Cutting Plane and Column Generation Methods Using Interior Point

Cutting plane and column generation

metho ds using interior p oint metho ds

John E. Mitchell

Mathematical Sciences,

Rensselaer Polytechnic Institute,

Troy NY 12180 USA

Visiting: TWI/SSOR,

Delft UniversityofTechnology,

2628 CD Delft,

The Netherlands

email: [email protected] or [email protected]

http://www.math.rpi.edu/~mitchj

TWI/SSOR, TU Delft

25 Novemb er, 1997

Abstract

Column generation metho ds can b e used to solveinte-

ger programming problems, sto chastic programming prob-

lems, crew scheduling problems, and multicommo dity net-

work ow problems, among others. These metho ds solve

a sequence of linear programming subproblems. The

rapid increase in computational p ower in the last few

years has made it p ossible to solvevery large instances.

For such problems, it is often more ecient to use an

interior p oint metho d to solve the linear programming

subproblems than to use the simplex metho d. We survey

b oth the theoretical and computational research in this

area. 1

Intro duction

Interested in linear programming problems with a

large numb er of constraints or a large num-

ber of variables.

Solve problems using constraint generation or

column generation metho ds.

Solve linear programming subproblems using inte-

rior p oint metho ds

These techniques are used to solve:

{ Integer programming problems

{ Multicommo dity network ow problems

{ Sto chastic programming problems

{ Semi-in nite programming problems

{ Airline crew scheduling problems

{ Bounded error parameter estimation

{ Adaptive ltering 2

Solving an integer program with cutting planes

Example:

min 2x x

1 2

s.t. x + 2x 7

1 2

2x x 3

1 2

x ;x 0; integer

1 2

H t t

3 H

H t t t

A *

t t t

t t

1 2 x 0

Linear programming relaxation

Solve LP relaxation.

Obtain x =2:6, x =2:2, value 7:4.

1 2 3

Cutting planes:

min 2x x

1 2

s.t. x + 2x 7

1 2

2x x 3

1 2

x + x 4

1 2

x 2

x ;x 0

1 2

H t t

3 H

@ H

@i H t t t

A *

t t t

- t t

1 2 0 x

After adding two cuts

Solve. Optimal solution is x =2,x =2,value 6.

1 2

Note: feasible in the original integer program, so

optimal, since optimal for a relaxation. 4

Convex hull of integer p oints:

t t H

H 3

p p p p p p H

p p p p p p p @ H

p p p p p p p p H

p p p p p p p p p @ H

p p p p p p p p p p H

p p p p p p p p p p p @ t t t H

p p p p p p p p p p p

p p p p p p p p p p p t t t

p p p p p p p p p p

p p p p p p p p p

p p p p p p p p

p p p p p p p

p p p p p p

t t

0 x 1 2

A cutting plane example

If we knew all the constraints of the convex hull of the

integer program then we could solve the integer program

by solving one linear program. Unfortunately, hard to

get such a description. 5

Basic structure of a cutting plane algorithm

1. Solve the linear programming relaxation.

2. If the solution to the relaxation is feasible in the inte-

ger programming problem, STOP with optimality.

3. Else, nd one or more cutting planes that separate

the optimal solution to the relaxation from the con-

vex hull of feasible integral p oints, and add a subset

of these constraints to the relaxation.

4. Return to step 1. 6

Notes:

Classically, rst relaxation is solved using the primal

simplex algorithm.

After the addition of cutting planes, the current pri-

mal iterate b ecomes infeasible.

However, the change to the dual problem is the addi-

tion of some columns with corresp onding variables.

If these extra dual variables are given the value 0 then

the current dual solution is still dual feasible.

Therefore, subsequent relaxations are solved using

the dual simplex metho d in the classical approach.

Dual simplex metho d is quick to reoptimize after ad-

dition of just a few constraints.

Notice that the values of the relaxations provide lower

b ounds on the optimal value of the integer program.

These lower b ounds can b e used to measure progress

towards optimality, and to give p erformance guaran-

tees on integral solutions. 7

Interior p oint metho ds for linear programs

Interior p oint metho ds generally faster than simplex

for large problems, ie, more than a thousand variables

and/or constraints.

