Radial Basis Function Surrogate Model Research Articles

Integration of expert knowledge into radial basis function surrogate models

A current application in a collaboration between Chalmers University of Technology and Volvo Group Trucks Technology concerns the global optimization of a complex simulation-based function describing the rolling resistance coefficient of a truck tyre. This function is crucial for the optimization of truck tyres selection considered. The need to explicitly describe and optimize this function provided the main motivation for the research presented in this article. Many optimization algorithms for simulation-based optimization problems use sample points to create a computationally simple surrogate model of the objective function. Typically, not all important characteristics of the complex function (as, e.g., non-negativity)—here referred to as expert knowledge—are automatically inherited by the surrogate model. We demonstrate the integration of several types of expert knowledge into a radial basis function interpolation. The methodology is first illustrated on a simple example function and then applied to a function describing the rolling resistance coefficient of truck tyres. Our numerical results indicate that expert knowledge can be advantageously incorporated and utilized when creating global approximations of unknown functions from sample points.

Sequential Optimization of Strip Bending Process Using Multiquadric Radial Basis Function Surrogate Models

Surrogate models are used within the sequential optimization strategy for forming processes. A sequential improvement (SI) scheme is used to refine the surrogate model in the optimal region. One of the popular surrogate modeling methods for SI is Kriging. However, the global response of Kriging models deteriorates in some cases due to local model refinement within SI. This may be problematic for multimodal optimization problems and for other applications where correct prediction of the global response is needed. In this paper the deteriorating global behavior of the Kriging surrogate modeling technique is shown for a model of a strip bending process. It is shown that a Radial Basis Function (RBF) surrogate model with Multiquadric (MQ) basis functions performs equally well in terms of optimization efficiency and better in terms of global predictive accuracy. The local point density is taken into account in the model formulation.

SO-MI: A surrogate model algorithm for computationally expensive nonlinear mixed-integer black-box global optimization problems

This paper introduces a surrogate model based algorithm for computationally expensive mixed-integer black-box global optimization problems with both binary and non-binary integer variables that may have computationally expensive constraints. The goal is to find accurate solutions with relatively few function evaluations. A radial basis function surrogate model (response surface) is used to select candidates for integer and continuous decision variable points at which the computationally expensive objective and constraint functions are to be evaluated. In every iteration multiple new points are selected based on different methods, and the function evaluations are done in parallel. The algorithm converges to the global optimum almost surely. The performance of this new algorithm, SO-MI, is compared to a branch and bound algorithm for nonlinear problems, a genetic algorithm, and the NOMAD (Nonsmooth Optimization by Mesh Adaptive Direct Search) algorithm for mixed-integer problems on 16 test problems from the literature (constrained, unconstrained, unimodal and multimodal problems), as well as on two application problems arising from structural optimization, and three application problems from optimal reliability design. The numerical experiments show that SO-MI reaches significantly better results than the other algorithms when the number of function evaluations is very restricted (200–300 evaluations).

Stochastic radial basis function algorithms for large-scale optimization involving expensive black-box objective and constraint functions

This paper presents a new algorithm for derivative-free optimization of expensive black-box objective functions subject to expensive black-box inequality constraints. The proposed algorithm, called ConstrLMSRBF, uses radial basis function (RBF) surrogate models and is an extension of the Local Metric Stochastic RBF (LMSRBF) algorithm by Regis and Shoemaker (2007a) [1] that can handle black-box inequality constraints. Previous algorithms for the optimization of expensive functions using surrogate models have mostly dealt with bound constrained problems where only the objective function is expensive, and so, the surrogate models are used to approximate the objective function only. In contrast, ConstrLMSRBF builds RBF surrogate models for the objective function and also for all the constraint functions in each iteration, and uses these RBF models to guide the selection of the next point where the objective and constraint functions will be evaluated. Computational results indicate that ConstrLMSRBF is better than alternative methods on 9 out of 14 test problems and on the MOPTA08 problem from the automotive industry (Jones, 2008 [2]). The MOPTA08 problem has 124 decision variables and 68 inequality constraints and is considered a large-scale problem in the area of expensive black-box optimization. The alternative methods include a Mesh Adaptive Direct Search (MADS) algorithm (Abramson and Audet, 2006 [3]; Audet and Dennis, 2006 [4]) that uses a kriging-based surrogate model, the Multistart LMSRBF algorithm by Regis and Shoemaker (2007a) [1] modified to handle black-box constraints via a penalty approach, a genetic algorithm, a pattern search algorithm, a sequential quadratic programming algorithm, and COBYLA (Powell, 1994 [5]), which is a derivative-free trust-region algorithm. Based on the results of this study, the results in Jones (2008) [2] and other approaches presented at the ISMP 2009 conference, ConstrLMSRBF appears to be among the best, if not the best, known algorithm for the MOPTA08 problem in the sense of providing the most improvement from an initial feasible solution within a very limited number of objective and constraint function evaluations.

A modified global optimization method based on surrogate model and its application in packing profile optimization of injection molding process

This article introduces a step-by-step optimization method based on the radial basis function (RBF) surrogate model and proposes an improved expected improvement selection criterion to better the global performance of this optimization method. Then it is applied to the optimization of packing profile of injection molding process for obtaining best shrinkage evenness of molded part. The idea is first, to establish an approximation function relationship between shrinkage evenness and process parameters by a small size of design of experiment with RBF surrogate model to alleviate the expensive computational expense in the optimization iterations. And then, an improved criterion is used to provide direction in which additional training samples could be added to better the surrogate model. Two test functions are investigated and the results show that stronger global exploration performance and more precise optimal solution could be obtained with the improved method at the expense of increasing the infill data properly. Furthermore the optimal solution of packing profile is obtained for the first time which indicates that the type of optimal packing profile should be first constant and then ramp-down. Subsequently, the discussion of this result is given to explain why the optimal profile is like that.