barrier function

7/23/2019 Barrier Function

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Chapter

19

BARRIER FUNCTION METHODS

1

Introduction

Penalty function methods generate a sequen ce of infeasible points

{xk

which

com e closer to the constraints as the iterations proceed. By contrast, barrier

function methods

-

which ca n only be applied to problems with inequalities but

no equality constraints

-

generate points which lie inside the feasible region.

Hence, for the remainder of the chapter, we consider the problem

Minimize F x )

19.1.1)

subject to c i x ) i=

1 ...,

m )

19.1.2)

2

Barrier functions

Definition On e form of barrier function for the problem 19.1. ) , 19.1.2) is

Because the barrier term inc ludes reciprocals of the constraints w e see that

B

is

very much greater than

F

when any c i x ) s near zero .e. wh en

x

is near the

boundary of the feasible region. Similarly,

B =

F when all the q x ) are much

greater than zero and x is in the interior of the feasible region.


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Definition A second (more popular) barrier function for (19.1.1), (19.1.2) is

m

B(x, r)

=

F (x)-

log(cj(x))

(19.2.2)

i

1

When 1 > ci(x)

>

0 then log(ci(x)

< 0.

Hence the second term on the right

of (19.2.2) implies B >>

F

when any of the constraint functions approaches

zero. N ote, however, that (19.2.2) is undefined when any c j(x ) is negative.

There is a relationship, similar to that for penalty functions, between uncon-

strained m inima of B(x , r ) and the solution if (19.1. I) , (19.1.2).

Proposition

Suppose that (19.1. l), (19.1.2) has a unique solu tion x,*

,

A*. Sup-

pose also that

p

is a positive constant and, for all rk

0. Hence (19.2.12) gives one positive and on e negative value for XI But

(19.2.11) means that x2 must have the same sign as xl. However, a solution

with

both

xl and x2 negative cannot satisfy (19.2.8). Therefore the uncon-

strained minimum of B(x ,

r

is at

Hence, as

r

0 we have xl

-+

and x2

.

These values satisfy the

optimality conditions for problem (19.2.7), (19.2.8).

Exercises

1. Deduce the Lag range multiplier fo r the worked exam ple above.

2. U se a log-barrier function approach to solve the problem

Minimize

xl +x2 sub jec t to

x; +x;

3. A log-barrier approach is used to solve the problem

Minimize

Ty

subject to yTQy

5 V,.

Suppose that the barrier parameter r is chosen s o the minimum of B(y,r ) oc-

curs where yTQy= kV,, where k

0 , x2 >

1

using the barrier function

(1

9.2.1).


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Barrier function methods

3 Numerical results with B-SUMT

We use B-SUMT to denote the SAMPO implementation of the barrier SUMT

algorithm using the log-barrier function

(1

9.2.2).

In

B-SUMT

he unconstrained

minim izations are done by QNw or QNp

A safeguard is needed in the line search for the unconstrained minimization

technique used in

B-SUMT.

The log-barrier function is undefined if any of the

constraints ci x)are non-positive and therefore the line search must reject trial

poin ts where this occurs. This can be done within the framework of the Arm ijo

line-search by re-setting

B ( x ,

r ) to a very large value at any point x which vio-

lates one or more of the constraints.

In this section w e consider the minimum-risk and maximum-return problems

in the forms Minrisk4 and Maxret4. Program

sample11

allows us to solve

such problems using

B-SUMT

and we begin with Problem T l

lb.

We note first

that B-SUMT must be started with a feasible point. Hence the automatic initial

guess

y = l l n , i =

1 .

..,n

(suitable for the other two SU M T algorithms) w ill

probably not be appropriate for B-SUMT. n the present case, it is fairly easy to

find a feasib le starting guess. Since we have the expected return fo r each asset

4.4.4),we can set

y3

= 0.95 and

yi

= 0.01, i = 1,2,4,5. This ensures

C y i

< 1

and also that the expected portfolio return exceeds

Rp l

).

The progress m ade

by

B-SUMT

s show n in Table

19.1.

For com parison, this table also summ arises

the behaviour of P-SUMT from the same starting point. In both cases the un-

constrained minimizer was QNw and the initial penalty parameter and rate of

reduction were given by

r = 0.1, P = 0.25.

B-SUMT P-SUMT

Table

19.1.

B-SUMT

and

P-SUMT solutions to problem

Tl

l b

Th e fact that P-SUMT converges much more quickly than B-SUMT s largely due


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to the fact that the minimum of B(x,O. 1 ) is very much further from the opti-

mu m than the m inimum of P (x , 0. 1) . For this problem, therefore, the initial

choice

rl

=0.1 is a bad o ne for the barrier approach.

The solution to problem T I l b obtained with

B-SUMT

s approximately

with

V w

1.083 and R

w

1.044 . Th e solution from

P-SUMT

s

giving V w 1.083 and

R

w 1.026 . These both lie between the two solutions

(17 .5.5) and (17.5.6). Both solutions are both equally valid because they are

feasible points giving the same value of the objective function V. Th e fact that

B-SUMT and P-SUMT erminate at different points seems to be due to the fact

that one w orks inside the feasible region while the other operates outside.

We now apply

B-SUMT

to the maximum-return problem T13b. We need to

find a feasible starting values for the

yi,

which is not so straightforward as

it was for

Minrisk4

However, if we have already solved

Minrisk4

then its

solution will be appropriate as an initial guess for

Maxret4

provided it gives

V

barrier function

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