This paper introduces a numerical 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 graphics processing units (GPUs), 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 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 11 benchmark test functions with scalable dimensions, including the well-known Ackley function, Griewank function, Levy function, Rastrigin function, and Rosenbrock 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 11 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：本文提出一种数值方法，用于封闭受限于变量简单边界的非线性函数的全局最小值。该方法结合区间分析与图形处理单元（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，并保留一个必须存在全局最小值的有限区域集合。由于区间分析的严格性，该方法即使在舍入误差存在的情况下也能保证封闭全局最小值。在效率方面，该方法采用一种新颖的基于GPU的单程序、单数据并行编程风格，以规避主要的GPU性能瓶颈；同时，方法中集成了变量循环技术以降低大规模非线性函数极小化时的计算成本。通过最小化11个具有可扩展维度的基准测试函数（包括著名的Ackley函数、Griewank函数、Levy函数、Rastrigin函数和Rosenbrock函数）验证了该方法的有效性。这些基准测试函数代表了全局优化中的重大挑战，而在文献中尚未报道过对维度超过80的此类函数实现保证全局最小值的封闭。我们的方法完全搜索可行域，并在合理计算时间内仅使用一块GPU成功封闭了这些维度高达10,000的11个基准测试函数的保证全局最小值，这远超文献报道的结果，得益于基于GPU架构的独特方法设计与实现。