An Efficient Steepest-Edge Simplex Algorithm for SIMD ... - CiteSeerX

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tableau representation of the constraint matrix. The parallel algorithm reduces .... software. On the other hand, efficient parallel simplex method implement ation .... BI?S x. If B is the basic submatrix of A associated with x, ~we set Xo. = B-lb, and.
An

Efficient

Steepest-Edge

Algorithm

for

Michael

E.

Simplex

SIMD

Thomadakis

Department

Computers

and

Jyh-Charn

of Computer

Texas College

A&M

Science

University

Station,

{miket,

TX

77843-3112

jcliu}~cs

Abstract

.temu.

and

edu

Lewis

simplex This

paper

and

Dual

lems

proposes Simplex

on massively

are based the

on

tableau

taining

local the

Edge the

method

work.

The

speedup that

Edge

1

and

MP-1 1000

Simplex

local

Pivot PEs

and

times, running

MP-2

of

before

pivot

increase models

on high-end

PEs.

and

mesh

respectively,

the

The

to search are

optimal

are in the over Unix

for the net-

results

order

of 100

Steepest-

workstations.

Introduction

Linear Programming (LP) is a significant area in the field of mathematical optimization, where a linear function z = Clxl + C2Z2 + . . . + c~r~ is optimized, subject to a set of linear equality or inequality constraints. The Simplex is the most widely used solution method for linear programming problems [2, 12, 14]. As in the solution of any large scale mathematical system, the computation time for large LP problems is a major concern. Recently, several researchers experimented with the parallelization of the Simplex algorithm, on vector, shared

in

They

reported

sizes.

tableau

and

computer. the

on shared

of the

inverse

Helgason

the the

on

[9] discussed

using

sparse

computer,

implementation.

scalability

simplex

et aL

Simplex

revised a shared

[17] studied

revised

MIMD

on

good

Stunkel

the

revised

memory

actual

and

[5].

Shu

The

[16] parallelized

early

performance. of

matrix

without

Recently,

dense

simplex

are

relatively

efficient quire

easy

elab-

Shu

and

Wu

focus Dual

PA, USA @1996 ACM 0-8979 l-8 fj3-7/9fj/05. .$3.50

286

SIMD

on

a CM-

algorithms

machines

On

the

implement and

with

other

ation

placement

to express

hand,

would

re-

low-level

in current

paper

is on the

methods

on

been

the

pivot

on the sequential

for

The

(SPARC and

Two-Phase

selection

data

data-parallel

design

SIMD

on

solution 1000,

MP-2)

comparisons

parallel

methods.

parallel

simplex

parallel solution matrices. Our

tableau

technique

a Sun effort

MP-1

time

so that

of scalable

multiprocessor

method and

it

and

has been

MP- 1 and MP-2 Ma.sPar SIMD maversion of the steepest-edge simplex

implemented

workstation.

Philadelphia,

on

method

of this

is based

implemented chines. A

(MP-I

prototypes

point

software.

to data

Simplex

Steepest-Edge

tation

interior

We present an efficient massively LP problems with dense constraint

algorithm

has

simplex

a disappointing

algorithms

implement

is dficult

main and

systems. to general

Permission to make digital/hard copies of all or part of thk material for personal or classroom use is granted without fee provided that the copies are not made or dktributed for profit or commercial advsntage, rhe. copyright notice, the title of the publication snd its dste appear, and notxe IS given that copyright is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers or to rdktribute to lists, requires specific permission andlor fee.

[1] had

and

[5] implemented

that

library

inverse

languages.

The Primal

to

attention

which

high-level

the

concluded

parallel

motion,

in

interior-point

available

careful

explicit

in

et al.

and

They

commercially

the

implementation

Eckstein

2 machine.

and distributed memory Multiple-Instruction Multiple-Data (MIMD) types of computer architectures. One of the earliest parallel tableau simplex methods on a small-scale distributed memory MIMD machines is by Finkel in [6]. His study showed that the overhead for distributing matrix elements for pivot operations, for finding minimum or maximum values among different PEs, and for inter-process synchronization, did not allow for any significant speedup. Wu

