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.

翻译：本文提出了一种基于GPU的完全搜索方法，用于确定带变量简单约束的非线性函数全局最小值的包含区间。该方法结合区间分析理论与GPU的计算能力及架构，通过迭代排除搜索域中不可能存在全局最小值的区域，最终保留有限个必须包含全局最小值的区域。在有效性方面，由于区间分析的严密性，即使存在舍入误差，该方法仍能保证包含非线性函数的全局最小值。在效率方面，该方法采用新颖的基于GPU的单程序多数据并行编程模式以规避主要GPU性能瓶颈，并集成变量循环技术以降低大规模非线性函数最小化的计算成本。通过最小化10个具有可扩展维度的多峰基准测试函数（包括著名的Ackley函数、Griewank函数、Levy函数和Rastrigin函数）验证了本方法的有效性。这些基准测试函数代表了全局优化的重大挑战，现有文献尚未报道对超过80维的此类函数实现有保证的全局最小值包含。我们的方法通过完全搜索可行域，仅使用单个GPU在合理计算时间内成功包含了这10个基准测试函数（最高达10,000维）的有保证全局最小值，基于GPU架构的独特方法设计与实现远超文献报道结果。