OPTIMIZING OVER THE EFFICIENT SET OF A CONVEX MULTIPLE OBJECTIVE PROBLEM TWO SPECIAL CASES1

Kỷ yếu công trình khoa học 2014 – Phần I
Trường Đại học Thăng Long 26
OPTIMIZING OVER THE EFFICIENT SET OF A CONVEX
MULTIPLE OBJECTIVE PROBLEM: TWO SPECIAL CASES
1

Assoc. Prof. Nguyen Thi Bach Kim
1
and M.Sc.Tran Ngoc Thang
2
,
1,2
School of Applied Mathematics and Informatics
Hanoi University of Science and Technology

1
Email:

2
Email:
Abstract. Optimizing over the efficient set is a very hard and interesting task in multiple
objective optimization. Besides, this problem has some important applications in finance,
economics, engineering, and other fields. In this article, we propose convex programming
procedures for solving the problem of minimizing a real function over the efficient set of a
convex multiple objective programming problem in the two special cases. Preliminary
computational experiments show that these procedures can work well.
AMS Subject Classification: 2000 Mathematics Subject Classification. Primary: 90
C29; Secondary: 90 C26
Key words: Global optimization, Optimization over the efficient set, Outcome set,
Convex programming.
1. Introduction

Let
X
be a nonempty, compact and convex set in
n
R
. Let
( ), =1, ,
i
f x i k
,
2k 
,
be convex functions defined on a suitable open set containing
X
. Then the convex multiple
objective programming problem may be written as

1
Min ( ) = ( ( ), , ( )) s.t. .
T
k
f x f x f x x X
(CMP)
When
= 2,k
problem
()CMP
is called a bicriteria convex programming problem. Such
problem are not uncommon and have received special attention in the literature (see, for
instance, H.P.Benson and D.Lee [2], J.Fulop and L.D.Muu [3], N.T.B.Kim and T.N.Thang [6],

H.X.Phu [10], B.Schandle, K.Klamroth and M.M.Wiecek [11], Y.Yamamoto [14] and
references therein).
Let
h
be a real-valued function on
n
R
. The central problem of interest in this paper is
the following problem

min ( ) s.t. ,
X
h x x E
(OP
0
)
where
X
E
is the efficient decision set for problem
()CMP
and defined as follows:

0 0 0
={ | such that ( ) ( ) and ( ) ( )}.
X
E x X x X f x f x f x f x    

1
THIS RESEARCH IS FUNDED BY VIETNAM NATIONAL FOUNDATION FOR SCIENCE AND TECHNOLOGY
DEVELOPMENT (NAFOSTED) UNDER GRANT NUMBER "101.01-2013.19"
Kỷ yếu công trình khoa học 2014 – Phần I
Trường Đại học Thăng Long 27
As usual, we use the notation
12
yy
, where
12
,
k
yyR
, to indicate
12
ii
yy
for all
=1, ,ik
.
It is well-known that, the set
X
E
is generally neither convex set nor given explicitly as
the form of a standard mathematical programming problems, even in the case of linear multiple
objective programming problem when the component functions
1
,,
k
ff

of
f
are linear and
X
is a polyhedral convex set. Therefore, problem
0
()OP
is one of hard global programming
problems. This problem has applications in finance, economics, engineering, and other fields.
Recently this problem has been attracted a great deal of attention from researcher (see e.g. [1,
2, 3, 4, 5, 6, 8, 9, 12, 14] and references therein). In this article, simple convex programming
procedures are proposed for solving two special cases of problem
0
()OP
. These special-case
procedures require quite little computational effort in comparison to ones required by
algorithms for the general problem
0
()OP
.
2 Preliminaries
Let
Q
be a nonempty subset in
k
R
. The set of all efficient points of
Q
is denoted by
ep

Q
and given by

0 0 0
={ | such that and }.
ep
Q q Q q Q q q q q    

It is clear that a point
0
ep
qQ
if
00
( ) ={ }
k
Q q q

R
, where
={ | 0, =1, , }
kk
i
y y i k

RR
.
Let

={ | = ( ) for some }.

