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Integer Linear Programming CONTACT@ADAMFURMANEK.PL HTTP://BLOG.ADAMFURMANEK.PL FURMANEKADAM 1 25.07.2020 INTEGER LINEAR PROGRAMMING - ADAM FURMANEK About me Experienced with backend, frontend, mobile, desktop, ML, databases. Blogger,

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Integer Linear Programming

CONTACT@ADAMFURMANEK.PL HTTP://BLOG.ADAMFURMANEK.PL FURMANEKADAM

INTEGER LINEAR PROGRAMMING - ADAM FURMANEK 25.07.2020

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About me

Experienced with backend, frontend, mobile, desktop, ML, databases. Blogger, public speaker. Author of .NET Internals Cookbook. http://blog.adamfurmanek.pl contact@adamfurmanek.pl furmanekadam

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Agenda

Declarative programming at a glance. Some theory. Real life example. Implementation consideration. Summary.

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Declarative programming

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Declarative programming

Paradigm that expresses the logic of a computation without describing its control flow. Declarative programming often considers programs as theories of a formal logic, and computations as deductions in that logic space. A high-level program that describes what a computation should perform. SELECT * FROM Orders

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Declarative programming — why?

Decouples problem formulation from solving process. Solving part can be replaced without modifying the formulation. Easier to optimise algorithms for because „the goal” is understandable for the machine. Can be used with little to no programming skills. Sometimes we don’t know how to solve it.

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Example — RTS game

We are playing RTS game. We are allowed to hire footmen and archers. Every footman costs 10 gold and 30 food. Every archer costs 20 gold, 25 food, and 10 wood. Footman’s attack is equal to 5, archer’s attack is equal to 7. Our population limit is set to 200 units. Every footman „costs” 1 unit, every archer „costs” 2 units. We have 1000 gold, 1000 food, and 200 wood. We want to get strongest possible army.

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Units

Gold Food Wood Footman 10 30 Archer 20 25 10

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Attack Unit count Footman 5 1 Archer 7 2 Available gold Available food Available wood Max units count Constraint 1000 1000 200 200

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Variables

Variables: 𝑔 − 𝑔𝑝𝑝𝑢𝑛𝑓𝑜 𝑏 − 𝑏𝑠𝑑ℎ𝑓𝑠𝑡 Constraints: 𝑔 + 2 ⋅ 𝑏 ≤ 200 // population 10 ⋅ 𝑔 + 20 ⋅ 𝑏 ≤ 1000 // gold 30 ⋅ 𝑔 + 25 ⋅ 𝑏 ≤ 1000 // food 10 ⋅ 𝑏 ≤ 200 // wood Target value: 5 ⋅ 𝑔 + 7 ⋅ 𝑏

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Example

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Example

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Example

Variables: 𝑇, 𝐹, 𝑂, 𝐸, 𝑁, 𝑃, 𝑆, 𝑍 ∈ 0, … , 9 and all different Constraints: 1000𝑇 + 100𝐹 + 10𝑂 + 𝐸 + 1000𝑁 + 100𝑃 + 10𝑆 + 𝐹 = 10000𝑁 + 1000𝑃 + 100𝑂 + 10𝐹 + 𝑍

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Example

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Real life examples

RTS game = factory production allocation:

  • We have available resources
  • We have facitilies
  • We want to plan work to maximize the production

Send more money = pattern recognition:

  • We know structure of a problem but doesn’t understand the latent factors

Floor tiling = microchip transistor layout:

  • We have transistors and gates of given size
  • We want to minimize heat emission and the board size

Others: BTS placement, power plant rooms layout, scheduling systems, items personalization and many more.

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Some theory

MIXED INTEGER LINEAR PROGRAMMING

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Mixed Integer Linear Programming

Programming in mathematics means finding a solution to

  • ptimization problem.

Linear Programming is a class of a problems with only linear constraints and with linear cost function. Mixed Integer Linear Programming is a class of a problems with integer constraint for some of variables.

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Constraints

We can add two variables: 𝑏 + 𝑐 We can multiply variable by constant: 5 ⋅ 𝑏 We can constrain variable with lower/upper bound: 𝑏 ≤ 7 We cannot multiply variables: 𝑏 ⋅ 𝑐 We cannot use strict constraings (greater than, less than, not equal) 𝑏 < 10

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Cost function

Notice that not hiring anyone was also a solution! Problem may have zero, one, multiple or infinitely many solutions. We need to compare them. We use cost function to specify which one of them is the best.

