Global Solutions to Nonconvex Problems by Evolution of Hamilton-Jacobi PDEs  

作　　者：Howard Heaton Samy Wu Fung Stanley Osher 

机构地区：[1]Typal Research,Typal LLC,Los Angeles,USA [2]Department of Applied Mathematics and Statistics,Colorado School of Mines,Golden,USA [3]Department Mathematics,UCLA,Los Angeles,USA

出　　处：《Communications on Applied Mathematics and Computation》2024年第2期790-810,共21页应用数学与计算数学学报（英文）

基　　金：partially funded by AFOSR MURI FA9550-18-502,ONR N00014-18-1-2527,N00014-18-20-1-2093,N00014-20-1-2787;supported by the NSF Graduate Research Fellowship under Grant No.DGE-1650604.

摘　　要：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(i.e.,proximal operators)are approximated via the Hopf-Lax formula for the viscous Hamilton-Jacobi equation.Our numerical examples illustrate global convergence.

关 键 词：Global optimization Moreau envelope HAMILTON-JACOBI Hopf-Lax-Cole-Hopf Proximals Zero-order optimization 

分 类 号：O175[理学—数学]