Parametric programming problems are problems whose structure (most often sparsity structure) is known.
Most quadratic and linear programming solvers assume a fully generic problem structure. You give them any matrix and they'll spit out an answer. This means that internally they must also use generic matrix operations.
If you know the full structure of the optimization problem and how it changes with respect to changes in parameters, you could in principle compile a unique solver that is fully specialized to the parameterized problem. If your problem is particularly simple, you could even enumerate all possible answers in a lookup table.
Now here is the problem: While it is not that difficult to find the sparsity pattern of the optimization problem and choose the appropriate sparse matrix library based on the problem, actually producing a unique program that does nothing but solve your parametric optimization problem is a pipe dream. Most approaches involve a lot of human ingenuity to find exploitable structures in specific problems that become useless the moment you change the problem even a little bit.
So here is my suggestion: figure out how to use genetic programming to produce fast quadratic programming solvers to specific problems.
Note that you don't necessarily need a perfect solution since you can always run a bunch of QP steps on an initial guess to nudge it towards convergence. The goal would be to reduce the number of these steps to as close to zero as possible.
Most quadratic and linear programming solvers assume a fully generic problem structure. You give them any matrix and they'll spit out an answer. This means that internally they must also use generic matrix operations.
If you know the full structure of the optimization problem and how it changes with respect to changes in parameters, you could in principle compile a unique solver that is fully specialized to the parameterized problem. If your problem is particularly simple, you could even enumerate all possible answers in a lookup table.
https://en.m.wikipedia.org/wiki/Parametric_programming
Now here is the problem: While it is not that difficult to find the sparsity pattern of the optimization problem and choose the appropriate sparse matrix library based on the problem, actually producing a unique program that does nothing but solve your parametric optimization problem is a pipe dream. Most approaches involve a lot of human ingenuity to find exploitable structures in specific problems that become useless the moment you change the problem even a little bit.
So here is my suggestion: figure out how to use genetic programming to produce fast quadratic programming solvers to specific problems.
Note that you don't necessarily need a perfect solution since you can always run a bunch of QP steps on an initial guess to nudge it towards convergence. The goal would be to reduce the number of these steps to as close to zero as possible.