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Surrogate Model Algorithms for Computationally Expensive BlackBox Global Optimization Problems
Abstract:
The algorithm SOM uses DempsterShafer theory to combine information derived from various model characteristics in order to determine the influence of individual models in the mixture. Extensions of SOM with respect to the sampling strategy (algorithms SOMc and SOMs) have been compared in numerical experiments, and it was found that whenever it is a priori unknown which surrogate model should be used, it is advisable to use a mixture model in order to prevent accidentally selecting the worst model. It could be shown that mixture models containing radial basis function interpolants generally work very well, whereas using only polynomial regression models should be avoided. Moreover, algorithms using mixture models often outperform the algorithms that use only the single models that are contributing to the mixture.
Although there are many computationally expensive blackbox optimization applications that have besides continuous also integer variables, or that have only integer variables, algorithms for solving these types of problems are scarce. In the second part of this thesis two algorithms, namely SOMI for mixedinteger problems, and SOI for purely integer problems have been developed and were shown to find accurate solutions for computationally expensive problems with blackbox objective functions and possibly blackbox constraints. The constraints were treated with a penalty approach and numerical experiments showed that the surrogate model based algorithms outperformed commonly used algorithms for (mixed) integer problems such as branch and bound, and genetic algorithms. Also NOMAD (Nonsmooth Optimization by Mesh Adaptive Direct Search) has been included in the comparison. NOMAD is suitable for integer and mixedinteger blackbox problems, but its performance for these problem types has not been studied in the literature. In the numerical experiments, NOMAD also proved superior as compared to branch and bound and the genetic algorithm, but it performed worse than SOI and SOMI for most test problems.
Lastly, the algorithm SOI has been further extended to directly handling constraints with a response surface. The algorithm, SOIc, has been developed specifically for a watershed management problem that has only one constraint, but SOIc is easily generalizable for problems with more constraints. In the considered application problem parts of the agricultural land in the Cannonsville reservoir watershed in upstate New York have to be retired in order to decrease the total phosphorus runoff to a given limit at minimal cost. A computationally expensive simulation model has to be used to compute the costs and phosphorus runoff. The performance of SOIc has been compared to a genetic algorithm, NOMAD, and the discrete dynamically dimensioned search algorithm on three problem instances with different sizes of the feasible region. The surrogate model based algorithm SOIc performed also for these problems significantly better than all other algorithms and could be shown to be the most robust.