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Spatial Multiobjective Optimization of Agricultural Conservation Practices using a SWAT Model and an Evolutionary Algorithm
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Gradient flows and proximal splitting methods: A unified view on accelerated and stochastic optimization.

Guilherme França1,2, Daniel P Robinson3, René Vidal2

  • 1University of California, Berkeley, California 94720, USA.

Physical Review. E
|June 17, 2021
PubMed
Summary

This study reveals that five proximal algorithms are discretizations of a single gradient flow. Applying similar schemes to Newton's equation yields accelerated proximal algorithms, unifying optimization methods as simulations of dissipative systems.

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Spatial Multiobjective Optimization of Agricultural Conservation Practices using a SWAT Model and an Evolutionary Algorithm
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Area of Science:

  • Optimization
  • Machine Learning
  • Statistics
  • Physics
  • Nonlinear Functional Analysis

Background:

  • Optimization is fundamental to machine learning, statistics, and physics.
  • Proximal algorithms are well-suited for complex optimization problems.
  • Existing accelerated methods are difficult to analyze and lack a guiding principle.

Purpose of the Study:

  • To unify proximal algorithms under a single framework.
  • To develop new accelerated proximal algorithms.
  • To connect optimization methods with classical dissipative systems.

Main Methods:

  • Demonstrating that five proximal algorithms are discretizations of Cauchy's gradient flow.
  • Applying discretization schemes to Newton's equation with a dissipative force (accelerated gradient flow).
  • Extending methods to stochastic settings, linking to Langevin and Fokker-Planck equations.

Main Results:

  • All five known proximal algorithms are shown to be discretizations of a single gradient flow.
  • New accelerated variants of proximal algorithms are derived using accelerated gradient flow.
  • Connections are established between optimization, stochastic processes, and classical dissipative systems.

Conclusions:

  • A unified framework is provided for several important optimization methods.
  • Proximal and accelerated optimization algorithms can be viewed as simulations of classical dissipative systems.
  • This work offers a new perspective on the analysis and development of optimization algorithms.