One more question: What is optimization/linear programming like? From what I've seen it looks like more complicated systems of linear equations, but I'm sure there's more to it than that.
Fun is what it is.
LP, NLP, QP, QCP, MILP, MINLP... once you know the basics, there are so many areas of application. Not just industrial, I know I use it when optimizing designs in games e.g. ship and missile designs in Aurora 4X.
minx,y f(x,y)
s.t.
g(x,y) <= 0
h(x,y) = 0
y ∈ ℤ
Above is a generic example, but it's not the only form (or class!) of optimization problems.
You'll obviously need to learn to model the scenario (or even relax it to make it tractable), figure out if the problem is convex or not and apply appropriate methods/algorithms. Be prepared to learn GAMS, as it's a widely used industrial software. There's a free version of it (limited as always
) but if you can make the problem small enough to run on the free version, you can scale it up and run on
NEOS servers for free. Excel also has solvers though I don't recall if any of them are global solvers, so you can only expect local optima. OpenOffice Calc has a MILP solver. Python's scipy I believe also has solvers.
That is, if you plan on going a bit more into the engineering side of things.
On the theoretical side of things, I imagine you'd be looking into creating algorithms, finding ways to rewrite non-convex expressions to convex expressions, or conditions for optima such as the Karush-Kuhn-Tucker conditions. I haven't seen much use for algebraic solutions for anything non-trivial (as numeric solutions are far vaster).