  We propose a gradient descent method for solving optimisation problems
arising in settings of tropical geometry - a variant of algebraic geometry that
has become increasingly studied in applications such as computational biology,
economics, and computer science. Our approach takes advantage of the polyhedral
and combinatorial structures arising in tropical geometry to propose a
versatile approach for approximating local minima in tropical statistical
optimisation problems - a rapidly growing body of work in recent years.
Theoretical results establish global solvability for 1-sample problems and a
convergence rate of $O(1/\sqrt{k})$. Numerical experiments demonstrate the
method's superior performance over classical descent for tropical optimisation
problems which exhibit tropical convexity but not classical convexity. Notably,
tropical descent seamlessly integrates into advanced optimisation methods, such
as Adam, offering improved overall performance.
