Introduction to Integer Programming Modeling and Methods

Michael TrickCarnegie Mellon UniversityCPAI-OR School, Le Croisic

2002

Some History Integer Programming goes back a long way:

Schrijver takes it back to ancient times (linear diophantine equations), Euler (1748), Monge (1784) and much more.

“Proper” study began in the 1950s Dantzig (1951): linear programming Gomery (1958): cutting planes Land and Doig (1960): branch and bound

Survey books practically every five years since Tremendous practical success in last 10-15 years

Scope This talk will not be

comprehensive! Attempt to get across main

concepts of integer programming Relaxations Primal Heuristics Branch and Bound Cutting Planes

Integer Program (IP)

Minimize cxSubject to Ax=b l<=x<=u some or all of xj integral

X: variables

Linear objective

Linear constraints

Makes things hard!

Rules of the Game Must put in that form! Seems limiting, but 50 years of

experience gives “tricks of the trade”

Many formulations for same problem

Example Formulations

Warehouse location: n stores, m possible warehouses; cost k[j] to open warehouse m; cost c[i,j] to handle store i out of warehouse j.

Minimize the total opening costs plus handling costs

Subject toEach store assigned to one open warehouse

Warehouse Formulation

Variables x[i,j] = 1 if store i served by

warehouse j; 0 otherwise y[j] = 1 if warehouse j open; 0

otherwise

Objective Minimize sum_j k[j]y[j]+sum_i,j c[i,j]x[i,j]

Warehouse Formulation Constraints:

sum_j x[i,j] = 1 for all isum_i x[i,j] <= ny[j] for all j

Binary Integer Programs Restrict variables to be 0-1 Many specialized methods OR people are real good at

formulating difficult problems within these restrictions

Key concepts Relaxation R(IP)

“easily” solved problem such that Optimal solution value is no more than that of IP If solution is feasible for IP then is optimal for IP If R(IP) infeasible then so is IP

Most common is “linear relaxation”: drop integrality requirements and solve linear program

Others possible: lagrangian relaxation, lagrangian decomposition, bounds relaxation, etc.

Illustration

Linear Relaxation

Why this fetish with Linear Relaxations? IP people are very focused on

linear relaxations. Why? Sometimes linear=integer Linear relaxations as global

constraints Duals and reduced costs

Linear=integer formulations Happens naturally for some

problems Network flows Totally unimodular constraint matrices

Takes more work, but defined for Matchings Minimum spanning trees

Closely associated with polynomial solvability

Duals and Reduced Costs

Associated with the solution of a linear program are the dual values--- one per constraint--- measures the marginal value of changing the right-hand-side of the constraint

--- Useful in many algorithmic ways

Dual exampleSum_i x[i,j]-y[j] <= 0

Suppose facility j* has cost 10 and y*[j*] = 0. The dual value of this constraint is 4. What

cost must facility j* have to be appealing?

Answer: no more than 10-4=6.

Dual Example 2 Products 1, 2, 3 use chemicals A, B

Maximize 3x1+2x2+2x3Subject to x1+x2+2x3 <= 10; (.667) 5x1+2x2+x3 <= 20; (.667)

Solution: x2=6.67 x1=1.67What objective must a product that uses 4 of A

and 3 of B have to be appealing: at least 4.67

Final advantage of linear relaxations: Global Linear relaxations are

Relatively easy to solve: huge advances in 15 years

Incorporate “global” information Often provide good bounds and guidelines

for integer program Variables with very bad reduced cost likely not in

optimal integer solution Rounding doesn’t always work, but often gets

good feasible solutions

Feasible solutions Solutions that satisfy all the

constraints but might not be optimal Generally found by heuristics Can be problem specific

Must have value greater than or equal to optimal value (for minimizing)

Feasible Solution

Feasible solution

Fundamental Branch and Bound AlgorithmSolve relaxation to get x*If infeasible, then IP infeasibleElse If x* feasible to IP, then x*

optimal to IPElse create new problems IP1 and IP2

by branching; solve recursively, stop if prove subproblem cannot be optimal to IP (bounding)

Branching Create two or more subproblems IP1,

IP2,… IPn such that Every feasible solution to IP appears in at

least one (often exactly one) of IP1, IP2, … IPn

x* is infeasible to each of R(IP1), R(IP2), … R(IPn)

For linear relaxation, can choose a fraction xj* and have one problem with xj <=[xj] and the other with xj >= [xj]+1 ([x]: round down of x)

Illustration

x*

IP1

IP2

Bounding Along way, we may find solution x’

that is feasible to IP. If any subproblem has relaxation value c* >= cx’ then we can prune that subproblem: it cannot contain the optimal solution. There is no sense continuing on that subproblem.

