Integer Linear Programming & The Google Optimization Toolset By Selma Ardelean 

> whoami ● Junior Software Developer @ NewVoiceMedia ● Interested in machine learning, artificial intelligence and optimization techniques @matah @s_ardelean [email protected] 

Lınear Optımızatıon or lınear programıng is the name given to computing the best solution to a problem modeled as a set of linear relationships. 

A bit of history ● Developed during World War 2 ● Released to the public in 1947 ● Active work going on ● Still quite a few open problems George Bernard Dantzig Leonid Kantorovich John von Neumann 

Ok..ok.. but how does it look like? 

Canonical Form Constants Variables 

Variables: x, y Objective function: 3*x + 4*y Constraints: x + 2*y <= 14 3*x - y >= 0 x - y <= 2 Basic Example 

Simplex Algorithm Let’s just call it magic ● Proposed by George Bernard Dantzig in 1947 ● average case polynomial complexity ● worst case exponential complexity ● used widely in practice 

Procedure 1. Simplify constraints 2. Represent constraints geometrically 3. Get coordinates of all vertices and replace into the objective function 

Applications ● Transportation problem ● Minimization of production costs ● Maximization of profit ● Telecommunications networks ● Cellular networks ● Scheduling ● Travelling Salesman 

Let’s get real 

Google Optimization Toolset ● Open Source under the Apache License ○ https://github.com/google/or-tools ● Written in C++ ● Used internally at Google ● Provides an APIs for various languages: C++, Java, Python and C# ● Can be used with other solvers ● Does provide its own open source linear solver: GLOP ● Actively being developed ● Documentation has come a long way 

NumVar - define the limits of our variables Setup the constraints 

Print out your results and enjoy! 

A bit more real 

The Problem: Stigler diet, named for economics Nobel laureate George Stigler, who computed an inexpensive way to fulfill basic nutritional needs given a set of foods The Constraints: The Variables: The Objective: minimize cost 

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Resources ● Discrete Optimization Course ○ https://class.coursera.org/optimization-003 ● https://en.wikipedia.org/wiki/Integer_programming ● Scheduling problem ○ https://github.com/evilqubit/lpSolve ● lp_solve API reference ○ http://lpsolve.sourceforge.net/5.5/ ● Online example of simplex algorithm ○ http://www.zweigmedia.com/RealWorld/simplex.html ● Gurobi solver reference ○ http://www.gurobi.com/ 

Conclusion I am by no means an expert in anything Linear programming is applicable to a variety of problems The Google Optimization toolset is easy to use and understand And provides much more than just a linear solver Math is fun! 

Thanks for listening! Questions?