k
Y y y f x x XR

As usual, the set
Y
is said to be the outcome set for problem
()CMP
; see, for instance,
[2, 6, 12, 15]. Since the functions
, =1, ,
i
f i k
are continuous and
n
X  R
is a nonempty,
compact set, the outcome set
Y
is also compact set in
k
R
. Therefore the efficient set
ep
Y
is
nonempty [7]. The relationship between the efficient decision set
X
E
and the efficient set
ep

Y

of the outcome set
Y
is described as follows.
Proposition 2.1
i) For any
0
ep
yY
, if
0
xX
satisfies
00
()f x y
, then
0
X
xE
.
ii) For any
0
X
xE
, if
00
= ( )y f x
, then
0

.
ep
yY




Proof. This fact follows from the definitions. 
Kỷ yếu công trình khoa học 2014 – Phần I
Trường Đại học Thăng Long 28

Let

= ={ | for some }.
kk
T Y z z y y Y

   RR

It is clear that
T
is a nonempty, full-dimension closed convex set. The following interesting
property of
T
(Theorem 3.2 in [15]) will be used in the sequel .

Proposition 2.2
=.
ep ep
TY

For each
=1,2, , ,ik
let
I
i
y
be the optimal value of the following convex
programming problem

min s.t. .
i
y y T

It is clear that
I
i
y
is also the optimal value of the following convex programming problem

min ( ) s.t. .
i
f x y X
(P
i
)
The point
1
= ( , , )
I I I

k
y y y
is called the ideal point of the set
T
. Notice that the ideal
point
I
y
need not belong to
T
.
Proposition 2.3 If
I
yT
then
={ }
I
ep
Yy
.
Proof. This fact follows from the definitions and Proposition 2.2. 

It is clear that the case of
={ }
I
ep
Yy
is a special case of problem
()CMP
. Let

={ | ( ) , =1, , }.
id I
ii
X x X f x y i k

By the definition,
id
X
is a convex set. The following corollary is immediate from Proposition
2.1 and Proposition 2.3.
Proposition 2.4 If
id
X
is not empty then
I
yT
and
=
id
X
EX
. Otherwise,
I
y
does
not belong to
T
.

The next discuss concerns with the case that

()CMP
is a bicriteria convex programming
problem, i.e.
=2k
, and the objective function
()hx
of the problem
()OP
is defined by

1 1 2 2
( ) = ( ) ( ) = , ( )h x f x f x f x
  
  
(1)

where
2
12
= ( , )
T
  
R
. The case could happen in certain common situations. For instance,
problem
0
()OP
represents the minimization of a criterion function
()
i

fx
,
{1,2}i
, of
problem
()CMP
over the efficient decision set
X
E
.
Let
Kỷ yếu công trình khoa học 2014 – Phần I
Trường Đại học Thăng Long 29

={ ( )| }.
YX
E f x x E

The set
Y
E
is called the efficient outcome set for problem
()CMP
. The outcome-space
reformulation of problem
0
()OP
can be given by

min , s.t. .

Y
y y E

  
(OP
1
)
By the definition, it is easy to see that
=
Y ep
EY
. Combining this fact and Proposition
2.2, problem
1
()OP
can be rewritten as follows

min , s.t. .
ep
y y T

  
(OP
2
)
Here, instead of solving problem
1
()OP
, we solve problem
2

()OP
.
Since
T
is a nonempty convex subset in
2
R
, it is well know [9] that the efficient set
ep
T
is homeomorphic to a nonempty closed interval of
1
R
. By geometry, it is easily seen that
the problem

2 1 1
min{ : , = }
I
y y T y y
(P
S
)
has an unique optimal solution
S
y
and the problem

1 2 2
min{ : , = }

I
y y T y y
(P
E
)
has an unique optimal solution
E
y
. If
I
yT
then
SE
yy
and the efficient set
ep
T
is a curve
on the boundary of
T
with starting point
S
y
and the end point
E
y
.

