Leveraging Space-Filling Curves and the DIRECT Algorithm. A Novel Approach to Derivative-Free Multi-Dimensional Global Optimization

Bachelor Thesis from the year 2018 in the subject Mathematics - Applied Mathematics, Nanyang Technological University, language: English, abstract: Imagine unlocking solutions to the most complex, multi-dimensional optimization problems with unprecedented efficiency. This groundbreaking work introduces a novel adaptation of the Dividing Rectangles (DIRECT) algorithm, ingeniously enhanced through the application of space-filling curves. Delve into a revolutionary approach that transforms intricate, high-dimensional challenges into simpler, one-dimensional equivalents, effectively sidestepping the notorious computational bottlenecks that plague traditional methods. Explore the depths of Lipschitzian optimization and witness the elegant integration of the Hölder condition, a cornerstone in the quest for global optima. Uncover the secrets behind the construction of the Hilbert curve and its transformative role in mapping points for optimized sampling. This research meticulously details the implementation of both the Hilbert Space Filling Curve module and the modified DIRECT module, providing a comprehensive blueprint for practical application. Through rigorous simulations and detailed analysis, witness firsthand the enhanced performance of the modified DIRECT algorithm, surpassing its standard counterpart in both speed and accuracy. Discover how this innovative technique achieves significant dimensionality reduction, opening new avenues for solving previously intractable problems in diverse fields ranging from engineering to finance. This study not only presents a theoretical framework but also provides concrete evidence of its effectiveness, paving the way for future advancements in global optimization and algorithm efficiency. This report offers a beacon of innovation in the realm of computational complexity, providing a pathway to more efficient and effective solutions in multi-dimensional optimization. The insights presented within will empower researchers and practitioners alike to tackle complex challenges with newfound confidence, leveraging the power of space-filling curves to unlock the full potential of the DIRECT algorithm. This approach promises a significant leap forward in the field, offering a practical and theoretically sound method for addressing the ever-present challenges of global optimization in high-dimensional spaces, leading to breakthroughs in various scientific and engineering domains.