I’m currently working on a constrained optimization problem (the problem comes from the European Power Market) where the constraints define a solutions space which forms a complex polytope with many facets. I’m interested in finding a simpler polytope that approximates the original solutions domain, but with fewer facets. At the end, I am looking for an algorithm that, given a set A of n constraints, returns a set B of m constraints, where m < n.
Below is my specific constraints set, but my question is much more general and can be applied to many different situations! The initial set of constraints is defined as below, you have approximately 130 constraints after pre-solving: $$ \sum_{z} PTDF_{z,cb} \cdot nex_z \leq RAM_{cb} \quad \forall cb \in CB. $$ In a matrix form: $$ PTDF \cdot Nex \leq RAM \\ \text{PTDF} \in \mathbb{R}^{130 \times 14} \\ \text{Nex} \in \mathbb{R}^{14 \times 1} \\ \text{RAM} \in \mathbb{R}^{130 \times 1} $$
This set of constraints defines a solution space which is a 14-Dimensional convex polytope. In 2D, it is a polyhedron like this: 2D polyhedron
The constraints are linear and the solution space is convex. I repeat: the set of constraints has already been passed through a pre-solver which removed redundant or useless constraints.
Question: How can you simplify the set of constraints so as to reduce down the number of constraints to let's say a dozens of them? In other words, how can you prune the complex polyhedron to get a simpler polyhedron with less facets but still approximating the complex one?
Any advice or references to relevant literature would be greatly appreciated. Thank you in advance for your help!