Interior p oint metho ds try to keep iterates well-

centered: good to keep x z for all com-

i i

p onents, where varies from iteration to iteration.

z =dual slack. If x z = for all comp onents then

i i

the p ointisonthe central tra jectory.

Duality gap is approximately n. n is number of

primal variables.

Harder to exploit a warm start with an interior

p oint metho d. 8

A closer lo ok at adding a constraint

Approximately solve primal-dual LP pair

T T

min c x max b y

s.t. Ax = b P s.t. A y c D

x 0

Approximate solution: x , y .

Use interior p oint metho d so x > 0, A y

Decide to add constraint g x h.

Mo di ed primal-dual pair:

min c x

s.t. Ax = b P

g x x = h

x; x 0

max b y + hy

s.t. A y + gy c D

0 0

y 0

Now g x

in P .

In D , y = y , y = 0 is feasible.

0 0 9

Restarting with interior p oint metho d:

Have dual feasibility, but not interior dual p oint.

Use ane direction to regain dual interior p oint:

T 1

y = AA Ag ; y =1:

This direction gives interior p oint with minimal dis-

ruption to dual feasibility. Adjustmentto A y + gy

is pro jection of g onto null space of A.

Not easy to regain primal interior feasible p oint.

Options:

{ Infeasible primal-dual metho d.

{ Backup to earlier iterate.

{ Dual metho d.

Would also liketogetawell-centered iterate. 10

Analytic Center Cutting Plane Metho d

ACCPM

Gon, Vial et al., late 80's onwards

Originally develop ed for nondi erentiable optimiza-

tion problems. Gon, Haurie, Vial 1992.

Subsequently applied to sto chastic programming prob-

lems, multicommo dity network ow problems.

Similar to the indep endently develop ed Mitchell-

To dd 1992 application of the primal pro jective al-

gorithm to solveinteger programming problems with

an interior p oint cutting plane metho d.

How the algorithm works

Find a weighted analytic center of an approximation

to the set of optimal solutions.

If this center is feasible, use a contour of the ob jective

function for a cut.

Else, cut o the center.

Return to the rst step. 11

Example of ACCPM

Want to nd lowest p oint on curve.

Use piecewise linear approximation.

Ob j B

B Analytic center

B Feasible region

B for relaxation

rr rr rr rr rr rr rr rr rr B

rr rr rr rr rr rr rr rr r B

rr rr rr rr rr rr rr rr B

rrrrrrrrrrrrrr B

rrrrrrrrrrrrr B

v rrrrrrrrrrrr B

rr rr rr rr rr B

rr rr rr rr r B

rr rr rr rr B

rrrrrr B

rrrrr B

rrrr B

rr B

r B

x 12

Analytic center infeasible, so cut it o :

Ob j B

Analytic center B

Feasible region B

for relaxation B

B rr rr rr rr rr rr rr rr rr

B rr rr rr rr rr rr rr r

B rr rr rr rr rr rr

B rr rr rr rr r

B rr rr rr

B rr r

x 13

Analytic center feasible, so cut on ob jective function:

Ob j B

Analytic center B

Feasible region B

for relaxation B

B H

B H rr rr rr rr rr rr

B H rr rr rr rr r

B H rr rrv rr

B H rr r

B H

Analytic center optimal for original problem, so STOP. 14

Features of Algorithm ACCPM:

The pictures corresp ond to the primal problem:

min c x

s.t. Ax b

Have an upp er b ound UB on optimal value.

So havea set of lo calization

L := fx : Ax b; c x UBg:

Corresp onding dual problem is

max b y

s.t. A y = c

y 0

The algorithm works in the dual space, generating a

primal solution once close to the analytic center.

Typically, only sp end ve or six iterations on each

subproblem.

Any feasible dual solution givesalower b ound LB

on the optimal value of the underlying problem. 15

The algorithm itself:

1. Havea set of lo calization for the primal problem,

and upp er and lower b ounds UB and LB .

2. Approximately minimize the p otential function

T m

y; U B:=m+1 log UBb y log y .

j =1

3. Once the gradient is small enough, generate a pri-

mal solution. The primal and dual solutions are

an approximate primal-dual analytic center.

4. Search for cutting planes:

If the primal p oint is infeasible in the relaxation,

improve the approximation to the function.

