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

Heaton, Howard; Fung, Samy Wu; Osher, Stanley

doi:10.1007/s42967-022-00239-5

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2023

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Cited by 4 publications

(1 citation statement)

References 35 publications

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“…For descent algorithms, our aim is to find nearly global optimal solutions using the Hamilton-Jacobi equation-based proximal operator (Heaton et al, 2023;Osher et al, 2023) [36,54].…”

Section: Discussionmentioning

confidence: 99%

Optimised Sparse Image Reconstruction Via Densification and Relocation

Kang,

Chizhov,

Augustin

2023

Preprint

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“…For descent algorithms, our aim is to find nearly global optimal solutions using the Hamilton-Jacobi equation-based proximal operator (Heaton et al, 2023;Osher et al, 2023) [36,54].…”

Section: Discussionmentioning

confidence: 99%

Optimised Sparse Image Reconstruction Via Densification and Relocation

Kang,

Chizhov,

Augustin

2023

Preprint

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A kernel formula for regularized Wasserstein proximal operators

Li,

Liu,

Osher

2023

Res Math Sci

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A Hamilton–Jacobi-based proximal operator

Osher

Heaton²,

Fung

2023

Proc. Natl. Acad. Sci. U.S.A.

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First-order optimization algorithms are widely used today. Two standard building blocks in these algorithms are proximal operators (proximals) and gradients. Although gradients can be computed for a wide array of functions, explicit proximal formulas are known for only limited classes of functions. We provide an algorithm, HJ-Prox, for accurately approximating such proximals. This is derived from a collection of relations between proximals, Moreau envelopes, Hamilton–Jacobi (HJ) equations, heat equations, and Monte Carlo sampling. In particular, HJ-Prox smoothly approximates the Moreau envelope and its gradient. The smoothness can be adjusted to act as a denoiser. Our approach applies even when functions are accessible only by (possibly noisy) black box samples. We show that HJ-Prox is effective numerically via several examples.

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

Cited by 4 publications

References 35 publications

Optimised Sparse Image Reconstruction Via Densification and Relocation

Optimised Sparse Image Reconstruction Via Densification and Relocation

A kernel formula for regularized Wasserstein proximal operators

A Hamilton–Jacobi-based proximal operator

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