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

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外文摘要：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 func-tion.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 Hamil-ton-Jacobi equation.Our numerical examples illustrate global convergence.

外文关键词：

Global optimizationMoreau envelopeHamilton-JacobiHopf-LaxCole-HopfProximalsZero-order optimization

作者：

Howard Heaton、Samy Wu Fung、Stanley Osher

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作者单位：

Typal Research,Typal LLC,Los Angeles,USA

Department of Applied Mathematics and Statistics,Colorado School of Mines,Golden,USA

Department Mathematics,UCLA,Los Angeles,USA

基金：

&&&&&&&&NSF Graduate Research Fellowship

项目编号：

AFOSR MURI FA9550-18-502ONR N00014-18-1-2527N00014-18-20-1-2093N00014-20-1-2787DGE-1650604

出版年：

2024

DOI：

10.1007/s42967-022-00239-5

应用数学与计算数学学报

上海大学

应用数学与计算数学学报

影响因子：0.165

ISSN：1006-6330

年,卷(期)：2024.6(2)