Optimization algorithms are pivotal in advancing various scientific and industrial fields but often encounter obstacles such as trapping in local minima, saddle points, and plateaus (flat regions), which makes the convergence to reasonable or near-optimal solutions particularly challenging. This paper presents the Steepest Perturbed Gradient Descent (SPGD), a novel algorithm that innovatively combines the principles of the gradient descent method with periodic uniform perturbation sampling to effectively circumvent these impediments and lead to better solutions whenever possible. SPGD is distinctively designed to generate a set of candidate solutions and select the one exhibiting the steepest loss difference relative to the current solution. It enhances the traditional gradient descent approach by integrating a strategic exploration mechanism that significantly increases the likelihood of escaping sub-optimal local minima and navigating complex optimization landscapes effectively. Our approach not only retains the directed efficiency of gradient descent but also leverages the exploratory benefits of stochastic perturbations, thus enabling a more comprehensive search for global optima across diverse problem spaces. We demonstrate the efficacy of SPGD in solving the 3D component packing problem, an NP-hard challenge. Preliminary results show a substantial improvement over four established methods, particularly on response surfaces with complex topographies and in multidimensional non-convex continuous optimization problems. Comparative analyses with established 2D benchmark functions highlight SPGD's superior performance, showcasing its ability to navigate complex optimization landscapes. These results emphasize SPGD's potential as a versatile tool for a wide range of optimization problems.

翻译：优化算法在推动科学与工业各领域发展中具有关键作用，但常面临陷入局部极小值、鞍点及平坦区域等障碍，这使得收敛至合理或近似最优解尤为困难。本文提出最陡扰动梯度下降算法（SPGD），该新颖算法创新性地将梯度下降法原理与周期性均匀扰动采样相结合，以有效规避这些障碍并在可能时导向更优解。SPGD的独特设计在于生成一组候选解，并选择相对于当前解具有最陡损失差异的解。它通过集成一种策略性探索机制来增强传统梯度下降方法，该机制显著提升了逃离次优局部极小值及有效遍历复杂优化景观的可能性。我们的方法不仅保留了梯度下降的定向效率，同时利用了随机扰动的探索优势，从而能在多样化问题空间中进行更全面的全局最优解搜索。我们通过求解三维元件排布这一NP难问题验证了SPGD的有效性。初步结果显示，相较于四种现有方法，SPGD在复杂地形响应曲面及高维非凸连续优化问题上均取得显著改进。与经典二维基准函数的对比分析突显了SPGD的卓越性能，展示了其驾驭复杂优化景观的能力。这些结果彰显了SPGD作为广泛优化问题通用工具的潜力。