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?