If the primal p oint is feasible, improve the up-

p er b ound UB.

5. If the gap b etween UB and LB small enough, STOP.

6. Else, lo op: return to step 2.

Note: The b ound in the p otential function is not

up dated while the relaxation is b eing solved. 16

Using ACCPM to solve sto chastic programs:

Bahn, Du Merle, Gon, Vial 1995

Two-stage sto chastic program with recourse

given by

min c Qx ; x +

0 0

s.t. A x b

0 0 0

with recourse function

Qx ; := inf fc x j B x b A x g

0 0

Get a set of constraints for each realization of the

uncertainty. These constraints are added as cut-

ting planes.

Solve problems with several thousand rows and vari-

ables.

Considerably faster than solving the full LP. Also

faster than Dantzig-Wolfe on some problems.

ACCPM has also b een successfully applied to mul-

ticommo dity network ow problems and non-

di erentiable optimization problems Gon,

Gondzio, Sarkissian, Vial 1997, Gon, Haurie, Vial

1992. 17

Integer programming problems

Mitchell 1997a, 1997b, Mitchell and Borchers 1996,

1997 used primal-dual interior p oint cutting plane

metho ds.

Solve LP relaxations approximately.

Primal heuristics: Round the interior iterate to

give a good integer feasible solution.

Use dynamically altered tolerance on duality gap to

decide how accurately to solve relaxations.

Also check primal and dual values against b est heuris-

tic value when deciding whether to search for cutting

planes.

Restart by backing up towards a known primal

feasible p oint. For example, for maximum cut prob-

lem, the p oint with every comp onent equal to 0.5

is feasible. Thus, restart at convex combination of

this p oint and nal iterate, typically 95 of wayto

b oundary.

Restart dual by returning to an earlier iterate.

Anyvariable to o close to its b ounds is set to 10 ,

say. 18

The ground state of an Ising Spin Glass

Problem in glassy dynamics in statistical physics

An Ising spin glass is a mo del of a magnetic mate-

rial, and it consists of a grid of magnetic spins.

Each spin S is in one of two states, whichwe call

\up" and \down"; we assign S the value +1 if the

spin is up and 1 if the spin is down.

We assume the grid is an L L square grid emb edded

on a torus.

Further, we assume that the interactions between

spins are restricted to neighb ours, that is, we con-

sider the short range mo del with Ising spins S .

i 19

Graph representation:

4 4 grid on a torus:

t t t t

Know no de interactions, 1. These are edge

weights.

Find state of eachvertex.

If edge b etween neighb ouring vertices has value J =

+1, wantvertices to b e in same state.

If edge b etween neighb ouring vertices has value J =

1, wantvertices to b e in opposite states. 20

Integer programming mo del

Minimize the Hamiltonian of the energy:

H := J S S

ij i j

neighb ours i;j

with S = 1.

Can b e mo delled as a Max Cut problem: all the

up vertices on one side of the cut, and all the down

vertices on the other side, with edge weights derived

from J .

So solve

minfc x : x is the incidence vector of a cutg

Here, x has one comp onent for each edge. The opti-

mal value of this problem is even. 21

Polyhedral theory:

Initial relaxation:

minfc x :0 x 1g

Cutting planes from cycles: Every cycle and

every cut intersect in an even numb er of edges. Gives

the following facet de ning inequality for every sub-

set F of o dd cardinalityofevery chordless cycle C :

xF xC n F j F j1

All the squares in the grid are chordless cycles.

There are also many other chordless cycles.

Anyintegral vector that satis es all the cycle inequal-

ities must b e the incidence vector of a cut.

There are also other families of valid inequalities. 22

Run times

Means with

Min to Max bars

Time

4 hours

1 hour

Slop es corresp onding

to p owers 4.64 and 5.8

1 second

10 20 40 60 80 100

L 23

Results

Solved 100 problems of each size 10 10, 20 20, ...,

100 100.

Problems of size 100 100 are solved in an average of

3hrs, 20 minutes on a Sun SPARCstation 20/71.

By comparison, a simplex cutting plane co de using

CPLEX 3.0 on a Sun SPARCstation 10 required up

to a day to solve problems of size 70 70. De

Simone et al, 1996.

