Consider a convex set \(C\). I want to approximate this set by some parametric inequalities of the form \(f_i \le 0 ~\forall i=1, ..., M\). Here \(f_i\) are parametrized by some parameters (e.g. could be hyperplanes etc..). I want to know if I can compute these \(f_i \forall i\) by some convex optimization. I am aware of outer approximation techniques for convex sets. However they are iterative in nature. I want to perform this step as a convex optimization. For example, I can say that I approximate the convex set by a linear inequality and I then run an optimization to compute the parameters of the linear inequality (while optimizing some metric).
asked
anon123 |

How is \(C\) defined/specified? Is it the solution set of a system of explicit inequalities?

Let us assume for the time being that C is characterized by some nonlinear inequalities which I know. Can I do something then? I can outer approximate each of these inequalities by taking derivatives. E.g if there are M inequalities I can then select M points and construct an outer approximation around those points. I want to know how to select these M points so that I arrive at the best possible approximation.

possibly more than M points too. But my goal is to cleverly select these points.

If the convex set is already characterized by known nonlinear inequalities, let's say c_i <= 0, then what's "wrong" or undesirable about them? You have to tell us what you don't like about the existing characterization (c_i <= 0) in order for us to tell you a "better" approximation. Your existing characterization is presumably sharp, so you seems to want to give up some sharpness in exchange for functions f_i which you "like" better than c_i. But you haven't told us what kind of functions f_i you like better than c_i.