Tropical gradient descent

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.