Fitting logT imeversus log L gives a slop e of 4.64.

4:64

This shows that runtime only grows at rate L .By

comparison, the simplex runtimes app eared to grow

at a rate of L .

The numb er of iterations p er linear program aver-

ages out to around 8. Fewer iterations are required

in earlier relaxations and more iterations for later re-

laxations.

One p ossible way to improve the algorithm would b e

to crossover from an interior p oint metho d to a

simplex metho d after a certain numb er of stages. 24

The linear ordering problem

Mitchell and Borchers, 1997.

Applications include

{ Triangulation of input-output matrices in eco-

nomics

{ Archeological seriation

{ Minimizing total weighted completion time in one-

machine scheduling

{ Aggregation of individual preferences

Example: comparing Dutchcheeses:

Each of a group of p eople p erform pairwise compar-

isons between the cheeses. Ob jective: determine

the overall preference order for the group.

In general:

{ Have p ob jects to place in order.

{ If place i b efore j ,pay cost g i; j .

{ Conversely, if place i after j ,pay cost g j; i.

{ Cho ose ordering to minimize the cost. 25

Mo delling the problem:

Variables: De ne

1 if i b efore j

xi; j =

0 otherwise

Eliminate variables:

Must have xi; j +xj; i = 1 for each pair 1 i<

j p. Use this to eliminate variables xj; i, j>i.

Ob jective:

xi; j , i

ci; j :=g i; j g j; i.

Initial relaxation:

P P

p1 p

min ci; j xi; j

i=1 j =i+1

sub ject to 0 xi; j 1; 1 i

Cutting planes

Use triangle inequalities to prevent

i b efore j b efore k b efore i

or: xi; j =xj; k =xk; i=1

Enforced by xi; j +xj; k +xk; i 2.

For 1 i

xi; j +xj; k xi; k 1

i k

xi; j xj; k +xi; k 0

? @

i k

If x is integral and satis es all the triangle inequalities

then it solves the linear ordering problem.

Other inequalities exist. We only used triangle in-

equalities. 27

Crossover

Three di erent algorithms:

1. Use interior p oint metho d exclusively to solve

the relaxations.

2. Use the simplex metho d exclusively to solve

the relaxations.

3. Combine the two metho ds: use the interior p oint

metho d to solve the rst few relaxations and use the

simplex metho d to solve the remaining relaxations.

Exp erimented with di erent p oints to p erform the crossover. 28

Randomly generated problems

Up to 250 ob jects.

{ For i

and 99.

{ For j

and 39.

Randomly p ermute so the primal heuristic is not at

an advantage.

Zero out a p ercentage of the entries.

Generated six di erent classes of problems. Five

problems in each class. For each problem within a

particular class, we used the same crossover criterion.

Crossover p oints:

{ r150.0 and r200.0: switch after two stages.

{ r150.1 and r250.0: switch after three stages.

{ r200.1: switch after add < 600 constraints in a

stage. On average, after 7 stages.

{ r100.2: switch after add < 100 constraints in a

stage. On average, after 7 stages. 29

Time to solve with interior p ointcode

vs time to solve with simplex co de

No crossover

Key:

y y

0 zero es

2 10 zero es

4 20 zero es

Int Pt time/

Simplex time

2000 4000 8000

Simplex time secs 30

Combination co de time vs Simplex time

6 y

Key:

0 zero es

2 10 zero es

Combo y 4 20 zero es

time/ y

CPLEX

y time

2 4

0.10

2000 4000 8000

31 Simplex time secs

Comments:

For suciently hard problems, combining the two

co des p erforms signi cantly b etter than either

co de individually.

For larger problems, the interior p oint and simplex

co des require comparable time. The interior p oint

solver is a research co de, and we b elieve based on our

exp erience in solving standard test problems that this

interior p oint solver is roughly half as fast as current

high qualityinterior p oint solvers.

Each of the criteria resulted in improvementover

pure simplex for almost every set of problems.

Final numb er of constraints for the pure integer co de

are approximately:

{ r100.2: 3000 constraints.

{ r150.0: 4000 constraints.

{ r150.1: 5000 constraints.

{ r200.0: 6500 constraints.

{ r200.1: 8000 constraints.

