Here's an excerpt from a comment I previously made on Hacker News:

I'm a Ph.D. student in operations research (OR). My suggestion would be to first build a strong foundation in linear programming. This will introduce you to the notion of duality, which is heavily emphasized in many mathematical programming courses. Here's a good open-source book on linear programming written by Jon Lee, the current editor of Mathematical Programming A: https://github.com/jon77lee/JLee_LinearOptimizationBook

Then I'd suggest studying more general methods for continuous and convex optimization. The book I see mentioned a lot is Convex Optimization by Boyd and Vandenberghe, although we didn't use this in our coursework. Instead, we used a lot of the material presented here: http://mitmgmtfaculty.mit.edu/rfreund/educationalactivities/

If you read the above (or any other two books on linear programming and convex optimization), you'll probably have a better idea of what you want to study next and how you want to go about it. The next natural step would be to study combinatorial (i.e., integer or mixed-integer) optimization. (Jon Lee has another book on this subject; I've also heard good things about the Schrijver book.)