Complexity Analysis of Iterated Local Search Algorithm in Experimental Domain for Optimizing Latin Hypercube Designs

Abstract

Computer experiments involve a large numbers of variables, but only a few of them have no negligible influence on the response. As is recognized by several authors, the choice of the design points for computer experiments should fulfill at lest two requirements - space-filling and non-collapsing. Unfortunately, randomly generated Latin Hypercube Designs (LHDs) almost always show poor space-filling properties. On the other hand, maximin distance designs have very well space-filling properties but often show poor projection properties under the Euclidean or the Rectangular distance. To overcome this shortcoming, Morris et al. have suggested to search for maximin LHDs when looking for "optimal" designs. It is shown that the Iterated Local search (ILS) approach not only able to obtain good LHDs in the sense of space-filling property but the correlations among the factors are acceptable i.e. multi-collinearity is not high. Anyway from the point of view of computational complexity the problem is open. When number of factors or number of design points is large then it requires hundreds of hours by the brute-force approach to find out the optimal design. So when numbers of factors as well as number of experimental points are large, the heuristic approaches also require a couple of hours or even more to find out a simulated optimal design. So time complexity is an important issue for a good algorithm. Specially for the need of real time solution, the time complexity of the ILS approaches is analyzed. The inner most view as well as the effect of the parameters of the algorithms have been observed and have been analyzed. After analyzing, the time complexity model of the algorithms for two optimal criterion namely Opt (D1, J1) as well as Opt (φ) has been developed. More over some experiments have been performed for higher dimension namely dimensions k >10. Some new maximin LHDs value are obtained from these experiments, as there are few maximin LHDs value available in the literatures for higher dimension, k >10. From these experiments, multi-collinearity property, maximin LHDs in Rectangular distance, mimimal D values, maximum pair-wise distance value of LHDs etc. are represented in this thesis.