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Research Papers

Methodology for Global Optimization of Computationally Expensive Design Problems

[+] Author and Article Information
Stefanos Koullias

Aerospace Systems Design Laboratory,
School of Aerospace Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: skoullias3@gmail.com

Dimitri N. Mavris

Professor
Aerospace Systems Design Laboratory,
School of Aerospace Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: dimitri.mavris@aerospace.gatech.edu

In the fictitious setup of describing a deterministic function $y(·)$ as a realization of a Gaussian random process, the deterministic observations y as well as the value y(x) are random variables, but in this manuscript they are written in the lower case to remind the reader that they are not true random variables.

It is understood that the expectation $EI(x)$ is the conditional expectation $EI|y(x)$.

Equality constraints can be written as two inequality constraints.

Henceforth, all parameters of the constraint function GP model cj shall be indicated using the relevant variables thus far but with a j subscript, where $j=1,…,m$

The implementation is identical to EGO except for the ISC. The P-algorithm maximizes the probability of improvement.

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received September 2, 2013; final manuscript received April 3, 2014; published online June 2, 2014. Assoc. Editor: Michael Kokkolaras.

J. Mech. Des 136(8), 081007 (Jun 02, 2014) (12 pages) Paper No: MD-13-1387; doi: 10.1115/1.4027493 History: Received September 02, 2013; Revised April 03, 2014

Abstract

The design of unconventional systems requires early use of high-fidelity physics-based tools to search the design space for improved and potentially optimum designs. Current methods for incorporating these computationally expensive tools into early design for the purpose of reducing uncertainty are inadequate due to the limited computational resources that are available in early design. Furthermore, the lack of finite difference derivatives, unknown design space properties, and the possibility of code failures motivates the need for a robust and efficient global optimization (EGO) algorithm. A novel surrogate model-based global optimization algorithm capable of efficiently searching challenging design spaces for improved designs is presented. The algorithm, called fBcEGO for fully Bayesian constrained EGO, constructs a fully Bayesian Gaussian process (GP) model through a set of observations and then uses the model to make new observations in promising areas where improvements are likely to occur. This model remedies the inadequacies of likelihood-based approaches, which may provide an incomplete inference of the underlying function when function evaluations are expensive and therefore scarce. A challenge in the construction of the fully Bayesian GP model is the selection of the prior distribution placed on the model hyperparameters. Previous work employs static priors, which may not capture a sufficient number of interpretations of the data to make any useful inferences about the underlying function. An iterative method that dynamically assigns hyperparameter priors by exploiting the mechanics of Bayesian penalization is presented. fBcEGO is incorporated into a methodology that generates relatively few infeasible designs and provides large reductions in the objective function values of design problems. This new algorithm, upon implementation, was found to solve more nonlinearly constrained algebraic test problems to higher accuracies relative to the global minimum than other popular surrogate model-based global optimization algorithms and obtained the largest reduction in the takeoff gross weight objective function for the case study of a notional 70-passenger regional jet when compared with competing design methods.

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Figures

Fig. 1

MLE fit (top) compared with fully Bayesian fit (bottom). Shaded area represents an uncertainty region of ±2 s.

Fig. 2

Bayesian penalization of unnecessarily complex models and overly simple models. Conditional models (solid lines) and weighted sum model (dotted line) through four observations (+).

Fig. 3

Results of nonlinearly constrained tests after N simplex gradients. N = 5, 10, and 20 shown in top, middle, and bottom panels, respectively.

Fig. 4

Results of nonlinearly constrained tests for all N in terms of equality constraint violation

Fig. 5

Mission profile for notional 70-passenger jet

Fig. 6

Baseline (solid) versus optimized (wireframe) configuration

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