基于布朗场和稀疏网格的高维仿真优化
High-Dimensional Simulation Optimization via Brownian Fields and Sparse Grids
摘要：
高维仿真优化是众所周知的挑战。我们提出了一种新的采样算法，收敛到一个全局最优解，并遭受最小的灾难的维度。该算法包括两个阶段。首先，我们采取稀疏网格试验设计的样本和近似的响应面通过核岭回归与布朗场内核。其次，我们遵循预期的改进策略-通过关键的修改来提高算法的采样效率-从稀疏网格的下一级迭代采样。在响应面光滑性和模拟噪声条件较好的情况下，建立了无噪声和有噪声模拟样本的收敛速度上界。这些上界在可行集的维度上仅略微恶化，并且如果已知目标函数具有高阶光滑性，则可以改进这些上界。大量的数值实验表明，该算法显着优于典型的替代品在实践中。
英文摘要：
High-dimensional simulation optimization is notoriously challenging. We propose a new sampling algorithm that converges to a global optimal solution and suffers minimally from the curse of dimensionality. The algorithm consists of two stages. First, we take samples following a sparse grid experimental design and approximate the response surface via kernel ridge regression with a Brownian field kernel. Second, we follow the expected improvement strategy -- with critical modifications that boost the algorithm's sample efficiency -- to iteratively sample from the next level of the sparse grid. Under mild conditions on the smoothness of the response surface and the simulation noise, we establish upper bounds on the convergence rate for both noise-free and noisy simulation samples. These upper bounds deteriorate only slightly in the dimension of the feasible set, and they can be improved if the objective function is known to be of a higher-order smoothness. Extensive numerical experiments demonstrate that the proposed algorithm dramatically outperforms typical alternatives in practice.
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