Lecture 1 integer programming

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ISyE/CS 720: Integer Programming


Jim Luedtke

Department of Industrial and Systems EngineeringUniversity of Wisconsin-Madison

January 21, 2014

ISyE/CS 720: Integer Programming 1


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Todays Outline

About this class About me About you

Integer Programs Why are they interesting?

First denitions and rst integer programming models Branch-and-bound: Framework for solivng integer programs



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Couse Details

Where and When

Class Overview

Office Hours (subject to change):Tuesday: 11AM12PM, Friday: 9-10AMBy Appointment

Textbook: Integer Programming , Laurence Wolsey, 1998 Course home page on Piazza:

https://piazza.com/wisc/spring2014/isyecs720/homeLecture slides, homework assignments, solutionsUse Q&A on this site to post questions about the course

Learn@UW course web page:https://learnuw.wisc.eduUsed only for posting grades, and possibly for submission of parts of some homework assignments

See syllabus for overview of course topics and objectives

https://piazza.com/wisc/spring2014/isyecs720/homehttps://learnuw.wisc.edu/https://learnuw.wisc.edu/https://learnuw.wisc.edu/https://piazza.com/wisc/spring2014/isyecs720/homehttps://piazza.com/wisc/spring2014/isyecs720/home


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Couse Details

Where and When

Goal of the coursePrimary course objective

Give students a solid understanding of theory of integerprogramming and methods to solve IP problems

What this course is NOT (primarily) about: We wont spend much time on how to model problems as anIP (see ISyE/CS 635)

We study what makes a model goodWe will see some advanced modeling techniques

We wont spend much time in lecture on software But you will be required to use someOverview in lecture, some details in office hour sessions, butyoull be expected to learn much on your own

We wont cover algorithms for easy integer programs (see

CS 577)ISyE/CS 720: Integer Programming 4


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Couse Details

Where and When

Expectations

I am expected to...

Teach lectures Be at my office hours Guide your learningprocess (assignments)

Give you feedback onhow you are doing in atimely fashion

You are expected to... Learn Attend lectures andparticipate (askquestions!)

Do the homework Know and followacademic conduct

guidelines Be polite, if possible.

SleepingCell PhonesLeaving in the middleof lecture



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Couse Details

Where and When

Assignments and gradingComponents of grade

Homework (20%) Midterm (35%) save the date: March 13, 7:15 - 9:15 p.m.

Final exam or course project (35%): May 14, 5:05 - 7:05p.m.

Class participation (10%)

Homework assignments

Can work in groups of up to 2 people Assignments may not be completely graded Students should self-check ungraded parts of assignments

Notify me this week if you have a conict with either exam time.ISyE/CS 720: Integer Programming 6


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Couse Details

Where and When

Warning on course difficulty

This is a Ph.D.-level course Much of the material in this courseis difficult

Requires very serious math Proofs will be done by me and by you (Gasp!!)

Prerequisite

Strong linear programming background is essential and will beassumed



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Couse Details

Where and When

Questions about the course?



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Couse Details

Getting to Know You...

About me...

B.S., ISyE, UW-Madison, 2001. M.S., OR, GA Tech, 2003. Ph.D., ISyE, GA Tech, 2007.

Fall 2007-Summer 2008: IBM Research Research Areas: Integer programming (linear and nonlinear),stochastic programming, applications

Married. Three children, Rowan (6), Cameron (4), Remy

(< 1). Interesting fact: Biked from Seattle to New York City insummer 2001.

Not so interesting fact: Cannot hear out of left ear



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Couse Details

Getting to Know You...

About you...



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Motivation for IP

Denitions

Programming? I Hate Programming!QuestionWhat does Programming mean in Mathematical Program-ming, Linear Programming, Stochastic Programming, In-teger Programming?

AnswerPlanning

Mathematical Programming (Optimization) is about decisionmaking. Integer Programming is about decision making with integers . More precisely, decision making in which some of the decisionscan take only certain integer values.


IS E/CS 720 I P i


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Motivation for IP

Denitions

Why is programming with integers important?

First answer

Many decisions involve deciding a quantity that is indivisible: Number of airplanes to produce Number of oors in a building What about number of cents to invest in a stock?

Sometimes a continuous approximation is good enough


IS E/CS 720: Integer Programming


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Motivation for IP

Denitions

Why is programming with integers important?

