I have a convex set of the form $$Z = \{x|x^TQ_ix+b_i^Tx+c_i\le0,i=1,\ldots,m\}$$ where each $Q_i\succeq0$, that I wish to approximate by a set of the form $$\hat Z = \{x|Ax\le b\}$$ We can further assume that each $x_i$ belongs to an interval $X_i$.
Additionally, if possible, I would like to find an approximation such that $\hat Z \subseteq Z$. Taking tangent hyperplanes to $Z$ in $\mathbb{R}^n$, at various points, to construct $A$ and $b$ would unfortunately create a set that contains points outside of $Z$.
Can anyone point me in the direction of some relevant literature?
Update: I was thinking I could select some points on the surface of the quadratic function and use these point to form hyperplanes that would define $A$ and $b$. The problem becomes, how would one choose these points (a mesh) on the surface to minimize the approximation error? Intuitively, I imagine that generating a mesh with finer points in the regions where the surface has high curvature would yield a good approximation, but how do I find these regions?
Update: For those interested, I have found a couple of references that I plan to study:
Approximation of convex sets by polytopes -- a fairly general treatment
Feasible Region Approximation Using Convex Polytopes -- a specific algorithm