As it turns out, convex optimization is being seen by a few folks as the new least square. As wwalker mentioned, convex optimization is currently being used in solving for measurements taken with hardware implementing compressive sensing (or compressed sensing or compressive sampling). The A/D converter is just one of the application of compressive sensing. One of the most well known example is that of the single pixel camera at rice but there are many others. I have listed most of the known hardware implementing compressive sensing here:
One should note that while convex optimization has given some real impetus to the field (by providing theoretical bounds), signal reconstruction is also using speedier techniques nowadays even though linear programming techniques remains some sort of gold standard. For those of you interested in the subject, I write a small blog on the subject of CS:
convex optimization covers a lot of different problem types. in general, you are trying to minimize (or maximize) some function (the objective function) over a set of variables subject to a set of constraints on those variables. 'convex optimization' deals specifically with optimization problems where the objective function is... convex. this essentially means you can efficiently find the global minimum (or maximum if the function is concave)
ok, so a simple example: lets say you're on a hacker's budget and you want to maximize your caloric intake given your limited funds. This is your objective function. BUT, if you're like me, you can't live on ramen and mountain dew... i have a basic set of nutritional requirements. These are your constraints. For example, sodium has to be less than A and protein greater than B and so on and so forth. So you are essentially optimizing over ALL possible diets.
My guess is you'd end getting back: kale, potatoes and beans. Now that's a real hacker's diet!
At my day job, we use linear programming to calculate virtual capacities on airplanes, i.e. overbooking. There's a few other optimization techniques involved, such as dynamic progamming, but if you've ever been asked to take a different plane, chances are an LP was involved.
