GPU-based complete search for nonlinear minimization subject to bounds

This paper introduces a GPU-based complete search method to enclose the global minimum of a nonlinear function subject to simple bounds on the variables. Using interval analysis, coupled with the computational power and architecture of GPU, the method iteratively rules out the regions in the search domain where the global minimum cannot exist and leaves a finite set of regions where the global minimum must exist. For effectiveness, because of the rigor of interval analysis, the method is guaranteed to enclose the global minimum of the nonlinear function even in the presence of rounding errors. For efficiency, the method employs a novel GPU-based single program, single data parallel programming style to circumvent major GPU performance bottlenecks, and a variable cycling technique is also integrated into the method to reduce computational cost when minimizing large-scale nonlinear functions. The method is validated by minimizing 10 multimodal benchmark test functions with scalable dimensions, including the well-known Ackley function, Griewank function, Levy function, and Rastrigin function. These benchmark test functions represent grand challenges of global optimization, and enclosing the guaranteed global minimum of these benchmark test functions with more than 80 dimensions has not been reported in the literature. Our method completely searches the feasible domain and successfully encloses the guaranteed global minimum of these 10 benchmark test functions with up to 10,000 dimensions using only one GPU in a reasonable computation time, far exceeding the reported results in the literature due to the unique method design and implementation based on GPU architecture.