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I have been looking into tropical algebra/geometry for a research problem I'm working on in optimization. Tropical math gets referenced a lot in the literature, but it seems to me that its mostly just rephrasing equations with "max" or "min" constraints into a tropical algebraic equation and then saying hey look we can do this. I haven't found that many examples where the tropical framework actually provides a new solution or application, so I was wondering if anyone had examples?

Here are two papers that I think did this well:

https://arxiv.org/pdf/1904.01082.pdf In this paper the authors rephrase a problem into tropical math and use it to come up with an approach to a generalization of an existing problem, which is (in my opinion) a lot more natural in this setting. They use this insight to come up with a new algorithm to solve the problem, which has trade-offs with the existing algorithms.

https://arxiv.org/pdf/1805.07091.pdf I would say that this paper is in the camp of just rephrasing things into the tropical setting, but its about neural networks, where a major open problem is coming up with interpretations, so I would say that this counts.

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There is recently more work on applying tropical geometry to phylogenetics. The key idea is to take advantage of the tropical convexity of the tree space.

One paper satisfying your requirements is Tropical medians by transportation which provides a new solution to the consensus problem from systematic biology.

Although it is proven that the formulation is equivalent to a transportation problem (which is well-studied), the application to biology wouldn't be possible without the tropical point of view.

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