Linear ProgrammingInterior-Point Methods

D. Eiland

Linear Programming Problem

LP is the optimization of a linear equation that is subject to a set of constraints and is normally expressed in the following form :

0

)(

x

bxA

xcxfMinimize :

Subject to :

Barrier Function

To enforce the inequality on the previous problem, a penalty function can be added to

0x)(xf

n

jjp xxcxf

1

)ln()(

Then if any xj 0, then trends toward

As , then is equivalent to

)(xf p

0 )(xf p )(xf

Lagrange Multiplier

To enforce the constraints, a Lagrange Multiplier (-y) can be added to )(xf p

)()ln(),(1

bxAyxxcyxLn

jj

Giving a linear function that can be minimized.

bxA

Optimal Conditions

Previously, we found that the optimal solution of a function is located where its gradient (set of partial derivatives) is zero.

That implies that the optimal solution for L(x,y) is found when :

0),(

0),( 1

bxAyxL

yAeXcyxL

y

Tx

Where :)1,...,1(

)(

e

xdiagX

Optimal Conditions (Con’t)

By defining the vector , the previous set of optimal conditions can be re-written as

eXz 1

0),,(

ezX

czyA

bxA

zyxL T

Newton’s Method

Newton’s method defines an iterative mechanism for finding a function’s roots and is represented by :

When ,

)('

)(1

n

nnn vf

vfvv

0)( 1 nvfnn vv 1

Optimal Solution

Applying this to we can derive the following :

zXe

zyAc

xAb

z

y

x

XZ

A

ATT

0

10

00

),,( zyxL

Interior Point AlgorithmThis system can then be re-written as three separate equations :

xeZzZXx

zyAcyAz

czyAZXeZxAxAbyAZXATT

TT

11

111 )(()(

Which is used as the basis for the interior point algorithm :1.Choose initial points for x0,y0,z0 and the select value for τ between 0 and 1

2.While Ax - b != 0a) Solve first above equation for Δy [Generally done by matrix

factorization]b) Compute Δx and Δzc) Determine the maximum values for xn+1, yn+1,zn+1 that do not violate the

constraints x >= 0 and z >= 0 from :

yayy

zazz

xaxx

nn

nn

nn

1

1

1

With : 0 < a <=1