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

arXiv: https://arxiv.org/abs/2202.11014 · HTML: https://mlsanity.com/paper/2202.11014/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Feb 2022.

## Built on by (most cited first, 6 of 6)

- [Diffusion at Absolute Zero: Langevin Sampling Using Successive Moreau Envelopes [conference paper]](https://mlsanity.com/paper/2502.01358/) (2025)
- [Gradient is All You Need? How Consensus-Based Optimization can be Interpreted as a Stochastic Relaxation of Gradient Descent](https://mlsanity.com/paper/2306.09778/) (2023)
- [HJ-sampler: A Bayesian sampler for inverse problems of a stochastic process by leveraging Hamilton-Jacobi PDEs and score-based generative models](https://mlsanity.com/paper/2409.09614/) (2024)
- [Thinking Out of the Box: Hybrid SAT Solving by Unconstrained Continuous Optimization](https://mlsanity.com/paper/2506.00674/) (2025)
- [Probabilistic Gaussian Homotopy: A Probability-Space Continuation Framework for Nonconvex Optimization](https://mlsanity.com/paper/2603.13546/) (2026)
- [Implicit Neural Optimal Transport via Fixed-Point Optimization](https://mlsanity.com/paper/2605.10792/) (2026)

## Abstract (arXiv)

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.