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Cutting plane method

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Algorithms

Basic algorithms:

  • Cutting plane methods – we solve the problem without integer constraints

(continuous version of the problem), and next we cut the plane to get smaller problem

  • Branch and bound – we solve the problem without integer constraints, and

next we bound some variables and perform next iteration

MILP is NP-complete and we sometimes use heuristics:

  • Tabu search
  • Simulated annealing
  • Ant colony optimization

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What can we do?

WE WILL IMPLEMENT

Boole’s logic. Multiplication. Comparison

  • perators.

WE WILL NOT IMPLEMENT

Arithmetic: division, remainder, exponentation, roots. Comparisons: min, max, absolute value. Number theory: factorial, GCD. Algorithms: if condition, sorting, loops, lexicographical comparisons, Gray’s code, linear regression. Set operators: SOS type 1 and 2, approximation. Graph operators: MST, vertex/edge cover, max flow, connectivity, shortest path, TSP. Turing machine.

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Conjunction — AND operator (&&)

We have two variables: 𝑏, 𝑐. We want to create variable 𝑦 which is 𝑦 = 𝑏 && 𝑐. Formula: 0 ≤ 𝑏 + 𝑐 − 2𝑦 ≤ 1

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Conjunction — AND operator (&&)

If both 𝑏 and 𝑐 are 1 then 𝑦 must be 1. If 𝑦 was 0: 0 ≤ 1 + 1 − 2 ⋅ 0 ≤ 1 0 ≤ 2 − 0 ≤ 1 0 ≤ 2 ≤ 1

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0 ≤ 𝑏 + 𝑐 − 2𝑦 ≤ 1

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Conjunction — AND operator (&&)

If both 𝑏 and 𝑐 are 1 then 𝑦 must be 1. If 𝑦 is 1: 0 ≤ 1 + 1 − 2 ⋅ 1 ≤ 1 0 ≤ 2 − 2 ≤ 1 0 ≤ 0 ≤ 1

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0 ≤ 𝑏 + 𝑐 − 2𝑦 ≤ 1

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Conjunction — AND operator (&&)

If either 𝑏 or 𝑐 is 1 then 𝑦 must be 0. If 𝑦 was 1: 0 ≤ 1 + 0 − 2 ⋅ 1 ≤ 1 0 ≤ 1 − 2 ≤ 1 0 ≤ −1 ≤ 1

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0 ≤ 𝑏 + 𝑐 − 2𝑦 ≤ 1

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Conjunction — AND operator (&&)

If either 𝑏 or 𝑐 is 1 then 𝑦 must be 0. If 𝑦 is 0: 0 ≤ 1 + 0 − 2 ⋅ 0 ≤ 1 0 ≤ 1 − 0 ≤ 1 0 ≤ 1 ≤ 1

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0 ≤ 𝑏 + 𝑐 − 2𝑦 ≤ 1

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Conjunction — AND operator (&&)

If both 𝑏 or 𝑐 are 0 then 𝑦 must be 0. If 𝑦 was 1: 0 ≤ 0 + 0 − 2 ⋅ 1 ≤ 1 0 ≤ 0 − 2 ≤ 1 0 ≤ −2 ≤ 1

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0 ≤ 𝑏 + 𝑐 − 2𝑦 ≤ 1

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Conjunction — AND operator (&&)

If both 𝑏 or 𝑐 are 0 then 𝑦 must be 0. If 𝑦 is 0: 0 ≤ 0 + 0 − 2 ⋅ 0 ≤ 1 0 ≤ 0 − 0 ≤ 1 0 ≤ 0 ≤ 1

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0 ≤ 𝑏 + 𝑐 − 2𝑦 ≤ 1

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Conjunction — AND operator (&&)

If both 𝑏 and 𝑐 are 1 then 𝑦 must be 1. If either 𝑏 or 𝑐 is 1 then 𝑦 must be 0. If both 𝑏 or 𝑐 are 0 then 𝑦 must be 0.

