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A Hamilton-Jacobi-based proximal operator.

Stanley Osher1, Howard Heaton2, Samy Wu Fung3

  • 1Department of Mathematics, University of California, Los Angeles, CA 90095.

Proceedings of the National Academy of Sciences of the United States of America
|March 29, 2023
PubMed
Summary
This summary is machine-generated.

We introduce HJ-Prox, a novel algorithm for approximating proximal operators used in optimization. This method, applicable to black-box functions, offers a flexible approach to complex optimization problems.

Keywords:
Cole–HopfHamilton–JacobiMoreauproximalzeroth-order

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Area of Science:

  • Optimization Algorithms
  • Numerical Analysis
  • Mathematical Modeling

Background:

  • First-order optimization algorithms rely on gradients and proximal operators.
  • Explicit proximal formulas are limited to specific function classes.
  • Approximating proximals is crucial for broader algorithm applicability.

Purpose of the Study:

  • To develop a novel algorithm, HJ-Prox, for accurately approximating proximal operators.
  • To provide a method applicable to functions accessible only via black-box samples.
  • To demonstrate the numerical effectiveness of the proposed approach.

Main Methods:

  • Derivation from relationships between proximals, Moreau envelopes, and Hamilton-Jacobi (HJ) equations.
  • Utilizing heat equations and Monte Carlo sampling for approximation.
  • Smooth approximation of the Moreau envelope and its gradient.

Main Results:

  • HJ-Prox accurately approximates proximal operators, even for black-box functions.
  • The algorithm's smoothness can be adjusted, functioning as a denoiser.
  • Numerical examples confirm the effectiveness of HJ-Prox.

Conclusions:

  • HJ-Prox offers a versatile solution for approximating proximal operators in optimization.
  • The method extends the applicability of optimization algorithms to a wider range of functions.
  • This work bridges theoretical concepts like HJ equations with practical numerical methods.