ICS’96,

basis

the LU decomposition of the basis simplex algorithms on iPSC/2 and Touchstone Delta computers. Both methods are very successful on uniprocessors, with the first being more suitable to LP problems with dense constraint matrices, and the second LP problems with sparse matrices. For up to medium scale LP problems the explicit form of inverse method attained reasonable speedup. The LU decomposition simplex on the Toughstone Delta attained some speedup only on large and sparse problems. Their sparse simplex parallelization experienced high communication to computation ratio on the Touchstone machine due to strong data dependencies between PEs. Simplex algorithms for general LP problems on Single Instruction Mulitple Data (SIMD) computers have et al. been reported by Agarwal et al. in [1], and by Eckstein [15]

obtained

sequential

of the

on

parallelizations

of the

problem

hypercube

orating

asymptotic

speedups

small

to implement

methods

effi-

utilizing

experimental

the

a way

Steepest-

columns steps,

haa

iPSC/2

along

interconnection

and

and The

two

form

machine.

on very

performance

by main-

tableau

information

rows

properties sizes

method

presented explicit

MIMD

results

algorithms

matrix.

[18]

with

memory

Primal prob-

overhead

one of the

parallelization

scalability

selection

constraint

portions

toroidal

as problem

by MasPar’s times,

to

pipelined,

proposed

and

key

the

The

communication

mainly

element.

broadcasted of

pivot the

on each

utilizes

pivot

geometry

show

of

of

Programming

computers.

of

reduces

replicas

Linear

SIMD

Steepest-Edge

sub-matrices

next

ciently

parallel

the

algorithm

implementation for

representation

parallel with

a new

algorithms

Liu

can

and

per be

Performance running

on

SPARC

server

in terms MP-2)

simplex made

and

between PE

Unix

compu-

communication

iteration

is measured sequential

and

show

that

our

achieves

ex-

experiments a 16,384

1000

of average

MP-2

ecution the

speedups

sequential This

the

paper

scalar

SIMD

of the

and

Simplex

on the 4 discusses

simplex

to attain

The

to

1,200

times

over

the

MP-2

factors

high

Sequential

optimal

MasPar

which

speedup

analyzes

execution

its

the

baaic

per-

time

models.

allow

the

way

Finally,

steepest-edge

results.

Steepest-Edge

X(,), one

that

Algorithm

obtains

the

column,

Al

A

only

if its

that

~O,orzj~O,

j=l,2,

...,

k.

Every

in the general form can be expressed form. In this [4, 14] in the standard

paper,

generality,

types

in the

we consider

standard

into

minimization

form,

defined

as

min

=

z

LP

without of LP

the The

of

XJ

problems,

c’x

(1)

x.

If

B

set

Xo

plex x~o,

the

A

cost

hand

is a m x n constraint coefficients

side

such

(m

(RHS)

that

Ax

polytope

row

vector,

column

{x

E

and

vector

= b is called

F =

coefficient

Rn :

b

E

c’

R m

of constants.

a feasible

Ax

matrix,

the

Any

point,

and

= b, x > O} forms

x

ith

and

basis

present

of this

B partitions

denoted A

in the

vector by

formed

A

=

by

(n

Given

l?(i)

matrix

A

into

[NIB].

N

is a m x (W -

– m)

column two

non-basic

which

a bssis

jth

B,

BFS

basic

the

(n — m)

columns

x

components

associated

with

we denote

a BFS

the

by x =

basic

Ax

= NXN

and

columns [x~

and

the

duced

cost

row

B

expresses

of the

are positive

and,

xB

=

B-lb.

The

x is given

+ BXB

cost

value

= BXB

= b,

z of the

objective

known

tex

(i.e.,

the

optimal

F. X(o)

The and

straint

[4,

a corner

LP.

the

choose

one

among

those

baais,

is

Once the

pivot

a new

termines

matrix

In

the

If

we have

every

of

convex

denoted

BI?S x,

n.

~we

Sim-

matrix ~’.

lower

_cl ~xB uses

to

m

X

X

is

the

rows

of column

row

Xj

...

X2

...

of BFS to

negative x:,

13

a pivot

column

column

pivot

xl

=

a re-

form

vectcms

of X

En

X. heurist-

introduce

into

cost

the

coefficient.

is selected,

z is the

(z)

~

Xn

selection

reduced

where

simplex

row

that

de-

satisfies

condition

Then

simplex

erates

a new

performs to

X

the

tableau

=

k=l,2,...,

m

,

(3)

it

gen-

}

a pivot

tableau

respect new

~:x~2>0, x kj

{

from

operation X.

resulting

X

new

are computed

xkl–zlcj~>

by

BFS

x

c

polytope x*,

is

F is a verF,

also

and

a vertex initial on the

fore

on Z,j

reflects basis

~.

and

a representation The

elements

of

as follows: k=O,l,...,m,~#ij

(4)

~ xi~

k=i,

9

n.

a pivot

step,

throughout

that

row,

of

in

BFS con-

Eq.

