Parameter optimization in multiquadric response surface approximations

Multiquadric (MQ) response surface approximation uses the function C_i|X-X_i|²+h^1/2 to interpolate a given set of data. The performance of MQ approximation depends on the shift parameter h. Efficient methods of computing the optimal shift parameter based on the leave-one-out cross validation technique are presented in this paper. We also proved that the condition number of the MQ coefficient matrix is an increasing function of the shift parameter h. Two numerical examples are included to illustrate the proposed formulation.     

Rapid design optimization of antennas using space mapping and response surface approximation models

A computationally efficient method for design optimization of antennas is discussed. It combines space mapping, used as the optimization engine, and response surface approximation, used to create the fast surrogate model of the optimized antenna. The surrogate is configured from the response of the coarse‐mesh electromagnetic model of the antenna, and implemented through kriging interpolation. We provide a comprehensive numerical verification of this technique as well as demonstrate its capability to yield a satisfactory design after a few full‐wave simulations of the original structure. © 2011 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2011.     

Design optimization through parallel-generated surrogate models, optimization methodologies and the utility of legacy simulation software

This paper presents the motivation development and an application of a unique methodology to solve industrial optimization problems, using existing legacy simulation software programs. The methodology is based on approximation models generated with the utility of design of experiments methodologies and response surface methods applied on high-fidelity simulations, coupled together with classical optimization methodologies. Several DOE plans are included, in order to be able to adopt the appropriate level of detail. The approximations are based on stochastic interpolation techniques, or on classical least squares methods. The optimization methods include both local and global techniques. Finally, an application from the plastic molding industry (process simulation) demonstrates the methodology and the software package. Received December 30, 2000     

Multipoint cubic surrogate function for sequential approximate optimization

Multipoint cubic approximations are investigated as surrogate functions for nonlinear objective and constraint functions in the context of sequential approximate optimization. The proposed surrogate functions match actual function and gradient values, including the current expansion point, thus satisfying the zero and first-order necessary conditions for global convergence to a local minimum of the original problem. Function and gradient information accumulated from multiple design points during the iteration history is used in estimating a reduced Hessian matrix and selected cubic terms in a design subspace appropriate for problems with many design variables. The resulting approximate response surface promises to accelerate convergence to an optimal design within the framework of a trust region algorithm. The hope is to realize computational savings in solving large numerical optimization problems. Numerical examples demonstrate the effectiveness of the new multipoint surrogate function in reducing errors over large changes in design variables.     

Two-level optimization of airframe structures using response surface approximation

In this paper, a two-level optimization approach is developed for the preliminary and conceptual design of airframe structures. The preliminary design, involving a single objective multidisciplinary optimization, constitutes the lower level where ASTROS (Automated STRuctural Optimization System) is employed for multidisciplinary optimization. The conceptual design, which is carried out at the upper level, aims mainly at configuration design. The multiple objectives are incorporated as a single objective function by using the K-S function formulation. The objective function and constraints at the upper level are modelled through response surface approximation. During the upper level optimization process, the branch and bound method is applied for solving the problem with discrete design variables. The proposed strategy is demonstrated by the optimization of an Intermediate Complexity Wing (ICW) model. Received June 23, 1999     

Multiobjective optimization of complex antenna structures using response surface models

A very efficient multiobjective (MO) design technique for complex antenna structures involving a large number of design parameters is presented. This design technique, multiobjective‐fractional factorial design (MO‐FFD), is very different from conventional Pareto‐based MO algorithms, which take a great deal of effort to balance the trade‐off between all the design specifications. By performing one single combination of simulations, all the response surface models of design goals are simultaneously built, and Derringer's desirability functions are readily applied to these models so that the optimum structure is obtained. Compared to classical MO algorithms such as Strength Pareto Evolutionary Algorithm 2, nondominated sorting particle swarm optimizer, and cultural MO particle swarm optimization, MO‐FFD yields more desirable performances yet the required number of simulations is reduced by 97%. This article thoroughly illustrates the mathematical development of MO‐FFD, deriving a novel application of ultrawideband (UWB) antennas because of its MO optimization capability. More explicitly, MO‐FFD overcomes all the design challenges of dual band‐notched UWB antennas including desired impedance characteristics, enhanced fidelity factors, and uniform peak gains over the passband, which are what conventional Pareto‐based algorithms cannot attain. The measured results show that all the performance criteria are met; especially, the time‐domain signal distortion is minimized. © 2015 Wiley Periodicals, Inc. Int J RF and Microwave CAE 26:62–71, 2016.     

