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Geometry optimization has applications in many fields and problems such as structure design and inverse scattering. Recently, a method for geometry design was developed using genetic programming. The method is based on a tree shaped chromosome that represents Boolean combinations of convex shapes. Genetic programming (GP) is used to optimize the structure. This approach proved successful but required a large amount of computing resources. The typical number of function evaluations required was around 200,000. While other genetic algorithm (GA) based inverse scattering methods were shown to require a similar number of evaluations for problems of similar complexity, the efficiency of the method can still be improved by using local optimization. Local optimization methods such as the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method can compliment a GA's global search capabilities with a more efficient local search mechanism. The resulting method should provide more accurate answers using fewer function evaluations. One drawback is that the local optimization scheme will require a number of function evaluations that cannot be parallelized as easily as a genetic algorithm. If several chromosomes are optimized simultaneously, each can be run in parallel. In addition, the local optimizer can be run for a small number of steps.