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0 ≤ 𝑏 + 𝑐 − 2𝑦 ≤ 1

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Conjunction & Disjunction

Two variables conjunction: 0 ≤ 𝑏 + 𝑐 − 2𝑦 ≤ 1 𝑜 variables conjunction: 0 ≤ 𝑏1 + 𝑏2 + 𝑏3 + … + 𝑏𝑜 − 𝑜𝑦 ≤ 𝑜 − 1 Two variables disjunction: −1 ≤ 𝑏 + 𝑐 − 2𝑦 ≤ 0 𝑜 variables disjunction goes the same way

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Negation, exclusive or, implication

Negation: 𝑦 = 1 − 𝑏 Implication: 𝑏 ⇒ 𝑐 ≡ ~𝑏 ∨ 𝑐 Exclusive or: ~𝑏 ∧ 𝑐 ∨ (𝑏 ∧ ~𝑐)

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Multiplication of two binary variables is their conjunction. 𝑏 ⋅ 𝑐 ≡ 𝑏 ∧ 𝑐

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Multiplication of integer variables

Multiplication is possible when we know the maximum possible value of a variable. We set the upper bound and perform the long multiplication. It is rather slow approach, it requires 𝑃(𝑜2) temporary variables.

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Value decomposition

We decompose the variable to extract digits. We assume the maximum possible value, because we need to know the number of digits. Decomposition is straghtforward: 𝑐 = 𝑐0 + 2 ⋅ 𝑐1 + 4 ⋅ 𝑐2 + 8 ⋅ 𝑐3 + … + 2𝑜−1 ⋅ 𝑐𝑜−1

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Multiplication

𝑏 ⋅ 𝑐 = 𝑏0 ∧ 𝑐0 + 2 ⋅ 𝑏0 ∧ b1 + 4 ⋅ a0 ∧ 𝑐2 + … + 2𝑜−1 ⋅ 𝑏0 ∧ 𝑐𝑜−1 +2 ⋅ 𝑏1 ∧ 𝑐0 + 2 ⋅ 𝑏1 ∧ 𝑐1 + 4 ⋅ 𝑏1 ∧ 𝑐2 + ⋯ 2𝑜−1 ⋅ (𝑏1 ∧ 𝑐𝑜−1) +4 ⋅ 𝑏2 ∧ 𝑐0 + 2 ⋅ 𝑏2 ∧ 𝑐1 + 4 ⋅ 𝑏2 ∧ 𝑐2 + … + 2𝑜−1 ⋅ 𝑏2 ∧ bn−1 + ⋯ + 2𝑜−1( 𝑏𝑜−1 ∧ 𝑐0 + 2 ⋅ 𝑏𝑜−1 ∧ 𝑐1 + 4 ⋅ 𝑏𝑜−1 ∧ 𝑐2 + … + 2𝑜−1 ⋅ 𝑏𝑜−1 ∧ 𝑐𝑜−1 ) It can be represented as: 𝑏 ⋅ 𝑐 = ෍

𝑗=0 𝑜−1

2𝑗 ෍

𝑘=0 𝑜−1

2𝑘𝑏𝑗 ∧ 𝑐

𝑘

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Comparison operators To calculate whether 𝑏 > 𝑐 we can use: 0 ≤ 𝑐 − 𝑏 + 2𝑜−1𝑦 ≤ 2𝑜−1 − 1 To check whether numbers differ: 𝑦 = 𝑏 > 𝑐 ∨ (𝑐 > 𝑏)

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There are libraries doing that!

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https://github.com/afish/MilpManager

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Scheduling System

REAL LIFE EXAMPLE

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Idea

At the beginning of every term students must be assigned to classes. It is hard to make them happy because some of them work, some of them prefer to sleep long, some of them prefer to have lots of days off. We can try to represent the requirements as a MILP program and find the optimal solution.

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Usage

We ask students to assign preferences points to every possible class. The more points assigned the more student wants to be assigned to that class. We need to take care of rooms occupancy limits, collisions etc.

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Schedule

We are given a schedule of classes in courses Every class has associated day (Monday – Friday), hour (e.g., 9:30 AM) and duration (e.g., 1:30 hrs). Every class has associated room with occupancy limit.

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Constraints

Variables. Exactly one class in one course. Collisions. Rooms occupancy limits. Cost function.

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Variables

For every person, every course, every class we declare binary variable. Value 1 means that the student is assigned to this class. We define: 𝑦𝑑𝑝𝑣𝑠𝑡𝑓,𝑑𝑚𝑏𝑡𝑡,𝑡𝑢𝑣𝑒𝑓𝑜𝑢

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Exactly one class for a student

Every student must attend exactly one class in course. For every course we need to assign student to exactly

  • ne class.