一种新颖的混合响应面优化方法

为减少黑箱优化过程中的评估次数,提出了一种新颖的混合响应面优化方法(HRSO),利用混合响应面建立高精度的近似模型作为代理模型,通过迭代更新响应面不断接近真实模型,从而完成优化。以Dixon-Szego函数类作为测试函数,以评估次数为方法性能优劣的评价指标,实验结果表明,与Gutmann-RBF、CORS-RBF两种方法相比,HRSO能够在较少的评估次数内满足相同的收敛条件,且向全局快速收敛,是一种适合求解黑箱优化问题的方法。     

Update strategies for kriging models used in variable fidelity optimization

Many optimization methods for simulation-based design rely on the sequential use of metamodels to reduce the associated computational burden. In particular, kriging models are frequently used in variable fidelity optimization. Nevertheless, such methods may become computationally inefficient when solving problems with large numbers of design variables and/or sampled data points due to the expensive process of optimizing the kriging model parameters in each iteration. One solution to this problem would be to replace the kriging models with traditional Taylor series response surface models. Kriging models, however, were shown to provide good approximations of computer simulations that incorporate larger amounts of data, resulting in better global accuracy. In this paper, a metamodel update management scheme (MUMS) is proposed to reduce the cost of using kriging models sequentially by updating the kriging model parameters only when they produce a poor approximation. The scheme uses the trust region ratio (TR-MUMS), which is a ratio that compares the approximation to the true model. Two demonstration problems are used to evaluate the proposed method: an internal combustion engine sizing problem and a control-augmented structural design problem. The results indicate that the TR-MUMS approach is very effective; on the demonstration problems, it reduced the number of likelihood evaluations by three orders of magnitude compared to using a global optimizer to find the kriging parameters in every iteration. It was also found that in trust region-based method, the kriging model parameters need not be updated using a global optimizer—local methods perform just as well in terms of providing a good approximation without affecting the overall convergence rate, which, in turn, results in a faster execution time.     

Fast simulation‐driven optimization of planar microstrip antenna arrays using surrogate superposition models

A technique for simulation‐driven design of excitation tapers for planar antenna arrays is presented. Our methodology exploits antenna array models constructed as a superposition of simulated radiation and reflection responses of the array under design, with only one radiator active at a time. Low computational costs of these models are ensured by using iteratively corrected electromagnetic‐simulation data computed with coarse meshes. Our technique allows for simultaneous control of the radiation pattern and the reflection coefficients of the array. Numerical efficiency as well as scalability of the technique is demonstrated using the design examples of various sizes and topologies, including a sixteen element and hundred element microstrip patch antenna arrays of the Cartesian lattice and a hundred element microstrip antenna array of the hexagonal lattice. The proposed technique is versatile as it also can be applied for simulation‐based optimization of antenna arrays comprising other types of individually fed elements, e.g., wires, strips, or dielectric resonator antennas. © 2014 Wiley Periodicals, Inc. Int J RF and Microwave CAE 25:371–381, 2015.     

Approximation methods in multidisciplinary analysis and optimization: a panel discussion

This paper summarizes the discussion at the Approximation Methods Panel that was held at the 9^thAIAA/ISSMO Symposium on Multidisciplinary Analysis & Optimization in Atlanta, GA on September 2–4, 2002. The objective of the panel was to discuss the current state-of-the-art of approximation methods and identify future research directions important to the community. The panel consisted of five representatives from industry and government: (1) Andrew J. Booker from The Boeing Company, (2) Dipankar Ghosh from Vanderplaats Research & Development, (3) Anthony A. Giunta from Sandia National Laboratories, (4) Patrick N. Koch from Engineous Software, Inc., and (5) Ren-Jye Yang from Ford Motor Company. Each panelist was asked to (i) give one or two brief examples of typical uses of approximation methods by his company, (ii) describe the current state-of-the-art of these methods used by his company, (iii) describe the current challenges in the use and adoption of approximation methods within his company, and (iv) identify future research directions in approximation methods. Several common themes arose from the discussion, including differentiating between design of experiments and design and analysis of computer experiments, visualizing experimental results and data from approximation models, capturing uncertainty with approximation methods, and handling problems with large numbers of variables. These are discussed in turn along with the future directions identified by the panelists, which emphasized educating engineers in using approximation methods.     

