A Hybrid Evolutionary Algorithm and Cell Mapping Method for Multi-Objective Optimization Problems

We present a method for multi-objective optimization problems based on evolutionary algorithms and the cell mapping methods developed by the late Professor C.S. Hsu of UC Berkeley in 1980s. Evolutionary algorithms with a limited number of random trial solutions usually converge quickly toward the neighborhood of the Pareto set, but they take a considerable amount of time to find the finely-structured global solution of the Pareto set with sufficient accuracy for high dimensional problems. However, the rough solutions obtained with evolutionary algorithms offer an excellent starting point to apply the cell mapping method. A covering set of cells in a cellular design space can be found to contain all the random points of evolutionary algorithms. The cell mappings are constructed over the covering set based on a local search algorithm over the cellular grids for optimal solutions. The classical sorting algorithm of the cell mapping method is then used to identify the periodic groups of the cell mappings, which represent the solutions of the multi-objective optimization problem. A recovery strategy is also developed to recoup the solutions inevitably missed by evolutionary algorithms. The hybrid method of evolutionary algorithms and the cell mapping method has been tested on a large number of mathematical benchmark problems, and several challenging engineering problems including quantitative design of nonlinear controls of mechanical systems and a structural-acoustic optimization for minimum sound radiation. The results show that the hybrid method performs well.