Stopping Technique can stop early with

solution within a provable percentage of optimal (compare to be relaxation value)

Can also modify to generate all solutions (do not prune on ties)

How to make work better? Better formulations Better relaxations (cuts) Better feasible solutions

(heuristics)

Formulations Different formulations of integer

programs may act very differently: their relaxations might have radically different bounds

“Good Formulation” of integer program: provides a better relaxation value (all else being equal).

Back to Warehouse Example Alternative formulation of “Only use if

open constraint”

x[i,j] <= y[j] for all i,j(versus)sum_i x[i,j] <= ny[j]

Which is better?

Comparing Positives to original

Fewer constraints: linear relaxation should solve faster

Positives to disaggregate formulation Much better bounds (consider having x[i,j]=1

for a particular i,j. What would y[j] be in the two formulations?)

(Almost) no comparison! Formulation with more constraints works much better.

Ideal

Formulation gives convex hull of feasible integer points

Embarrassing Formulations Some things are very hard to formulate

with integer programming: Traveling Salesman problem: great success

story (IP approaches can optimize 15,000 city problems!), but best IP approaches begin with an exponentially sized formulation (no “good” compact formulation known).

Complicated operational requirements can be hard to formulate.

Further approaches Branch and Price

Formulations with exponential number of variables with complexity in generating “good” variables (see Nemhauser): heavy use of dual values

Branch and Cut Improving formulations by adding

additional constraints to “cut off” linear relaxation solutions (more later)

Algorithmic Details

PreprocessingPrimal HeuristicsBranchingCut Generation

Preprocessing Process individual rows to

detect infeasibilities detect redundancies tighten right-hand-sides tighten variable bounds

Probing: examine consequences of fixing 0-1 variable If infeasible, fix to opposite bound If other variables are fixed, inequalities

Preprocessing Much like simple Constraint

Programming

Improving Coefficients

3x1-2x2 11. Convert to with pos. coefficients with y1

= 1-x1

3y1 + 2x2 22. Note that constraint always satisfied when

y1 = 1, so change coefficient 3 to 22y1 + 2x2 2

3. Convert back to originalx1 -x2 1

Improving (?) Coefficients

1

1

Cuts off (1/2, 1/4) and others

Manual or Automatic? Modeling issue

Automatic identification not foolproof

Generally easy to see

Can provide problem-knowledge to further reduce coefficients

Automatic issue Many

opportunities will only occur within Branch and Bound tree as variables are fixed

“More foolproof” as models change

Identifying Redundancy and InfeasibilityUse upper and lower bounds on

variables: Redundancy

3x1 - 4x2 + 2x3 6 (max lhs is 5) Infeasibility

3x1 - 4x2 + 2x3 6 (max lhs is 5)

While very simple, can be used to fix variables, particularly within B&B tree

PP: Fixing VariablesSimple idea: if setting a variable to a

value leads to infeasibility, then must set to another value

3x1-4x2+2x3-3x4 3

Setting x4 to 1 leads to previous infeasible constraint, so x4 must be 0

PP: Implication Inequalities

Many constraints embed restrictions that at most one of x and y (or their complements) are 1.

This can lead to implication inequalities.

PP: Implication InequalitiesFacility location

x1+x2+…+xm mx0

x0 = 0 x1 = 0

x0 = 0 x2 = 0, etc.

(1-x0) + x1 1 (or x1 x0)

x2 x0 , etc.

Automatic disaggregation (stronger!)

PP: Clique InequalitiesThese inequalities found by “probing” (fix

variable and deduce implications).

These simple inequalities can be strengthened by combining mutually exclusive variables into a single constraint.

Resulting clique inequalities are very “strong” when many variables combined.

Example: Sports SchedulingProblem: Given n teams, find an “optimal”

(minimum distance, equal distance, etc.) double round robin (every team plays at every other team) schedule.

A: @B @C D B C @DB: A D @C @A @D CC: @D A B D @A @BD: C @B @A @C B A

Sports Scheduling

Many formulations (not wholly satisfactory)

One method: One variable for every “home stand” (series of home games) and “away trip” (series of away games).

Variables (team A)

Some variables:

2 3 41

H

@B @C

@C @D

H H

@E @F

y1

x1

x2

y1

x3

Constraints Can only do one thing in a time slot

y1+x1+x2 1x1+x2+y2 1

No “Away after Away”x1+x2+x3 1

No “Home after Home”y1+y2 1

Additional constraints link teams

Improving Formulation

Create Implication Graph

2 3 41

H

@B @C

@C @D

H H

@E @F

y1

x1

x2

y2

x3

Find cliques Cliques in graph: can only have

one

2 3 41

H

@B @C

@C @D

H H

@E @F

y1

x1

x2

y2

x3

Constraints Can only do one thing in a time slot

y1+x1+x2 1x1+x2+y2 1

No “Away after Away”x1+x2+x3 + y2 1

No “Home after Home”y1+y2 + x1 + x2 1

Additional constraints link teams

Clique Inequalities Resulting formulation is much tighter

(turns formulation from hopeless to possible)