Polynomial genetic programming for response surface modeling Part 2: adaptive approximate models with probabilistic optimization problems

This is the second in a series of papers. The first deals with polynomial genetic programming (PGP) adopting the directional derivative-based smoothing (DDBS) method, while in this paper, an adaptive approximate model (AAM) based on PGP is presented with the partial interpolation strategy (PIS). The AAM is sequentially modified in such a way that the quality of fitting in the region of interest where an optimum point may exist can be gradually enhanced, and accordingly the size of the learning set is gradually enlarged. If the AAM uses a smooth high-order polynomial with an interpolative capability, it becomes more and more difficult for PGP to obtain smooth polynomials, whose size should be larger than or equal to the number of the samples, because the order of the polynomial becomes unnecessarily high according to the increase in its size. The PIS can avoid this problem by selecting samples belonging to the region of interest and interpolating only those samples. Other samples are treated as elements of the extended data set (EDS). Also, the PGP system adopts a multiple-population approach in order to simultaneously handle several constraints. The PGP system with the variable-fidelity response surface method is applied to reliability-based optimization (RBO) problems in order to significantly cut the high computational cost of RBO. The AAMs based on PGP are responsible for fitting probabilistic constraints and the cost function while the variable-fidelity response surface method is responsible for fitting limit state equations. Three numerical examples are presented to show the performance of the AAM based on PGP.     

Multilevel microwave design optimization with automated model fidelity adjustment

A computationally efficient algorithm for electromagnetic (EM)‐simulation‐driven design optimization of microwave structures is proposed. Our technique exploits variable‐fidelity EM simulations and the multilevel design approach where an approximate optimum of the lower accuracy but faster EM model of the structure under design is used as a starting point for optimizing a more accurate model. Several enhancements of the basic multifidelity method are introduced, including an efficient algorithm of optimizing EM models that is based on local response surface approximations, as well as automated adjustment of model fidelity. Convergence of the procedure to the optimum design is ensured by defaulting to the higher fidelity model whenever the prediction given by the lower fidelity fails to improve the design. Distribution of the computational effort between the models of different fidelity allows for making larger steps in the design space at a low cost, as well as substantial reduction of the number of high‐fidelity model evaluations, because the high‐fidelity model is only referred to in the last design stage. The article provides comprehensive numerical verification of our technique. Substantial computational savings are demonstrated in comparison to the benchmark methods: over 40% on average as compared to a basic version of the multifidelity optimization approach and over 95% as compared to direct optimization of the high‐fidelity model. © 2013 Wiley Periodicals, Inc. Int J RF and Microwave CAE 24:281–288, 2014.     

Sequential kriging optimization using multiple-fidelity evaluations

When cost per evaluation on a system of interest is high, surrogate systems can provide cheaper but lower-fidelity information. In the proposed extension of the sequential kriging optimization method, surrogate systems are exploited to reduce the total evaluation cost. The method utilizes data on all systems to build a kriging metamodel that provides a global prediction of the objective function and a measure of prediction uncertainty. The location and fidelity level of the next evaluation are selected by maximizing an augmented expected improvement function, which is connected with the evaluation costs. The proposed method was applied to test functions from the literature and a metal-forming process design problem via finite element simulations. The method manifests sensible search patterns, robust performance, and appreciable reduction in total evaluation cost as compared to the original method.     

Multi-criteria shape optimization of a funnel in cathode ray tubes using a response surface model

One purpose of simulation describing the behaviors of structures is to optimize the performances within specific functional requirements and customers needs with respect to the design variables. For reduction of the volume of cathode ray tubes, the design of the glass geometry, especially funnel geometry, is essential while maintaining the internal vacuum pressure of the cathode ray tube. In order to describe the three-dimensional geometry of the funnel in cathode ray tubes, a higher-order response surface model is employed in the simulation model instead of non-uniform rational B-splines (NURBS) or Bezier curves because the response surface model is more robust for understanding the geometry change in finite element analysis. We formulate the design problem as a multi-criteria optimization because minimization of both volume and maximum stress is required. Using the response surface model of the geometry of the funnel and sequential quadratic programming within the process integration framework, the shape optimization of a funnel is successfully performed and the maximum stress level of the funnel is decreased by almost half.     

