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Tropical gradient descent.

Roan Talbut1, Anthea Monod1

  • 1Department of Mathematics, Imperial College London, London, UK.

Journal of Global Optimization : an International Journal Dealing with Theoretical and Computational Aspects of Seeking Global Optima and Their Applications in Science, Management and Engineering
|November 3, 2025
PubMed
Summary
This summary is machine-generated.

We introduce a new gradient descent method for tropical geometry optimization problems. This approach enhances accuracy and efficiency, especially for problems with tropical convexity but not classical convexity.

Keywords:
Gradient descentLocation problemsPhylogeneticsStatistical optimizationTropical projective torusTropical quasi-convexity

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Area of Science:

  • Tropical geometry
  • Optimization
  • Computational mathematics

Background:

  • Tropical geometry is a growing field with applications in biology, economics, and computer science.
  • Optimization problems in tropical geometry are challenging due to unique mathematical structures.
  • Existing methods may not fully leverage the properties of tropical geometry.

Purpose of the Study:

  • To develop a novel gradient descent method tailored for tropical geometry optimization.
  • To approximate local minima in tropical statistical optimization problems effectively.
  • To enhance the performance of optimization algorithms in tropical settings.

Main Methods:

  • A gradient descent algorithm exploiting polyhedral and combinatorial structures of tropical geometry.
  • Theoretical analysis establishing global solvability for 1-sample problems.
  • Integration with advanced optimization methods like Adam.

Main Results:

  • The proposed method achieves convergence rates comparable to classical gradient descent.
  • Demonstrated superior performance over classical gradient descent for tropical convex problems.
  • Showcased seamless integration with Adam, leading to improved accuracy.

Conclusions:

  • The developed gradient descent method is a versatile tool for tropical optimization.
  • This approach offers significant advantages for problems exhibiting tropical convexity.
  • The method shows promise for advancing research and applications in tropical geometry.