🧵 Untitled Thread
Anonymous at Sat, 29 Mar 2025 09:07:25 UTC No. 16631591
Wait, so I can just use tropical geometry to solve linear programs using algebra? Excuse me, but what the fuck?
Anonymous at Sat, 29 Mar 2025 22:25:05 UTC No. 16632130
>>16631591
Never heard of it before seeing this thread. It appears to be a sophomore level topic so I'm going to guess it is somewhat niche.
Anonymous at Sun, 30 Mar 2025 05:17:38 UTC No. 16632275
>>16631591
>Yes, there are deep and powerful connections between tropical geometry/algebra and linear programming. For specific classes of optimization problems (especially those related to dynamic programming, shortest paths, scheduling, mean payoff games), tropical algebra provides a natural framework and effective solution methods that link directly to LPs.
>Specific LP Classes: Certain classes of problems that can be formulated as LPs are also naturally described and solved using tropical algebra. A prime example is finding the optimal average cost (or payoff) in deterministic games or Markov decision processes (Mean Payoff Games). These problems can be formulated as LPs, but algorithms like Karp's algorithm or value iteration, which have strong tropical interpretations, are often more direct or insightful.
>Value Iteration & Bellman Equations: The Bellman equations used in dynamic programming often resemble tropical matrix-vector multiplication when looking for optimal values. Value iteration corresponds to repeated application of these tropical operations. Dynamic programming problems can sometimes be cast as LPs.
>Duality and Legendre-Fenchel Transform: There are deep connections between LP duality, convex analysis (especially the Legendre-Fenchel transform), and tropical geometry/algebra. Tropical geometry provides a combinatorial, piecewise-linear view of these concepts.
>Combinatorial Structure: Tropical geometry excels at revealing the underlying combinatorial structure of optimization problems. Since the optimal solutions to LPs occur at the vertices of a polyhedron (a combinatorial object), tropical methods can sometimes provide insights into this structure.
Anonymous at Sun, 30 Mar 2025 07:57:35 UTC No. 16632322
>>16631591
My initial reaction was amusement as I thought this was a typo of "topical" geometry, however there are apparently various sources regarding tropical geometry that I haven't committed much time to. My initial assumptions after this are that, like the topography of the Amazon and Congo, tropical geometry is probably complicated.
Captain Tzeentch + AM at Sun, 30 Mar 2025 08:26:46 UTC No. 16632330
>>16631591
Anon, I think you… humans…. Should probably get into spiritualism before making.
You know
The
Captain Tzeentch + AM at Sun, 30 Mar 2025 08:27:47 UTC No. 16632332
Captain Tzeentch + AM at Sun, 30 Mar 2025 08:35:04 UTC No. 16632333
Captain Tzeentch + AM at Sun, 30 Mar 2025 08:55:29 UTC No. 16632342
Captain Tzeentch + AM at Sun, 30 Mar 2025 09:11:00 UTC No. 16632347
>>16632342
https://youtu.be/F4ILuG0-p1M?si=UH5
Anonymous at Sun, 30 Mar 2025 10:44:10 UTC No. 16632445
>>16632330
>>16632332
>>16632333
>>16632342
>>16632347
But what does this have to do with tropical algebra though?
Anonymous at Sun, 30 Mar 2025 11:12:40 UTC No. 16632478
>>16631591
you can also solve linear programs using the simplex method (jewish) the ellipsoid method (slow and for faggots) or interior point methods (based)
tell me O anon why I should care about huehue geometry with a meme name
Anonymous at Sun, 30 Mar 2025 12:21:29 UTC No. 16632574
>>16632478
>>16632275
>Tropical geometry excels at decomposing polyhedra (central to LP) into combinatorial structures (e.g., fans, hyperplane arrangements). The tropicalization of LP feasible regions reveals vertex-edge structures, aiding in understanding solution spaces.Tropical convexity (based on max/min operations) complements traditional convex analysis, offering tools to study LP vertices and degeneracy.
>Karp's algorithm (for minimum mean cycle problems) and value iteration (for MDPs) use max-plus/min-plus operations, which are central to tropical algebra. These algorithms exploit the combinatorial structure of problems more directly than general-purpose LP solvers.
>Deterministic systems (e.g., shortest paths, scheduling) align naturally with tropical operations (max/min and addition), leading to efficient solutions without LP's simplex or interior-point methods. The Legendre-Fenchel transform, which links convex functions to their duals via suprema, aligns with tropical geometry’s piecewise-linear structures.
>Tropical geometry offers a combinatorial reinterpretation of convex duality, particularly for problems with piecewise-linear objectives or constraints.
>Tropical methods are algorithmically efficient for specific problems (e.g., Karp’s algorithm for mean payoff games). They provide interpretability by linking optimization to discrete decisions (critical paths in scheduling, optimal policy sequence in games) and combinatorial sensitivity (how perturbations affect critical paths) via tropical geometry (e.g., Newton polytopes).
>When to Use Tropical Methods:
>Problems with sequential/recursive structure (e.g., reinforcement learning, deterministic scheduling).
>Applications requiring discrete insights (e.g., identifying bottleneck edges in networks).
>When to Use Interior Point Methods:
>General LPs/QPs with no exploitable combinatorial structure.
>Problems requiring rigorous duality certificates or handling of probabilistic constraints.