1 Adaptive Response Surface Method Using Inherited Latin Hypercube Design Points G. Gary Wang, Assistant Professor Dept. of Mech. and Indus. Engineering University of Manitoba Winnipeg, MB, Canada R3T 5V6 Tel: 204-4749463 Fax: 204-2757507 Email: [email protected]Abstract This paper addresses the difficulty of the previously developed Adaptive Response Surface Method (ARSM) for high-dimensional design problems. The ARSM was developed to search for the global design optimum for computation-intensive design problems. This method utilizes Central Composite Design (CCD), which results in an exponentially increasing number of required design experiments. In addition, the ARSM generates a complete new set of CCD samples in a gradually reduced design space. These two factors greatly undermine the efficiency of the ARSM. In this work, Latin Hypercube Design (LHD) is utilized to generate saturated design experiments. Because of the use of LHD, historical design experiments can be inherited in later iterations. As a result, ARSM only requires a limited number of design experiments even for high-dimensional design problems. The improved ARSM is tested using a group of standard test problems and then applied to an engineering design problem. In both testing and design application, significant improvement in the efficiency of ARSM is realized. The improved ARSM demonstrates strong potential to be a practical global optimization tool for computation- intensive design problems. Inheriting LHD samples, as a general sampling strategy, can be integrated into other approximation-based design optimization methodologies. Keywords: Response Surface Method, Latin Hypercube Design, Computation-intensive Design, Design Optimization, Global Optimization Transactions of the ASME, Journal of Mechanical Design, Vol. 125, pp. 210-220, June 2003.
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From the beam design problem one can see that the solutions obtained through three different
methods all achieved a solution that satisfies the constraints. The improved ARSM reaches the
best solution as the same as that obtained with Matlab™ by trying various starting points
(possibly the global optimum). As a global optimization strategy that is independent on the
starting point, the number of function evaluations required by the improved ARSM is even much
smaller than that by Matlab™. The improved ARSM outperforms the previous ARSM in terms
of both the minimum objective function value and the efficiency, i.e., the number of objective
function evaluations. It is also observed that for the improved ARSM method, a few new points
are added at each iteration, since many previous design points can be inherited. As a result, the
optimization result improves slowly and thus needs more design iterations than the previous
ARSM method, for which a complete new set of points are generated. As shown in the example,
the improved ARSM method requires 15 iterations to reach the optimum, whereas the previous
ARSM method only needs 8 iterations. If the maximum amount of parallel computation or, the
minimum optimization time, is desired, the increase of number of design iterations may be of
concern.
Discussion
The work aims to improve the efficiency of the Adaptive Response Surface Method (ARSM) by
virtue of the Latin Hypercube Design. From the test problems and the design example, one can
see that the improved ARSM is more efficient that the previous ARSM, for either inheritance
method. For the ARSM with Method II, the algorithm reached very high accuracy with fewer
function evaluations than that of the previous ARSM.
Applicability of the ARSM to Various Functions
In (Wang et al. 2001), it is mentioned that ARSM works well for overall-convex functions. It
should be clarified that ARSM works better in general than the conventional response surface
method (RSM) for non-concave functions instead of overall-convex functions. That is to say, no
matter how complicated the function is, as long as there is a convex section in the design space,
ARSM has high potential to identify that section and locate the optimum. This observation is
taken from the testing of highly complicated functions mostly non-convex as described in
the Testing the ARSM section. For pure concave functions, ARSM performs the same as the
conventional RSM, because the optimum is at the boundary and the design space cannot be
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reduced. The improved ARSM includes a small utility function to identify concave functions.
This utility function makes ARSM applicable to all types of functions, even though ARSM bears
no advantages over conventional RSM for pure concave functions.
Constrained Optimization Problems
The ARSM solves constrained optimization problems as defined in Eqs. (6) - (9). In that model,
constrains can be given explicitly or be approximated by surrogates. If constraints are
computationally expensive, they can be approximated at each design iteration, similar to the
objective function. If constrains are explicitly given or are of first-order or second-order for
which accurate surrogates are easy to obtain, the optimum found from ARSM by using the model
defined in Eqs. (6) - (9) is guaranteed to satisfy the constraints described in Eqs. (3) - (5). If the
constraints are highly nonlinear and computation intensive, the satisfaction of constraints
depends on the accuracy of the surrogate in the neighborhood of the obtained optimum. In
general, ARSM works very well if the constraints are active in the neighborhood of the
unconstrained global optimum, where ARSM yields very accurate surrogates. If the constraints
are active in the region far from the neighborhood of the unconstrained global optimum, ARSM
might only give a mediocre solution as the errors may come from the inaccurate fitting for both
the objective and constraint functions. ARSM, however, bears advantages over the conventional
RSM as the chance of finding the constrained optimum is still higher than the latter because of
the smaller design space in ARSM. Though explicitly given constraints are assumed for the
constrained GC function and the design problem, the achieved constrained global optimum and
the high efficiency have testified the capability of ARSM for constrained optimization problems.
Further research is needed to improve the space reduction strategy to consider the objective as
well as constraint functions.
Though significant improvements have been made on ARSM, the selection of the threshold
value (cutting plane) is still ad hoc. Another observation of the improved ARSM is that if the
cutting is too conservative, most previous design points can then be inherited. Thus, few new
points are added for the next iteration. If only one or two points are added to the design group at
each iteration, the optimization result improves slowly. Therefore, an appropriate trade-off
between the aggressiveness of the cutting and the process speed is to be developed. From the
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author's experience, a 5~10% reduction on either bound of any of the variables indicates that an
appropriate threshold was chosen and such a reduction should be accepted for the next iteration.
Conclusion
The improved ARSM, using the Latin Hypercube Design (LHD) instead of the Central
Composite Designs (CCD), significantly increases the efficiency of the previous ARSM and
enables ARSM be used for high-dimensional problems. For a second-order response model, a
saturated experimental design becomes possible. The nature of LHD also makes the design
inheritance possible in ARSM, which further improves the efficiency of ARSM. From the
testing and the design example, the improved ARSM demonstrates greatly enhanced efficiency
over the previous ARSM. With the improved ARSM, a global design solution can be obtained
with very modest computation cost for computation-intensive design problems. Though
limitations exist, ARSM at the current development stage demonstrates strong potential to be a
global optimization tool for design problems involving computation-intensive function
evaluations. The method of inheriting Latin Hypercube Design points could be integrated to
other move-limits methods in which the design space is varied. It can also be used for step-by-
step sampling in a same design space to gradually improve the approximation accuracy. As a
general sampling method, it might find more applications in approximation-based design
optimization.
Acknowledgement
The author extends special thanks to Prof. R. Haftka at the University of Florida for his early
comments on D-optimal designs and Latin Hypercube Designs. The author also thanks Prof. W.
Chen at the University of Illinois at Chicago, Prof. J. Renaud at the University of Notre Dame,
and all the reviewers for their value feedback. Comments and source codes from Prof. J. S. Park
at Chonnam National University are also very much appreciated. In addition, research funding
from the Natural Science and Engineering Research Council (NSERC) of Canada is gratefully
acknowledged.
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