X.

and

the

parts

pivot

. . . . x(~)

of

1,2,...,

O is equal

c1

simplex

with

pivot

the

function

called

problem

RHS

with

tableau

consists

X.

profitable

6,=~=min xt~

non-

simplex pivot steps, it generates a of BFSS’. For the values of the at the BFSS we have cost function 20, ZI, . . . . zk evaluated that z, s z,_l, so that after a finite number k of pivot steps 1Ignoring

A,

xc~),x[~),

the

a given

vector

remaining

—z

step

minimum-ratio

the

simplex algorithm starts from some by applying successive transformations

sequence

the

row row

which

and

with

= c’B-lb.

that

point)

solution,

the

=

to

ic

~kl

14]

and

= O — c~B-lb

forl=O,l,...,

is well

where

c’

each

{ It

E’,

At

by Z = C~XB

1) matrix

in

. . . . n,

to introduce

associated

and

Simplex 1,2,

B

numbers. then

form

O, 1, . . . . n,

A3. =

A,

B-l AJ, j =

=

Z3

ame

a representation

matrix of A

by

component

B-l

B associated

a compact

=

m components,

IXB] d = [olxB]~,

X~

B-l[+]

-.2

where,

of

which

rest

in j

‘=

B,

of the m columns with

and

(n +

with

at a BFS

B-lb,

basis

where

amount

O, j

maintains

submatrix

—z

the

columns

(BFS)

basic

1) x

given

submatrix

associated

to zero

that Any

m basic

Rn

c

of the

zero,

the

columns

of constraint

Xo,xl,..., x., asfollows:

Similarly,

of the

method

AJ

rightconvex

N

m)

columns.

consists a vector

are equal

columns

ba-

profitable

components

=

EJ ~

profitable

be

than

basic

a

~ Rn feasible

submatrices

RHS b of (1) as a linear combination a basic feasible solution of A, is called any

basis.

a m x (n + 1) submatrix

the

of A.

no other

current

to

when

Xj

when

in

profitable

the

CJ is

where X*

c Rn

the

= j means

is the

a m x m submatrix of A.

discussion

bssis

–Xj,

rest

profitable

ZJ is less

the

a ncmletting

a “less” into

changes

=

x(,-l)

operation

a “more” A,

while

BFS

there

+

space of the given LP problem, for w~ich we assume F # 0. Any collection of m linearly independent columns of matrix : i = 1,2,. ... m}, is called a basis A, denoted by B = {A~(i) of A,

unit,

quantity

Xj,

C’X*

pivot

is considered

value

one

optimal

maintains

removing

Cj – c&Xj.

=

simplex

columns

by

the

A

introducing

cost coefficient

Aj

current

=

=

while

adjusting

@ L?(i _l ) of

function

is the

and

f? B of A

b in terms

z*

positive,

feasibility.

L?(,,,

tableau

of the

vector

Ax=b

where,

by the

zero,

and

Ai

AJ

is, when

into

basis

by

i.e.,

) to become

to

f3(, _l ),

cost

is found,

. .

primal

reduced

adjusted

LP

loss

c

of x increases

problem

an equivalent

new

the

andz~

drop

— C~B-l

which

reaches

to

column

Ej =c,

x(k)

of X(i_l

column

sis.

Linear Programming (LP) problems are usually formulated in the general form, where a linear function in k variables, z = CIZ1 +CZZZ +. . .+ckZk, is minimized or maximized, while it satisfies a set of linear equality or inequality constraints,

=

When simplex moves from BFS basis B(, ) is formed, by flowing

maintains

non-basic

Simplex

x*

component

a basic

mea-

BFS c F}. a new

: x

to BFS

2 overviews 3 develops

it

presents

and

the

Section

algorithm,

and

MP-1

Section

method.

scalability,

section

2

1,000

sa follows.

Simplex

Steepest-Edge

surements

of

min{c’x

is organized

Primal

formance

order

version.

a,

(4)

it

depends tableau. ods

is the

b.

287

(1 .a) shows columns

reduced

column In

Fig.

row,

is clear

cost

with row

X$,x:

pivot

element,

element

Zit, most

Dantzig’s

rule

~kt

and

[4],

where

top

shaded Z,J

are

respectively.

of the

simplex

the most 0 : k =

X

be-

are scattered

6’ is the

z;l, fCkj, and Zkl widely used pivot

Xj the one with E, = min{?~