Constraining an ellipse to an ellipse

1

I’ve been stuck on this for quite some time and I really need to know if this is possible or not under DCP.

The optimization problem is just finding the maximum volume inscribed ellipsoid subject to linear inequality constraints. For reasons not relevant to the issue I need to do this multiple times, each time additionally constraining the ellipse to the ellipse from the previous iteration. I cannot figure out how to make the constraint so that the entire ellipse being optimized for stays in the other ellipse while also being DCP compliant.

For the sake of clear communication, the variables are
B: an NxN PSD matrix being the ellipse; an optimization variable
d: an N-dim vector being the ellipse center; an optimization variable
P: an NxN PSD matrix being constraining the ellipse; a parameter
q: an N-dim vector being the center of the constraining ellipse; a parameter

2

Read section 8.4.2 “Maximum volume inscribed ellipsoid” of Convex Optimization – Boyd and Vandenberghe

After carefully studying and thinking about that section, you can also look at http://web.cvxr.com/cvx/examples/cvxbook/Ch08_geometric_probs/html/min_vol_union_ellip.html .

Amd then add your linear constraints.

3

Wow, thank you so much. I’ve read Convex Optimization – Boyd and Vandenberghe before and I have no idea how I missed the “Maximum volume ellipsoid in an intersection of ellipsoids” section. Looks like this is exactly what I need.