Idea generalized to variables and their complements

Can be found automatically, but may be a huge number (and clique generally hard)

On divide of “automatic” and “manual” modeling issue

Primal Heuristics Feasible solutions at B&B nodes

can greatly decrease the solution time

Most common: problem-specific heuristics embedded with B&B

Some general purpose heuristics: LP Diving, Pivot and Complement

LP-Diving

1. Solve LP2. Stop if infeasible or integral3. Fix all integral variables (or all 1’s)4. Select fractional variable and fix

to integer5. Go to 1

Branching Two decisions: which B&B node to

branch on and how to divide into two problems

Node selection Depth First: try for integral solution Best Bound: explore “appealing” nodes Adaptive: Depth First first, then best

bound

Branching: How to branch Priorities on variables and sets is

extremely important Normal branching is on a variable

equals 0 or equals 1, but much more complicated branching possible:

x1 + x2 + x3 + x4 1Could lead to two problems:a) x1 + x2 = 0

b) x3 + x4 =0

Branching as Modeling

Specially Ordered Sets of Type 2 (no more than 2 positive, must be adjacent): used to model piecewise linear functions:

x1,x2,x3,… xm

can lead to two problems:1 k-2 k-1 k+1 k+2 mk

Either all of the blue or all of the red are 0

Adding constraints Constraints can strengthen

formulation: we have seen clique inequalities already

Can be added on an “as needed” basis during calculations (generally to move away from current fractional solution).

Separation Problem

Given a fractional solution to the LP relaxation, find an inequality that is not satisfied by the fractional solution. Algorithm should be- fast- yield strong inequality

- heuristics acceptable

Types of Constraints

1. Feasibility: A large number of constraints is used in formulation- connectivity constraints (i.e. TSP)

- nonlinearities2. Problem specific facets - blossom inequalities for matching - comb inequalities for TSP3. General IP constraints

Gomory Constraints Any fractional solution from the simplex

method can be separated:xb + 2.5 x1 - 3.2 x2 = 2.1

where current sol. is xb =2.1, x1 = x2 = 0

xb + 2 x1 - 4 x2 - 2 = .1 - .5 x1 -.8 x2

LHS is integer so RHS must be also.1 - .5 x1 -.8 x2 0

is valid and violated by current solution

“Real Version” of Gomory Cuts

y+ sum_j ajxj =d Let d=[d]+f; let aj=[aj] + fj t=y+sum_(j:fj<=f) [aj]xj + sum_(j:fj>f) ([aj]+1)xjSo d-t = sum_(j:fj<=f) fjxj+sum_(j:fj>f) (fj-1)xjEithert<=[d] => sum_(j:fj<=f) fjxj >= ft>=[d]+1 => sum_(j: fj>f) (1-fj)xj >= 1-fDivide through to get RHS of 1 in each case. Result

givessum(j:fj<=f) (fj/f)xj + sum(j: fj>f) (1-fj)/(1-f)xj >= 1

Working through example

Previous example becomes

.5x1+.2x2 >= .9, which is a good, strong constraint

(can extend all this to mixed integer programs easily)

Gomory Constraints (modeling) Are we done? Very good to have

available but likely not the only tool in the arsenal.

0-1 Knapsack Covers

For problems that contain constraints like:

j N ajxj b

C N is a cover if j C aj > b

thenj C xj |C| - 1

is valid

Separation of Cover Inequalities

Given fractional LP solution x*, is there a cover C for which x* violates the cover inequality?

Solvable by a binary knapsack problem (one constraint IP): can be solved heuristically, or exactly by dynamic programming

Lifting of Cover Inequalities

Constraints can be strengthened:20x1+16x2+15x3+10x4+30x5 40

Cover on {1,2,3} leads to x1+x2+x3+0x4+0x5 2

Is the “0” coefficient on x4 the best possible? Can solve knapsack to get

x1+x2+x3+x4+0x5 2

Lifing of Cover Inequalities

We could continue to x5 to get

x1+x2+x3+x4+x5 2

If we had done x5 first, we would have got

x1+x2+x3+0x4+2x5 2

Different lifting sequences lead to different inequalities

Modeling implications Cover inequalities generally part of the

software, rather than the modeling (but if software has capability, do not include in model).

Decision is whether to use the inequalities or not essentially numerical question

Lifting is extremely important, as is relationship of covers in subproblems to full problem

Conclusions There is lots to integer

programming! Interesting interplay between

formulations and algorithms Much intelligence embedded in

software, making formulations a bit more “fool-proof”

Papers to Read

Progress in Linear Programming Based Branch and Bound Algorithms: An exposition, Ellis Johnson, George Nemhauser, and Martin Savelsbergh

MIP: Theory and Practice – Closing the Gap, Robert Bixby, Mary Fenelon, Zonghao Gue, Ed Rothberg, and Roland Wunderling