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I have a series of similar linear programs that depend on an input vector $a\in A$ and whose solution is an output vector $b\in B$. I can solve them individually, but this is wasteful. I suspect that for a small $A$ there are linear mapping from elements in $A$ to elements in $B$. I want to discover this mapping to avoid solving the linear program every time.

What can I do to discover this formula? My current approach is to solve various of these problems and use them solutions to solve a system of linear equations. It works, but it is not very elegant. Are there any standard approaches to accomplish this?

Edit: Just to be clear, I think robust linear programming does not cut it, there the problem is still to find a single feasible solution for a range of parameters, not an solution that depends on the input.

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You should look into the topic of "parameteric linear programming". In certain situations where the problem data vary linearly with parameters, you can derive formulas for the optimal solution and optimal value as a function of the parameters.

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