{ r250.0: 10000 constraints. 32

Infeasible interior p oint metho ds

Gondzio 1996 has investigated using an infeasi-

ble primal dual interior p oint metho d in a column

generation setting.

Imp ortant to restart from an approximate solution

to the relaxation. Can solve to optimality but should

backuptoapoint with only three digits of accuracy,

say.

Intro duces infeasibilities in b oth primal and dual when

restarting, but maintains -complementarity.

Use target following algorithm to recover feasi-

bility, so initially emphasize reducing infeasibil-

ities.

Once infeasibilities are suciently reduced, use in-

feasible primal-dual interior p oint metho d to get new

approximate solution.

Go o d results on multicommo dity network ow prob-

lems and other problems Gondzio and Sarkissian,

1996. 33

Other applications

Bounded error parameter estimation,Ye et

+ v , for input q , output al., 1997. Have y = q

i i i

y , noise v and parameter . Want to estimate .

i i

Set up as quadratic program, generate new columns

with each new observation. Successfully used interior

p oint column generation metho d, with guarantees on

p erformance. Obtain more centered nal solution

than when using least squares approach.

Adaptive ltering, Luo et al., 1997. Quadratic

mo del to minimize estimated error. Data available

over time. As more data b ecomes available, the solu-

tion is improved using a column generation approach

with an interior p oint metho d. Gives b etter results

than recursive least squares for problems with low

signal to noise ratio, and comparable results under

less adverse conditions.

Semi in nite programming, Kaliski et al., 1996.

Solve with nite approximation in cutting plane scheme,

as in Den Hertog et al. 1995. Use logarithmic bar-

rier metho d to solve subproblems. Implemented in

parallel with go o d results. 34

Theoretical prop erties

The ellipsoid algorithm can b e used to solvea

column generation problem in p olynomial time if cut-

ting planes can be generated in p olynomial time.

Grotschel et al., 1988.

An exactly analogous result has not b een proved

for an interior p oint metho d.

Early results showed that various algorithms are p oly-

nomial in the numb er of added cuts. Den Hertog et

al. 1995, Mitchell 1994, etc.

Gon et al. 1996 showed that ACCPM is fully

p olynomial, ie, the numb er of iterations to get a

duality gap of dep ends on a p ower of .

Atkinson and Vaidya 1995 develop ed a p olyno-

mial algorithm that requires that unimp ortant con-

straints b e dropp ed.

Mitchell and Ramaswamy 1993 extended the feasi-

bility algorithm of Atkinson and Vaidya to follow the

central tra jectory and solve the problem to optimal-

ity. Further, they showed that a long step variant

had the same complexity. 35

Many cuts, stronger cuts

All the results mentioned earlier are for when just

one weak cut is added at a time, that is, it is

easily satis ed by the current p oint.

For the Atkinson and Vaidya approach, the results

can b e extended to the case where many cuts are

added right through the current iterate Ramaswamy

and Mitchell, 1994.

For algorithms of the form ACCPM, results have

b een proven for adding many weak cuts, or for

adding a single cut through the center Gon et

al. 1993, Luo 1997.

Gon and Vial 1996 showed that the ACCPM still

converges even when a single cut that makes the cur-

rent iterate infeasible is added.

Gon and Vial 1997 showed that the ACCPM is

fully p olynomial if two cuts are added through the

current iterate. 36

Conclusions

Go o d for problems which are

Large thousands of constraints and/or variables

Add many constraints at once at least tens, prefer-

ably hundreds or thousands

Only need to solve approximately.

Also useful if the linear programs are degenerate.

Should b e implemented to

Only solve relaxations approximately.

Restart with an emphasis on b eing centered.

Exp end e ort to obtain go o d heuristic solution, so

can exploit go o d b ounds.

Occasionally solve relaxations to optimality.

Theoretically:

Best complexity comparable with ellipsoid metho d.

Still no p olynomial interior p oint cutting plane al-

gorithm which do es not need to drop unimp ortant

constraints.

ACCPM still to b e analyzed with many cuts added

at current p oint. 37

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[28] S. Ramaswamy and J. E. Mitchell. On up dating the analytic center

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Rensselaer Polytechnic Institute, Troy, NY 12180, Octob er 1994. 41