An Exploration And Exploitation Pareto Approach To Surrogate Optimization

Abstract

The experiments or simulations conducted by computers can be a tedious task,requiring substantial computational time. To find a global solution using a computerexperiments process, we usually need to perform many function evaluations of thecomputer model. This research focuses on developing an optimization method to find aglobally optimal solution efficiently using surrogates. The surrogates represent thecomputer model or the system that reads the inputs and generates the output responsesof interest, so that these surrogate models can be used in place of time-consumingsimulations runs. In surrogate based optimization, we iteratively build a surrogate model(a.k.a, approximate model or a metamodel) and conduct an optimization step, addingpoints in each iteration only as needed.The proposed surrogate optimization method, Exploration and ExploitationPareto Approach (EEPA), combines the notions of exploration and exploitation to seekthe best solution with fewer function evaluations. Exploration is used to explore the pointsin an unexplored region. Exploration does not use the surrogate to look for new points.Four different exploration methods were used in this research, specifically maximindistance (Johnson, et. al., [43]), cosine (Corley et.al. [20]), Sobol′ (Sobol', [89])visequence and Monte Carlo method (Niederreiter [56]). Maximin looks for points that areat maximin distance from the existing points, cosine looks for the maximum angulardistance between points, Sobol' looks for points that are evenly spaced and Monte Carlolooks for points randomly in the input space.Exploitation is used to explore promising areas in the input space. Exploitationrequires a surrogate or metamodel to be built in order to look for new points. Differentsurrogate or metamodel models, including multivariate adaptive regression splines, radialbasis functions, and treed regression, are used in this study. The minimum responsemethod (Regis and Shoemaker [61]; Regis and Shoemaker [62] is used as theexploitation method in this study. Response metric helps to look for points that possiblygive better solution.Exploration and exploitation are combined in EEPA by obtaining a Paretofrontier. A Pareto frontier represents the non-dominating solutions given two or moresolutions. The Pareto frontier is obtained by balancing the tradeoff between maximizingthe exploration metric and minimizing the predicted response. . Points are then chosenfrom this Pareto frontier at which additional function evaluations using the computermodel are executed.Various test functions were used to compare EEPA to pure exploitation and pureexploration methods. In addition, a green building test function is also considered. Thesetest functions are of different dimensions and structure. The green building data issimulated from a computer model called eQUEST. The results showed that EEPAreached the best solution faster than pure exploration or exploitation methods. Also,among the exploration methods, the cosine method performed well.