设计

将可靠性优化设计理论与可靠性灵敏度分析方法相结合,讨论了机械零部件稳健优化设计的问题.系统地推导了基于鞍点逼近的可靠性灵敏度公式,并把可靠性灵敏度计算结果融入可靠性稳健优化设计模型之中,将可靠性稳健优化设计归结为满足可靠性要求的多目标优化问题.在基本随机参数概率分布已知的前提下,应用鞍点逼近技术,得到极限状态函数的分布函数与概率密度函数,并且将此结果应用到机械零部件的可靠性灵敏度分析中,进而实现了机械零部件的可靠性稳健优化设计.通过与Monte-Carlo方法计算所得的结果相比可知,应用鞍点逼近技术可以迅速、准确地得到机械零部件可靠性稳健设计信息．     

A sequential approximation method using neural networks for engineering design optimization problems

There are three characteristics in engineering design optimization problems: (1) the design variables are often discrete physical quantities; (2) the constraint functions often cannot be expressed analytically in terms of design variables; (3) in many engineering design applications, critical constraints are often ‘pass–fail’, ‘0–1’ type binary constraints. This paper presents a sequential approximation method specifically for engineering optimization problems with the three characteristics. In this method a back-propagation neural network is trained to simulate a rough map of the feasible domain formed by the constraints using a few representative training data. A training data point consists of a discrete design point and whether this design point is feasible or infeasible. Function values of the constraints are not required. A search algorithm then searches for the optimal point in the feasible domain simulated by the neural network. This new design point is checked against the true constraints to see whether it is feasible, and is then added to the training set. The neural network is trained again with this added information, in the hope that the network will better simulate the boundary of the feasible domain of the true optimization problem. Then a further search is made for the optimal point in this new approximated feasible domain. This process continues in an iterative manner until the approximate model locates the same optimal point in consecutive iterations. A restart strategy is also employed so that the method may have a better chance to reach a global optimum. Design examples with large discrete design spaces and implicit constraints are solved to demonstrate the practicality of this method.     

Update-based evolution control: A new fitness approximation method for evolutionary algorithms

Evolutionary algorithms are robust optimization methods that have been used in many engineering applications. However, real-world fitness evaluations can be computationally expensive, so it may be necessary to estimate the fitness with an approximate model. This article reviews design and analysis of computer experiments (DACE) as an approximation method that combines a global polynomial with a local Gaussian model to estimate continuous fitness functions. The article incorporates DACE in various evolutionary algorithms, to test unconstrained and constrained benchmarks, both with and without fitness function evaluation noise. The article also introduces a new evolution control strategy called update-based control that estimates the fitness of certain individuals of each generation based on the exact fitness values of other individuals during that same generation. The results show that update-based evolution control outperforms other strategies on noise-free, noisy, constrained and unconstrained benchmarks. The results also show that update-based evolution control can compensate for fitness evaluation noise.     

Simultaneous distribution and sizing optimization for stiffeners with an improved genetic algorithm with two-level approximation

This article presents an improved genetic algorithm with two-level approximation (GATA) to optimize the distribution and size of stiffeners simultaneously. A novel optimization model of stiffeners, including two kinds of design variables, is established. The first level approximation problem transforms the original implicit problem to an explicit problem which involves the topology and size variables. Then, a genetic algorithm (GA) addresses the mixed variables. The individuals in the GA are coded by topology variables, and when calculating an individual’s fitness, the second level approximation problem is embedded to optimize the size variables. Considering the stiffeners’ optimization, several aspects of the initial GATA are updated, including the relationship between two kinds of variables, the weight and its sensitivity calculation and the GA strategy, to optimize the stiffeners’ size and distribution simultaneously. Numerical examples show that the improved GATA is effective in optimizing the stiffened shells’ topology and size variables simultaneously.     

An adaptive wavelet method for solving mixed-integer dynamic optimization problems with discontinuous controls and application to alkali–surfactant–polymer flooding

This article presents an adaptive rationalized Haar function approximation method to solve dynamic optimization with mixed-integer and discontinuous controls. Three measures are taken to deal with the discontinuity. First, the problem is converted into a multi-stage optimization problem by non-uniform control vector parameterization. Secondly, an adaptive strategy is proposed to regulate the interval division and the order of Haar function vectors. Thirdly, a structure detection method is presented to refine the subintervals, in which adjacent arcs with the same input type are merged into one to modify redundant subintervals. During this approximation solution, the mixed-integer restriction is realized by the integer truncation strategy. Combined with the Hamiltonian function, a validation principle is shown to verify the optimality of the solution. Finally, the proposed method is applied to solve the enhanced oil recovery for alkali–surfactant–polymer flooding. The effectiveness of the method is illustrated through simulation.     

线性空间中一类向量优化问题的最优性条件

在序线性空间中,引入(u,0_2;Y_ )-广义次似凸映射,建立了此映射下的一个择一定理。利用此定理,得到了带广义不等式约束的向量优化问题的最优性必要条件和充分条件。     

A new efficient convergence criterion for reducing computational expense in topology optimization: reducible design variable method

A new efficient convergence criterion, named the reducible design variable method (RDVM), is proposed to save computational expense in topology optimization. There are two types of computational costs: one is to calculate the governing equations, and the other is to update the design variables. In conventional topology optimization, the number of design variables is usually fixed during the optimization procedure. Thus, the computational expense linearly increases with respect to the iteration number. Some design variables, however, quickly converge and some other design variables slowly converge. The idea of the proposed method is to adaptively reduce the number of design variables on the basis of the history of each design variable during optimization. Using the RDVM, those design variables that quickly converge are not considered as design variables for the next iterations. This means that the number of design variables can be reduced to save the computational costs of updating design variables. Then, the iteration will repeat until the number of design variables becomes 0. In addition, the proposed method can lead to faster convergence of the optimization procedure, which indeed is a more significant time saving. It is also revealed that the RDVM gives identical optimal solutions as those by conventional methods. We confirmed the numerical efficiency and solution effectiveness of the RDVM with respect to two types of optimization: static linear elastic minimization, and linear vibration problems with the first eigenvalue as the objective function for maximization. Copyright © 2012 John Wiley & Sons, Ltd.     

A novel approach for efficient stance detection in online social networks with metaheuristic optimization

In the 19th and 20th centuries, social networks have been an important topic in a wide range of fields from sociology to education. However, with the advances in computer technology in the 21st century, significant changes have been observed in social networks, and conventional networks have evolved into online social networks. The size of these networks, along with the large amount of data they generate, has introduced new social networking problems and solutions. Social network analysis methods are used to understand social network data. Today, several methods are implemented to solve various social network analysis problems, albeit with limited success in certain problems. Thus, the researchers develop new methods or recommend solutions to improve the performance of the existing methods. In the present paper, a novel optimization method that aimed to classify social network analysis problems was proposed. The problem of stance detection, an online social network analysis problem, was first tackled as an optimization problem. Furthermore, a new hybrid metaheuristic optimization algorithm was proposed for the first time in the current study, and the algorithm was compared with various methods. The analysis of the findings obtained with accuracy, precision, recall, and F-measure classification metrics demonstrated that our method performed better than other methods.