Global optimization tailored for graphics processing units: Complete and rigorous search for large-scale nonlinear minimization

TL;DR

This paper presents a GPU-accelerated numerical method using interval analysis for guaranteed global minimization of large-scale nonlinear functions, successfully enclosing minima in high-dimensional benchmarks.

Contribution

The paper introduces a novel GPU-based interval analysis method that guarantees enclosure of the global minimum for large-scale nonlinear functions, surpassing previous high-dimensional results.

Findings

Enclosed global minima for 11 benchmark functions with up to 10,000 dimensions.

Achieved results using a single GPU within reasonable computation times.

Outperformed existing methods in high-dimensional global optimization.

Abstract

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.