Operations Research resources

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People with different backgrounds join Google's Operations Research team. Some are PhDs and well-known in their field; others are excellent software engineers enthusiastic about learning mathematical optimization.

Sometimes the software engineers ask the OR experts how to learn more about OR. We started to collect our answers in a document, excerpted below. These are opinions of individual Googlers, not official Google endorsements. We hope you enjoy eavesdropping on our team conversation!


Course Author Notes Comments
Coursera class on Discrete Optimization van Hentenryck MIP & CP Kvothe@: I loved this. Still haven't finished the final problem set though.
Basic Modeling for Discrete Optimization Lee & Stuckey More of a focus on CP
Advanced Modeling for Discrete Optimization Lee & Stuckey
Solving Algorithms for Discrete Optimization Lee & Stuckey
Modeling and Solving AI Problems in Picat Barták
OR(1): Models and Applications Kung Zaphod@: These and the next two are a great introduction to all things LP/IP.
OR(2): Optimization Algorithms Kung
OR(3): Theory Kung

LP & MIP Basics

Cover Title Author Comments
Cover of Introduction to Linear Optimization Introduction to Linear Optimization Bertsimas & Tsitsiklis BlackLotus@: For LP (and to a lesser extent MIP), I think this book is best.

Patrick@: Downvoting Bertsimas-Tsitsiklis as it's more for a "Second Course" on linear programming, and for that it's probably best together with Introduction to Linear Optimization.

BadBoy@: I need to have a look at this one. I usually do not like the way these guys present stuff, but I may be wrong.

Kvothe@: Chapters 10 ("Integer programming formulations") and 11 ("Integer programming methods") are great.
Cover of Linear Programming Linear Programming Vanderbei
Cover of Combinatorial Optimization Combinatorial Optimization: Polyhedra and Efficiency Schrijver SpiderWoman@: I remember liking Schrijver's "Combinatorial Optimization" way back when, but it's very mathematical and not something I'd recommend to someone joining the team for example…
Cover of Theory of Linear and Integer Programming Theory of Linear and Integer Programming Schrijver BadBoy@: Cool to show off in your library, when doing an interview or to impress someone. You most likely will not read it, and you won’t like it, unless you have a PhD in pure, twice-distilled math. So not the thing to start LP or MIP with. This being said, it contains a wealth of proofs and interesting information. Things like totally unimodular matrices and what they entail. And the bibliography is incredibly well detailed, with citations in the original languages. It’s a kind of Knuth’s Art of Computer Programming. Only this one is not digestible.

Kvothe@: Haven't read it, but distrust it based on typeface alone.
Cover image of A First Course in Linear Optimization A First Course in Linear Optimization Lee Freely Available under a CC license!
Cover of Introduction to Mathematical Optimization Introduction to Mathematical Optimization Fischetti BadBoy@: I went through the Italian version. Looks very good. I love what Fischetti does in general.
Cover of Linear Programming Linear Programming Chvatal BadBoy@: I don't like the book but it's where I learnt everything LP, and the notation is great.
Cover of Combinatorial Optimization Combinatorial Optimization Papadimitriou & Steiglitz BadBoy@: I loved it. It’s outdated, but you should read it.

Kvothe@: A bit dry for my taste.
Cover of Integer Programming Integer Programming Wolsey Unicorn@: Very terse, but covers most of the interesting parts of the field (from a solver perspective)
Cover of Integer Programming Integer Programming Conforti, Cornuéjols, & Zambelli Patrick@: Probably the most up-to-date book on MIP theory/methodology.
Cover of Facets of Combinatorial Optimization Facets of Combinatorial Optimization Jünger & Reinelt Patrick@: More on the theoretical side and biased towards the work by former ZIB director Martin Grötschel (it's from his 65th birthday celebration), but it includes what I think is the latest version of this computational MIP survey: "Tobias Achterberg and Roland Wunderling. Mixed integer programming: Analyzing 12 years of progress".
Cover of 50 Years of Integer Programming 50 Years of Integer Programming: 1958-2008 Jünger et al., ed. Patrick@: Slightly outdated, but a very good review of history and MIP state-of-the-art.
Cover of Network Flow Algorithms Network Flow Algorithms Williamson Unicorn@: A good book with many very recent results about network flows, while still being intuitive. Only for network flows, though, so not that generic. More complete review in French.
Cover of Algorithms Illuminated Algorithms Illuminated: Algorithms for NP-Hard Problems Roughgarden Unicorn@: Probably not the most advanced book of the pack! Still, it provides an intro to some OR algorithms (from the point of view of an algorithms course). Very readable! More complete review in French.
Cover of Practical Optimization Practical Optimization Gill, Murray, & Wright Unicorn@: Old reference book about continuous optimization. If you need any explanation about this family of algorithms, this book has you covered. (More complete review in French.)
Cover of Introduction to Optimization and Hadamard Semidifferential Calculus Introduction to Optimization and Hadamard Semidifferential Calculus Delfour Unicorn@: Very formal book about semidifferential optimization. Not easy to get into. More complete review in French.
Cover of The Moment-SOS Hierarchy The Moment-SOS Hierarchy: Lectures in Probability, Statistics, Computational Geometry, Control and Nonlinear PDEs Henrion, Korda, & Lasserre Unicorn@: If you’re optimizing with polynomials or wondering how far you can go with these, you will get the basics of the SoS hierarchy and unfamiliar applications. More complete review in French.
Cover of Introduction to Operations Research Introduction to Operations Research Hillier & Lieberman Kvothe@: A nice mix of theory & practice. A good first text for people new to the field, with worked-out examples and lots of exercises, some with answers in the back of the book. Downsides: the book tries a bit too hard to funnel users to its website, and it uses obsolete solvers.

