Research Papers

Convex Estimators for Optimization of Kriging Model Problems

[+] Author and Article Information
Karim Hamza

 Research Fellow Dept. of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109khamza@umich.edu

Mohammed Shalaby

 Structures Lab, General Electric-Global Research, Niskayuna, NY 12309shalaby@ge.com

J. Mech. Des 134(11), 111005 (Oct 03, 2012) (11 pages) doi:10.1115/1.4007398 History: Received July 23, 2011; Revised July 31, 2012; Published October 02, 2012; Online October 03, 2012

This paper presents a framework for identification of the global optimum of Kriging models that have been tuned to approximate the response of some generic objective function and constraints. The framework is based on a branch and bound scheme for subdivision of the search space into hypercubes while constructing convex underestimators of the Kriging models. The convex underestimators, which are the key development in this paper, provide a relaxation of the original problem. The relaxed problem has two main features: (i) convex optimization algorithms such as sequential quadratic programming (SQP) are guaranteed to find the global optimum of the relaxed problem and (ii) objective value of the relaxed problem is a lower bound within a hypercube for the original (Kriging model) problem. As accuracy of the convex estimators improves with subdivision of a hypercube, termination of a branch happens when either: (i) solution of the relaxed problem within the hypercube is no better than current best solution of the original problem or (ii) best solution of the original problem and that of the relaxed problem are within tolerance limits. To assess the significance of the proposed framework, comparison studies against genetic algorithm (GA), particle swarm optimization (PSO), random multistart sequential quadratic programming (mSQP), and DIRECT are conducted. The studies include four standard nonlinear test functions and two design application problems of water desalination and vehicle crashworthiness. The studies show the proposed framework deterministically finding the optimum for all the test problems. Among the tested stochastic search techniques (GA, PSO, mSQP), mSQP had the best performance as it consistently found the optimum in less computational time than the proposed approach except on the water desalination problem. DIRECT deterministically found the optima for the nonlinear test functions, but completely failed to find it for the water desalination and vehicle crashworthiness problems.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

Convex underestimators of Kriging models improve in accuracy as their domain of definition gets smaller in size: (a) function space as 1 region, (b) function space divided into 4 regions, and (c) function space divided into 16 regions

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Figure 2

Illustration of concave overestimators of covariance functions: (a) H in concave region, (b) H in nonconcave region, and (c) H in both concave and nonconcave regions

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Figure 3

Single-stage reverse osmosis water desalination

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Figure 4

Half-body vehicle structure subject to frontal crash

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Figure 5

Tested algorithms percentile of successful runs verses ratio of number of Kriging model evaluations compared to proposed approach



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