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Updated: Dec 6, 2025

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Physics successfully implements Lagrange multiplier optimization.

Sri Krishna Vadlamani1, Tianyao Patrick Xiao2, Eli Yablonovitch1

  • 1Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720; srikv@berkeley.edu eliy@eecs.berkeley.edu.

Proceedings of the National Academy of Sciences of the United States of America
|October 13, 2020
PubMed
Summary
This summary is machine-generated.

Physical machines can solve complex optimization problems by leveraging principles like minimum power dissipation. This research reveals these physical optimization methods are equivalent to mathematical Lagrange multiplier techniques for constrained problems.

Keywords:
Ising solvershardware acceleratorsphysical optimization

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Area of Science:

  • Physics
  • Computational Science
  • Applied Mathematics

Background:

  • Optimization is a fundamental aspect of both mathematics and physics, with principles like Least Action and Minimum Power Dissipation inherent in physical systems.
  • Physical annealing and adiabatic principles, including quantum annealing, predate their computational counterparts and demonstrate physics' capacity for solving optimization problems.
  • Recent advancements have seen various physical machines successfully optimize the Ising magnetic energy, a significant computational challenge.

Purpose of the Study:

  • To demonstrate that numerous physical optimization machines operate based on Onsager's Principle of Minimum Power Dissipation.
  • To establish the equivalence between physical optimization methods and Lagrange multiplier optimization for constrained problems.
  • To elucidate the role of physical gain coefficients in these optimization systems.

Main Methods:

  • Analysis of diverse physical optimization machines, particularly those optimizing Ising magnetic energy.
  • Application of Onsager's Principle of Minimum Power Dissipation to understand the underlying physics of these machines.
  • Mathematical comparison of physical optimization with established Lagrange multiplier techniques for constrained optimization.

Main Results:

  • It is demonstrated that most physical machines used for optimization function according to the Principle of Minimum Power Dissipation.
  • The optimization performed by these physical systems is shown to be mathematically equivalent to Lagrange multiplier optimization for problems with constraints.
  • Physical gain coefficients within these systems are identified as playing a role analogous to Lagrange multipliers.

Conclusions:

  • Physical optimization machines inherently utilize fundamental physical principles like minimum power dissipation.
  • The observed physical optimization processes are mathematically rigorous and equivalent to standard constrained optimization techniques.
  • Understanding the role of gain coefficients deepens the connection between physical systems and mathematical optimization theory.