Research Reviews

Review Author Comments
175 Years of Linear Programming Chandru & Rao BadBoy@: It's a great series of articles. I was exposed to this at IBM in the early 1990s. I do not know who first had the idea of presenting linear programming like this, but Vijay Chandru and Jean-Louis Lassez were involved, too.

The nice thing about it is that you need only entry-level linear algebra to understand it, and you can prove almost any important theorem in LP with the basics. The best would be a book on LP with this, plus some Chvatal, some Vanderbei, and then implementation issues and references to the relevant books. Chvatal and Vanderbei lack a bit of solid mathematical ground.

It's old, and should soon be renamed 200 years of Linear Programming. It's possible there were earlier attempts.

Research Articles

Article Author Comments
A new polynomial-time algorithm for linear programming Karmarkar BadBoy@: Karmarkar’s paper on Karmarkar’s algorithm. The example of how a paper should not be written. It took years to get to a working implementation, and in the meantime they discovered it was yet another interior point method.



Cover Title Author Comments
Cover of Model Building in Mathematical Programming Model Building in Mathematical Programming Williams A focus on LP and MIP.

Temere@: I really did not like it. The structure is weird (and artificially boosting the number of pages). And it's heavily rooted in "classical OR applications" (focus on economy or almost toy-like looking planning) with little relevance to MIP models we usually do at Google

Azalee@: Agreed.

BadBoy@: I still think the book was great back in the days. I had a look at it maybe 2 years ago, and gosh. It is outdated. Also, I’ve known the author since 1990, and we reconnected at ISMP 2015. He is a great guy, retired, traveling to conferences on his money and still making great presentations. His papers were great, especially on Fourier elimination. He has a very broad vision of what LP is. He was instrumental in starting XpressMP.
Cover of Applications of Optimization with XpressMP Applications of Optimization with XpressMP Guéret, Prins, Sevaux, & Heipcke

Solver-issued Modeling Guides

Guide Description Comments
MOSEK Modeling Cookbook Focuses on conic convex optimization. Unicorn@ A real reference for me when doing nonlinear modelling.
MOSEK Portfolio Cookbook Conic models for portfolio optimization

Research Reviews: MIP

Review Author Description
Mixed integer linear programming formulation techniques Vielma Focuses on the strength and size of mixed-integer formulations for unions of polyhedra-like piecewise linear functions. More on the theoretical side, but includes some practical techniques like incremental formulations in section 8.
Nonconvex piecewise linear functions: Advanced formulations and simple modeling tools. Huchette & Vielma More recent techniques for piecewise linear functions that are not included in the above review.

Research Reviews: MINLP

Review Author Description
Mixed-integer convex representability Lubin, Vielma, & Zadik For convex relaxations only.

Optimization Under Uncertainty

Stochastic Optimization

Cover Title Author Comments
Cover of Lectures on Stochastic Programming Lectures on Stochastic Programming: Modeling and Theory Shapiro, Dentcheva, & Ruszczynski
Cover of Introduction to Stochastic Programming Introduction to Stochastic Programming Birge & Louveaux Unicorn@: A more theoretical introduction to the topic. I don’t recommend it as much as Lectures on Stochastic Programming.

Research Reviews

Review Author
Optimization of Conditional Value-at-Risk Rockafellar & Uryasev

Robust Optimization

Cover Title Author Comments
Cover of Robust Optimization Robust Optimization Ben-Tal, El Ghaoui, & Nemirovski PDF.
Unicorn@: A great reference if the reviews below aren't detailed enough. A large part is devoted to nonlinear problems (typically not presented in the reviews).
I really like its Section 1.1.2, because it shows numerically that small coefficient deviations can make large infeasibilities.

Research Reviews

Review Author
A Practical Guide to Robust Optimization Gorissen, Yanıkoğlu, & den Hertog
Theory and Applications of Robust Optimization Bertsimas, Brown, & Caramanis

Research Articles

Article Author
Tractable Stochastic Analysis in High Dimensions via Robust Optimization (PDF) Bandi & Bertsimas


What are good reference books for introduction to operations research?

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