I am somewhat new to OR. I have taken up courses on Linear Programming, Game Theory and Queueing Theory and I know a bit of Dynamic Programming. Soon I shall be doing a project on OR or Optimization and haven't yet decided on the topic. So before selecting the topic, I wish to get an overview (i.e., what kinds of problems are studied) of the different areas of OR and Optimization. Could you provide some good reference(s) for the above. I repeat, I am not asking for resources to study the various areas in depth.
Note: I am interested more in the math of the subject than the applications.
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3 Answers
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Optimization is a very large area in terms of the types of problems and models. You have taken a course on linear programming, in which (a) the problem is deterministic (you assume that you know at the outset everything required to find an optimal solution), (b) the variables are real-valued and "continuous" (perhaps more correctly, "divisible"), meaning fractional values are allowed, (c) there is a single criterion (objective) function, (d) all constraints (and the objective function) are linear, and (e) there are no not-equal-to constraints (no $a^T x \neq b$). Pretty much every combination of those assumptions can be dropped, leading to another branch of the optimization family tree.
If you make all (some) variables integer-valued but leave all the other assumptions alone, you get integer (mixed-integer) linear programming. Let the object function be quadratic (all other assumptions holding) and you have quadratic programming. Let some variables be integer, the objective quadratic and some constraints quadratic (of the right form) and you have MIQCQP (mixed-integer quadratically constrained quadratic programming).
The assumption of a deterministic problem can also be dropped, leading to stochastic optimization, including among its forms chance constrained programming and stochastic programming with recourse (which differ in how the lack of determinism is handled). Drop the assumption of a single criterion and you have multicriterion optimization, which includes special cases such as goal programming, fuzzy goal programming, Pareto optimization, archimean weighted objective functions and other approaches.
That's not an exhaustive list by any means. I've intentionally not even touched optimal control theory. Importantly, that's just optimization. Stochastic processes, simulation and other OR areas are also out there (and also prone to forking into progressively more specific subareas with progressively longer acronyms).
You said you were not primarily interested in the applications, and this is already a very long answer, so I won't go into those. I will say that for just about every type of OR problem or model, there are real-world applications to be found.
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I would suggest checking out the blog posts of Prof Powell here. Although these mostly refer to stochastic optimization, he illustrates a problem that persists in the field of optimization, which is that different fields (dynamic programming, reinforcement learning, mathematical programming) have different notation and study similar things under different names. In the blog you can also find a link for an introductory book which is application oriented, so you can see what interests you.
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Informs has a list of case studies of different area of OR applications: https://pubsonline.informs.org/page/ited/cases
