论文标题
通过汉密尔顿 - 雅各比PDES的进化，全球解决非凸问题的解决方案
Global Solutions to Nonconvex Problems by Evolution of Hamilton-Jacobi PDEs
论文作者
论文摘要
计算任务通常可以作为优化问题。现实世界情景的目标函数通常是非convex和/或非不同的。解决这些问题的最新方法通常仅保证融合到本地最小值。这项工作介绍了基于汉密尔顿 - 雅各布基的Moreau自适应下降（HJ-MAD），这是一种零级算法，并保证了目标函数的连续性，并保证了与全球最小值的融合。核心思想是用自适应平滑参数计算物镜的莫罗包络的梯度（即“零件凸”）。 Moreau Invelope \ rev {（（\ ie近端操作员）}的梯度通过粘性汉密尔顿 - 雅各比方程的Hopf-Lax公式近似。我们的数值示例说明了全球融合。
Computing tasks may often be posed as optimization problems. The objective functions for real-world scenarios are often nonconvex and/or nondifferentiable. State-of-the-art methods for solving these problems typically only guarantee convergence to local minima. This work presents Hamilton-Jacobi-based Moreau Adaptive Descent (HJ-MAD), a zero-order algorithm with guaranteed convergence to global minima, assuming continuity of the objective function. The core idea is to compute gradients of the Moreau envelope of the objective (which is "piece-wise convex") with adaptive smoothing parameters. Gradients of the Moreau envelope \rev{(\ie proximal operators)} are approximated via the Hopf-Lax formula for the viscous Hamilton-Jacobi equation. Our numerical examples illustrate global convergence.
