I was looking at the book LECTURES ON MODERN CONVEX OPTIMIZATION, by Ben-Tal and Nemirovski, which covers a lot of material on conic optimization or conic programming. The ideas I seem to get, but I am having a hard time understanding the practical examples that the authors give. They provide two examples in chapter 2 of the book: Synthesis of arrays of antennae, and Stability analysis for an uncertain linear time-varying dynamic system; however, these examples require some mathematical sophistication and the authors don't really take the time to develop/explain those examples fully. I mean you can't explain the theory of Lyapunov functions in an example or footnote. This is not a criticism of the text, just a practical issue in their choice of examples.
What I would appreciate are essentially the "Hello World" problems for conic programming and SOCP. I get that the setup of a conic program is usually the same as a Linear Program--at least in this book. So the objective is still $c^Tx$ with the usual convex constraints $Ax \leq b$, however the only difference is that $x \in K$, where $K$ is a cone. Now I understand what an LP is, such as the diet problem or the location of a factory problem. I am trying to understand what the introduction of this conic constraint allows me to do that I could not do before? Another way of saying this, is what does a conic programming or SOCP version of the diet problem or the location of a factory problem look like? What question could I answer that I could not answer before?
Any help is appreciated.
