Proving ceil(ceil(x)/a) = ceil(x/a) using Optimization

While flipping through the recursion section of some algorithms text, I got curious as to whether Indeed, it is true. Below is a proof I came up with that interestingly enough exploits the fact that any ceiling function can be written as an optimization problem. Proof. Using simple algebra, we may rewrite the nested ceiling …

Advertisements

Alpha, Beta, Sharpe: An Accessible Overview of Finance Jargon

First, a bit of background. Alpha, Beta, Sharpe, and R2 are metrics that were proposed over 50 years ago by academics (like Eugene Fama and William Sharpe) that sought to understand financial markets. It was part of a model known as CAPM (Capital Asset Pricing Model) which is one of the only unified frameworks unique …

Why Convexity?

Etymologically, the term convex originates from the latin word convexus—which translates to arched. It is not uncommon for one to wonder why convexity seems to accompany the word "optimization" in almost every context? Why are convex functions, sets, and formulations so important? Why is there a convex relaxation (such as LASSO) craze sweeping the machine learning community, both in academia and industry? T. Rockafellar’s Convex …