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Linear Programming is strongly entwined with linear algebra, as are many of its generalizations under the heading of mathematical programming / convex optimization.

What analogies are there for convex optimization techniques outside of vector spaces dealt with in linear algebra. For example, Gaussian Elimination is generalized by Buchberger's algorithm for finding Groebner bases (or so I'm told); is there any algorithm that has a relationship with Buchberger's analogous to the Simplex Method's relationship with Gaussian Elimination?

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Here is an explicit example: Max-plus control and tropical algebra

(using the above words as keywords leads to several other relevant results).

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