ChapterZero

In a fortuitous turn, we happened to be introduced to convex sets on Monday with a view towards proving the classical result that a convex set is the intersection of the half spaces containing it. I say fortuitous because, although I have heard this mentioned before, I never made the connection between that fact and the often stated fact that it is easy to deal with convex optimization.

I just had the thought that this result has a great deal to do with optimization, because to find a suitable vector in the convex set, you can start off with a vector outside of the set, and project it onto half spaces containing the set, and refine each approximation by projecting it onto another half space. Of course, that doesn’t deal so much with the optimization part of the problem so much as it does with ensuring that the answer converged to is in the right set. Maybe optimization comes in with the choice of projection operators.