Sequential approximate optimization using dual subproblems based on incomplete series expansions

In this paper, dual formulations for nonlinear multipoint approximations with diagonal approximate Hessian matrices are proposed; these approximations for example derive from the incomplete series expansion (ISE) proposed previously. A salient feature of the ISE is that it may be used to formulate strictly convex and separable (recast) primal approximate subproblems for use in sequential approximate optimization (SAO). In turn, this allows for the formulation of highly efficient dual formulations, and different combinations of direct, reciprocal, and exponential intervening variables for the objective and the constraint functions may be used. Two frequently encountered problems in structural optimization, namely the weight minimization problem with sizing design variables and the minimum compliance topology optimization problem, are degenerate cases of the formulations we present. Computational experiments confirm the efficiency of our proposed methodology; to this end, comparative results for the method of moving asymptotes (MMA) are presented. Based on the paper entitled “Duality in Convex Nonlinear Multipoint Approximations with Diagonal Approximate Hessian Matrices Deriving from Incomplete Series Expansions,” presented at the 11th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Portsmouth, VA, USA, September 2006, paper no. AIAA-2006-7090.     

An approximate optimization strategy using refined hybrid metamodel

An approximate model called metamodel or surrogate model is a mathematical model that numerically approximates response of a system during an engineering simulation process or test. The introduction of a metamodel makes it possible to express response defined in the design problem as a simple mathematical function of design variables. A metamodel can be built with response surface method (RSM), kriging, neural network, radial basis function, and so on. Each method has its advantages and disadvantages. A combined metamodel called hybrid model, ensemble model, or multiple surrogates has been developed to maximize each metamodel's strength. The hybrid model of this research includes RSM and kriging. Besides, a strategy to refine the hybrid metamodel is implemented by reducing design space. In this process, information related to Hessian is utilized for an unconstrained optimization problem, on the contrary feasibility for a constrained optimization problem. This research presents a new hybrid metamodel-based optimization strategy called refined hybrid metamodel. Five mathematical test problems, two-bar design, spring design, and propeller shaft design problems are solved with the suggested method, verifying its usefulness. Most of the optimal results with the proposed method are closer to exact solutions with smaller function evaluations than existing methods.     

Rapid design and size reduction of microwave couplers using variable‐fidelity EM‐driven optimization

In this article, fast electromagnetic (EM) simulation‐driven design optimization of compact microwave couplers is addressed. The main focus is on explicit reduction of the circuit footprint. Our methodology relies on the penalty function approach, which allows us to minimize the circuit area while ensuring equal power split between the output ports and providing a sufficient bandwidth with respect to the return loss and isolation around the operating frequency. Computational efficiency of the design process is achieved by exploiting variable‐fidelity EM simulations, local response surface approximation models, as well as suitable response correction techniques for design tuning. The technique described in this work is demonstrated using two examples of compact rat‐race couplers. The size‐reduction‐oriented designs are compared with performance‐oriented ones to illustrate available design trade‐offs. Final design solutions of the former case illustrate ～92% of miniaturization for both coupler examples (with corresponding fractional bandwidths of 16%). Alternative design solutions pertaining to the latter case show a lesser size reduction (～90% for both examples), but present a much wider bandwidths (～25% for both couplers). The overall computational cost of the design procedure corresponds to about 20 and 10 high‐fidelity coupler simulations for the first and second design example, respectively. Numerical results are also validated experimentally. © 2015 Wiley Periodicals, Inc. Int J RF and Microwave CAE 26:27–35, 2016.     

Improving surrogate-assisted variable fidelity multi-objective optimization using a clustering algorithm

Surrogate-assisted evolutionary optimization has proved to be effective in reducing optimization time, as surrogates, or meta-models can approximate expensive fitness functions in the optimization run. While this is a successful strategy to improve optimization efficiency, challenges arise when constructing surrogate models in higher dimensional function space, where the trade space between multiple conflicting objectives is increasingly complex. This complexity makes it difficult to ensure the accuracy of the surrogates. In this article, a new surrogate management strategy is presented to address this problem. A k-means clustering algorithm is employed to partition model data into local surrogate models. The variable fidelity optimization scheme proposed in the author's previous work is revised to incorporate this clustering algorithm for surrogate model construction. The applicability of the proposed algorithm is illustrated on six standard test problems. The presented algorithm is also examined in a three-objective stiffened panel optimization design problem to show its superiority in surrogate-assisted multi-objective optimization in higher dimensional objective function space. Performance metrics show that the proposed surrogate handling strategy clearly outperforms the single surrogate strategy as the surrogate size increases.     

基于响应面的概率集群优化方法

面对实际应用中的大规模优化问题,基于响应面估计的概率集群优化方法以设计变量的概率分布作为优化对象,而非直接对设计变量值进行优化,可适应连续、离散及混合的设计变量类型。采用响应面构建概率集群评估函数的近似模型,并采用置信区间方法在迭代优化过程中不断更新响应曲面以确保近似精度。实验结果表明算法对解决复杂优化问题有效。