发表机构
Department of Applied Mathematics, University of Colorado, Boulder; School of Industrial and Systems Engineering, Georgia Tech（科罗拉多大学博尔德分校应用数学系; 佐治亚理工学院工业与系统工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出Powell风格的基于模型的信赖域导数无优化方法变体，推导其复杂度界限，并在随机子空间和噪声场景下验证，实现理论与实践性能的对比。
AI 中文摘要
我们提出了基于模型的信赖域导数无优化算法的变体，这些变体最接近Powell最初提出并实现的方法。这些方法依赖于低次多项式插值，并仔细维护插值集的几何结构。我们能够为这些方法推导出复杂度界限，使其在理论上与其他导数无优化方法具有竞争力。在随机生成的子空间中应用这些方法，我们恢复了我们认为接近紧致的复杂度。本文建立在最近的结果之上，其中推导了Powell方法的极大简化版本的复杂度。在这里，我们将分析扩展以完全纳入Powell的几何处理方式，并进行广泛的数值比较，将基于模型的信赖域方法的实践与理论性能联系起来。我们还将子空间基于模型的信赖域方法的分析扩展到函数评估存在噪声的情况。
英文摘要
We propose variants of model-based trust region derivative-free algorithms that are closest to methods initially proposed and implemented by Powell. These methods rely on low degree polynomial interpolation and carefully maintain geometry of the interpolation sets. We are able to derive complexity bounds for these methods that make them theoretically competitive to other derivative-free methods. Applying these methods in randomly generated subspaces recovers what we believe to be nearly tight complexity. This paper builds on recent results where complexity of a much simplified version of Powell's methods was derived. Here, we extend the analysis to fully incorporate Powell's geometry handling approach and conduct an extensive numerical comparison of the model-based trust region methods connecting practical and theoretical performance. We also extend the analysis of subspace model-based trust region methods to the case of noisy function evaluations.