ሥ

𝑑𝑝𝑣𝑠𝑡𝑓,𝑡𝑢𝑣𝑒𝑓𝑜𝑢

෍

𝑑𝑚𝑏𝑡𝑡

𝑦𝑑𝑝𝑣𝑠𝑡𝑓,𝑑𝑚𝑏𝑡𝑡,𝑡𝑢𝑣𝑒𝑓𝑜𝑢 = 1

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Collisions

Student cannot be in two places at the same time. ሥ 𝑑𝑝𝑣𝑠𝑡𝑓1, 𝑑𝑚𝑏𝑡𝑡1 , 𝑑𝑝𝑣𝑠𝑡𝑓2, 𝑑𝑚𝑏𝑡𝑡2 𝑑𝑝𝑚𝑚𝑗𝑒𝑗𝑜𝑕 ሥ

𝑡𝑢𝑣𝑒𝑓𝑜𝑢

𝑦𝑑𝑝𝑣𝑠𝑡𝑓1,𝑑𝑚𝑏𝑡𝑡1,𝑡𝑢𝑣𝑒𝑓𝑜𝑢 + 𝑦𝑑𝑝𝑣𝑠𝑡𝑓2,𝑑𝑚𝑏𝑡𝑡2,𝑡𝑢𝑣𝑒𝑓𝑜𝑢 ≤ 1

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Room occupancy limit

Every room has an occupancy limit. ሥ

𝑑𝑝𝑣𝑠𝑡𝑓,𝑑𝑚𝑏𝑡𝑡

෍

𝑡𝑢𝑣𝑒𝑓𝑜𝑢

𝑦𝑑𝑝𝑣𝑠𝑡𝑓,𝑑𝑚𝑏𝑡𝑡,𝑡𝑢𝑣𝑒𝑓𝑜𝑢 ≤ 𝑡𝑑𝑝𝑣𝑠𝑡𝑓,𝑑𝑚𝑏𝑡𝑡 where 𝑡𝑑𝑝𝑣𝑠𝑡𝑓,𝑑𝑚𝑏𝑡𝑡 means the size of the room

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Preferences cost function

Every student assigned points to classes. We want to maximize the sum of those points. ෍

𝑑𝑝𝑣𝑠𝑡𝑓,𝑑𝑚𝑏𝑡𝑡,𝑡𝑢𝑣𝑒𝑓𝑜𝑢

𝑞𝑑𝑝𝑣𝑠𝑡𝑓,𝑑𝑚𝑏𝑡𝑡,𝑡𝑢𝑣𝑒𝑓𝑜𝑢 ⋅ 𝑦𝑑𝑝𝑣𝑠𝑡𝑓,𝑑𝑚𝑏𝑡𝑡,𝑡𝑢𝑣𝑒𝑓𝑜𝑢 where 𝑞𝑑𝑝𝑣𝑠𝑡𝑓,𝑑𝑚𝑏𝑡𝑡,𝑡𝑢𝑣𝑒𝑓𝑜𝑢 means number of points

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What now?

Students: 150 Courses: 15 Classes: 9 per course Problem size: ~40 000 variables. Solving time: 2 seconds. This solution is optimal, we won’t get any better than that!

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Implementation consideration

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Licenses

Various approaches:

  • License per workstation
  • License per person
  • Licensing server with tickets

Licenses for universities and research work. Licenses for students.

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https://neos-server.org/neos/solvers/index.html

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Beyond MILP

SAT/SMT. Quadratic programming. Prolog. Specialized models for various problems.

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Autopromotion

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Summary

Declarative programming allows you to focus on a problem, not on an algorithm! If something is slow — just replace the solver. There are many models, choose as powerful as you can (to make modelling easy) and as primitive as possible (to make solving fast).

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Q&A

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References

Alexander Schrijver — „Theory of Linear and Integer Programming” Dennis Yurichev — „SAT/SMT by example” Adam Furmanek – „.NET Internals Cookbook” http://blog.adamfurmanek.pl/2015/08/22/ilp-part-1/ — a lot about ILP https://github.com/afish/MilpManager — library for modelling https://neos-server.org/neos/solvers/index.html — NEOS cloud for solving problems https://tore.tuhh.de/bitstream/11420/2548/1/201905-scopes-oehlert.pdf — Practical usage of ILP versus QP

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Thanks!

CONTACT@ADAMFURMANEK.PL HTTP://BLOG.ADAMFURMANEK.PL FURMANEKADAM

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