Stephen J. Leary

On the Design of Optimization Strategies Based on Global Response Surface Approximation Models

Striking the correct balance between global exploration of search spaces and local exploitation of promising basins of attraction is one of the principal concerns in the design of global optimization algorithms. This is true in the case of techniques based on global response surface approximation models as well. After constructing such a model using some initial database of designs it is far from obvious how to select further points to examine so that the appropriate mix of exploration and exploitation is achieved. In this paper we propose a selection criterion based on the expected improvement measure, which allows relatively precise control of the scope of the search. We investigate its behavior through a set of artificial test functions and two structural optimization problems. We also look at another aspect of setting up search heuristics of this type: the choice of the size of the database that the initial approximation is built upon.

A Knowledge-Based Approach To Response Surface Modelling in Multifidelity Optimization

This paper is concerned with approximations for expensive function evaluation – the expensive functions arising in an engineering design context. The problem of reducing the computational cost of generating sufficient learning samples is addressed. Several approaches of using a priori knowledge to achieve computational economy are presented. In all these, the results of a cheap model are treated as knowledge to be incorporated in the training process. Several approaches are described here: in particular, we focus on neural based systems. This approach is then developed as a new knowledge-based kriging model which is shown to be as accurate as neural based alternatives while being much easier to train. Examples from the domain of structural optimization are given to demonstrate the approach.

A constraint mapping approach to the structural optimization of an expensive model using surrogates (in special issue on surrogate modelling and space mapping for engineering optimization)

The use of response surface methods are well established in the global optimization of expensive functions, the response surface acting as a surrogate to the expensive function objective.In structural design however, the change in objective may vary little between the two models: it is more often the constraints that change with models of varying fidelity. Here approaches are described whereby the coarse model constraints are mapped so that the mapped constraints more faithfully approximate the fine model constraints. The shape optimization of a simple structure demonstrates the approach.

Optimal orthogonal-array-based latin hypercubes

The use of optimal orthogonal array latin hypercube designs is proposed. Orthogonal arrays were proposed for constructing latin hypercube designs by Tang (1993). Such designs generally have better space filling properties than random latin hypercube designs. Even so, these designs do not necessarily fill the space particularly well. As a result, we consider orthogonal-array-based latin hypercube designs that try to achieve optimality in some sense. Optimization is performed by adapting strategies found in Morris & Mitchell (1995) and Ye et al. (2000). The strategies here search only orthogonal-array-based latin hypercube designs and, as a result, optimal designs are found in a more efficient fashion. The designs found are in general agreement with existing optimal designs reported elsewhere.

Global Approximation and Optimization Using Adjoint Computational Fluid Dynamics Codes

Approximation methods have found increasing use in the optimization of complex engineering systems. The approximation method provides a surrogate model that, once constructed, can be used in lieu of the original expensive model for the purposes of optimization. These approximations may be defined locally, for example, a low-order polynomial response surface approximation that employs trust region methodology during optimization, or globally, by the use of techniques such as kriging. Adjoint methods for computational fluid dynamics have made it possible to obtain sensitivity information on the model’s response without recourse to finite differencing. This approach then allows for an efficient local optimization strategy where these sensitivities are utilized in gradient-based optimization. The combined use of an adjoint computational fluid dynamics code with approximation methods (incorporating gradients) for global optimization is shown. Several approximation methods are considered. It is shown that an adjoint-based approximation model can provide increased accuracy over traditional nongradientbased approximations at comparable cost, at least for modest numbers of design variables. As a result, these models are found to be more reliable for surrogate assisted optimization.

Screening and approximation methods for efficient structural optmization

In this paper we discuss two statistical techniques for achieving computational economy during the optimization process. The first, the use of approximization methods is often applied when optimizing expensive computational models of complex engineering systems: the idea is to replace the expensive analysis code by a cheap surrogate model for the purposes of optimization. There are many approximation methods available in the literature, we focus here on kriging. Teh second, screening experiments, has received much attention in the statistics community. This statistical tool has been applied to the problem of structural optimization her. Indeed, one purpose of this paper is to increase awareness of these tools in the structural optimization community. In particular, a focus here is on screening multiple responses, as a structural optimization problem typically requires optimization of at least one objective subject to at least one constraint. Finally, both approaches are combined in order to provide an algortihm which appears very efficient for large dimensional strucutral optimization problems. A structural optimization case study of industrial interest demonstrates the approach.