Branch and Price

8/10/2019 Branch and Price

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Intro Basics Dantzig-Wolfe Lower bound Block-angular structure Identical blocks Summary

An Introduction to Branch-and-PricePart I: Theory

Adam N. Letchford

Department of Management ScienceLancaster University

September 2011
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Outline

1 The basic idea

2 Dantzig-Wolfe decomposition3 The resulting lower bound

4 Exploiting block-angular structure

5 The case of identical blocks
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The basic idea

Lets recall how branch-and-cut works:

We formulate a COP as an ILP.

We derive a stronger formulation with a huge number of constraints.

We cant include all of the constraints in the LP...

... so we start with a few and generate others as needed.

To generate constraints we need separation algorithms.

After adding new constraints, we re-optimise with dual simplex.

When we cant find any more constraints, we branch.
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The basic idea (cont.)

Branch-and-price is a dual version of branch-and-cut:

We formulate a COP as an ILP.

We derive a stronger formulation with a huge number ofvariables.

We cant include all of the variablesin the LP...

... so we start with a few and generate others as needed.

To generate variables we need pricingalgorithms.

After adding new variables, we re-optimise with primalsimplex.

When we cant find any more variables, we branch.

I B i D i W lf L b d Bl k l Id i l bl k S
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The basic idea (cont.)

Question: How can we strengthen an ILP formulation byincreasing the number of variables?!

Answer: We use the decomposition technique of Dantzig& Wolfe (1960).

(For simplicity, Ill concentrate on 0-1 LPs.)

I t B i D t i W lf L b d Bl k l t t Id ti l bl k S
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Dantzig-Wolfe decomposition

Suppose we have a 0-1 LP of the form:

min {cx : Ax b, Cxd, x {0, 1}n} .

Suppose also that the constraints Axb are nice, whereas theconstraints Cxd are nasty.

That is, we could solve the 0-1 LP quickly (somehow) if theconstraints Cxd were not present.

Intro Basics Dantzig Wolfe Lower bound Block angular structure Identical blocks Summary
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Dantzig-Wolfe decomposition (cont.)

One way to handle 0-1 LPs of this type is to remove the nastyconstraints from the problem, but modify the objective function.

This is Lagrangian relaxation, which you have already seen.

An alternative (which usually works just as well or even better) isto use Dantzig-Wolfe decomposition.



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We define the two polyhedra:

P1 ={x[0, 1]n :Axb}

P2 ={x[0, 1]n :Cxd} .

The 0-1 LP can now be written as:

min

cx:x P1

P2

, x {0, 1}

n.

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Now consider the integral hull ofP1:

P1I =conv {x {0, 1}n :Ax b} .

By definition, P1I is the convex hull of a finite number of vertices.

Let v1, . . . , vt {0, 1}n be these vertices.

Note: the number of vertices can be huge (exponential).

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Any point in P1Ican be written as a convex combination(weightedaverage) of the vertices v1, . . . , vt.

More formally, ifx P1I, then there exist non-negative multipliers1, . . . , t, summing to one, such that:

x =t

k=1

kvk.



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g g y


This leads to the following 0-1 LP formulation of our problem:

min cx

s.t. Cxd

x=t

k=1vk

ktk=1k = 1 ()

{0, 1}t

x {0, 1}n.

This 0-1 LP, called the masterproblem, can have a huge numberof variables (one for each vertex ofP1I).

The constraint (*) is called a convexity constraint.



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g g y


It is usual practice to eliminate the x variables, so that the masterproblem takes the form:

min c

s.t. Cdt

k=1k = 1 ()

{0, 1}t.

Here, ck denotes the cost of vertex vk and Ckj denotes thecontribution of vertex vk to the jth constraint in the systemCxd.

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The resulting lower bound

So thats how we convert a 0-1 LP into a different 0-1 LPwith a huge number of variables.

Ill explain how to actually solvethe master problem in thenext session.

For now, lets try to understand why the master problem is astronger formulation than the original problem.

To do that, we look at the lower bounds that we get when wesolve the LP relaxations of the two problems.

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The resulting lower bound (cont.)

The LP relaxation of the original problem is:

min

cx : xP1 P2

.

Geoffrion (1974) showed that solving the LP relaxation of themaster problem is equivalent to solving:

min

cx : xP1I P2

.

So, ifP1I =P1, the LP relaxation of the master problem will give a

better bound than that of the original problem.

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The resulting lower bound (cont.)

Geoffrion (1974) also showed that solving the LP relaxation of themaster problem gives the best possiblelower bound that could be

obtained with Lagrangian relaxation.

That is, if we used the best possiblemultipliers in a Lagrangianrelaxation, we would get the same bound.

This is what makes Dantzig-Wolfe decomposition attractive.

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Block-angular structure

Dantzig-Wolfe decomposition is particularly effective when applied

to ILPs that have a block-angularconstraint matrix.This means that, if the nasty constraints Cx d are deleted, theresulting constraint system Axb decomposes into a number ofmuch smaller systems (blocks).

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Block-angular structure (cont.)

}

Nice system

Ax b

Nasty system

Cx d

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In this situation, we can substantially reduce the number ofvariables in the master problem.

Instead of one integral hull P1I, we have one integral hull for

each block.

Since each integral hull involves a much smaller number ofvariables, it is likely to have a much smaller number ofvertices.

In the master problem, we have one set of variables perblock and one convexity constraint per block.

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This is the resulting master:

min s

j=1cj j

s.t.

sj=1 Cjj dtj

k=1jk = 1 (j = 1, . . . , s)

j {0, 1}tj (j = 1, . . . , s).

Here, s is the number of blocks, tj is the number of vertices of thejth integral hull and vjk is the kth vertex of the jth integral hull.

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Lots of block-angular ILPs arise in Combinatorial Optimisation!

For example:one block for each facility / depot / factory

one block for each worker / machine / vehicle

one block for each product type / commodity

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The case of identical blocks

Moreover, in Combinatorial Optimisation it often happensthat the blocks are identical.

E.g., the machines/vehicles may all be of the same type.In this case, all of the integral hulls are the same.

So, we need only one set of variables.

This dramaticallyreduces the number of variables (though it

can still grow exponentially).

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The case of identical blocks (cont.)

This is the resulting master:

min c

s.t.

Cdtj=1j =s ()

{0, 1}t.

It looks just like the master for the general case (not

block-angular), except that the convexity constraint now has aright-hand side ofs (the number of blocks).

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SUMMARY

Dantzig-Wolfe decomposition is a well-known technique forreformulating ILPs.

It is best used when the ILP has block-angular structure...

... especially if all of the blocks are identical.

Such ILPs arise frequently in Combinatorial Optimisation.

In the next session, I will explain how to solve the masterproblem using the branch-and-pricemethod.

Ill also give some examples of COPs that can be solvedquickly using branch-and-brice.
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Branch and Price