Slam dunk answerWe can use 0-1 (binary) variables for a variety of purposes.

Modeling yes/no decisions. Enforcing logical conditions. Modeling xed costs. Modeling piecewise linear functions.

Usually a continuous approximation is not good enough




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Motivation for IP

Denitions

Integer programming is everywhere!

Design a supply chain network: decide where to open facilitiesand which customers they should serve

Portfolio optimization: limit number of stocks held

Stochastic optimization with risk: ensure percent of outcomeswhich are bad is small

LTL vehicle routing: which loads should be combined ontrucks and in what sequence should the loads be delivered?

Sports scheduling: assign games to days Chemical processing network design: which process typesshould be used and how should they be connected?

Television scheduling




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Motivation for IP

Mathematical Details

So what is an integer program?A generic mixed-integer program (MIP):

max cT x + hT yAx + Gy b

x 0y Z p+

Instance:

A is an m n matrix G is an m p matrix bR m

cR n

h R p

AssumptionAll data is rational.

Important, but notrestrictive.




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Motivation for IP


My friend in Math says IP is trivial Consider the following simple binary integer program:

max

n

i =1

c i y i

subject to

n

i =1a i y i b, y {0 , 1 }

n

Simple: enumerate all possible 0-1 vectors y, check if it satises theconstraint, keep the best

But how long will it take? Lets use an IBM BlueGene/L:

n Solutions to check Time3 8 010 1024 050 1 .1 10 15 2 sec110 1 .3 10 33 69 billion years

Thats four times the age of the universe!ISyE/CS 720: Integer Programming 16



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Motivation for IP


So how hard is integer programming? Could we be more clever and solve integer programmingproblems efficiently? (After all, we can solve linear programs...)

Solving general integer programs can be much more difficult

than solving linear programs.There is a whole theory (complexity) that supports this claimYou will learn some of the gory details

What if we ignore the integer restrictions?This is a relaxed version of the problem: the linearprogramming relaxationIt gives us an upper bound on the optimal valueRounding to a feasible integer solution may be difficult orimpossibleThe optimal solution to the LP relaxation can be arbitrarily far

away from the optimal solution to the MIP.So we have somethin to learn in this courseISyE/CS 720: Integer Programming 17 ISyE/CS 720: Integer Programming


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y g g g

Motivation for IP


Classes of integer programming problemsMixed-integer programming

Continuous and integer variables

Pure integer programming

All variables are integer: max{cx : Ax b, xZ n

+ }Binary integer programming

All variables are binary: max{cx : Ax b, x {0, 1}n }

Mixed binary programming (Mixed 0 1 programming) All integer variables are binary:max {cx + hy : Ax + Gy b, xR n+ , y {0, 1} p}

Linear programming No integer variables




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Motivation for IP


Closely related: Combinatorial Optimization A combinatorial optimization problem CP = ( N, F ) consistsof

A nite ground set N ,A set F 2N of feasible solutionsCosts c

j j N

The cost of S F is c(S ) = j S c j . The combinatorial optimization problem is then

min{c(S ) : S F}

Many COPs can be written as binary integer programsBinary decision variables: xi for i N S = {i N : x i = 1}Objective: iN ci x i




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Integer Programs

Simple Binary Models

Example Model: The Knapsack Problem

You are choosing what to bring in

your backpack on yourcross-country unicycling tour.You can carry at most b pounds.You have n possible items. Item iwould give benet of value ci and

and weight ai . What items to pack?




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Integer Programs

Simple Binary Models

Example Model: Machine assignment

Given m machines and n jobs, nd a least cost assignment of jobs to machines not exceeding the machine capacities Each job j requires aij units of machine is capacity bi Cost of assigning job j to machine i is cij




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Initial Examples

The usefulness of binary variablesSuppose for a set of elements to choose from, i I , we have thebinary variables:

yi = 1 if element i chosen0 otherwise.

How can we model these restrictions?

1. If we do choose i, must choose j2. We must either choose both i and j , or neither3. We can choose at most k items from a set S of items4. If we choose any one item in a set S , we must choose k

(k /S )5. If we choose all items in a set S , we must choose k (k /S )




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Initial Examples

Next Topic: Branch-and-bound

See Branch-and-bound slides.


Lecture 1 integer programming

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