Direct computation shows that the equation of the line through
S

y
and
E
y
is
,=ay


, where

12
1 1 2 2 1 1 2 2
11
= ( , ) and = .
EE
E S S E E S S E
yy
a
y y y y y y y y


   
(2)
It is clear that the vector
a
is strictly positive. Now, let

*
={ | , } and = \ ( , ),
SE

M y T a y M M y y

    

where
( , ) ={ = (1 ) |0 < <1}
S E S E
y y y ty t y t
and
M
is the boundary of the set
M
. It
is clear that
M
is a compact convex set. By the definition and geometry, we can see that
*
M

contains the set of all extreme points of
M
and

*
=.
ep
TM
(3)
Consider following convex problem

min , s.t. ,y y M

  
(OP
3
)
that has the explicit information as follows
Kỷ yếu công trình khoa học 2014 – Phần I
Trường Đại học Thăng Long 30

min ,
s.t. ( ) 0
,
,,
y
f x y
xX
ay





  

where vector
2
aR
and the real number


is determined by
(2)
.
The relationship between the optimal solutions to problem
0
()OP
and the optimal
solutions to problem
3
()OP
is presented in the following proposition.
Proposition 2.5 Suppose that
**
( , )xy
is an optimal solution of the problem
3
()OP
.
Then
*
x
is an optimal solution of problem
0
()OP
.
Proof. It is well known that a convex programming problem with the linear objective
function has an optimal solution which belongs to the extreme point set of the feasible solution
[13]. This fact and
(3)
implies that

*
y
is an optimal solution of problem
2
()OP
. Since
==
ep Y ep
T E Y
, by definition, we have
*
,,yy

    
for all
Y
yE
and
*
ep
yY
. Then

*
, , ( ) , .
X
y f x x E

      
(4)

Since
**
( , )xy
is a feasible solution of problem
3
()OP
, we have
**
()f x y
. By
Proposition 2.1,
*
X
xE
. Furthermore, we have
*
()f x Y
and
*
ep
yY
. This infers that
**
= ( )y f x
. Combining this fact and (4) prove that
*
x
is an optimal solution of problem
0

()OP
.
3 Solving Two Special Cases
Case 1. The ideal point
I
y
belongs to the outcome set
Y
and the objective function
()hx
of problem
0
()OP
is convex.
By Proposition 2.4, to detect whether the ideal point
I
y
belongs to
T
and solve problem
0
()OP
in this case, we solve the following convex programming problem

min ( ) s.t. .
id
h x x X
(CP
I
)

Namely, the procedure for this case is described as follows.
Procedure 1.
Step 1. For each
=1, ,ik
, find the optimal value
I
i
y
of problem
()
i
P
.
Step 2. Solve the convex programming problem
()
I
CP
.
If problem
()
I
CP
is not feasible Then STOP (the Case 1 does not apply).
Else Find any optimal solution
*
x
to problem
()
I

CP
. STOP
Kỷ yếu công trình khoa học 2014 – Phần I
Trường Đại học Thăng Long 31
(
*
x
is an optimal solution to problem
0
()OP
).

Below we present two numerical examples to illustrate Procedure 1.
Example 1. Consider problem
0
()OP
, where
X
E
is the efficient solution set to the
problem

1 2 1 2
Vmin ( ( ), ( )) = ( , )f x f x x x


22
12
s.t. ( 2) ( 2) 4xx   

12
2xx

  


12
2xx




12
2xx   

and
22
1 2 1 2
( ) = min{0.5 0.25 0.2;2 4.6 5.8}h x x x x x   
.
In the case
=0


Step 1. Solving problem
1
()P
and
2

()P
, we obtain the ideal point
= (0.6667.0.6667)
I
y
.
Step 2. Solving problem
()
I
CP
, we find that it is not feasible and the algorithm is
terminated. It means that the ideal point
I
y
does not belong to
T
.
In the case
=1


Step 1. Solving problem
1
()P
and
2
()P
, we obtain the ideal point
= (1,1)
I

y
.
Step 2. Solving problem
()
I
CP
, we find an optimal solution
*
= (1,1)x
. Then
*
x
is the
optimal solution to problem
0
()OP
and the optimal value
*
( ) = 0.9500.hx


Example 2. Consider problem
0
()OP
, with
22
1 1 2
( ) =10 (1 2 3 )h x x x x  
and
X

E
is
the efficient solution set to problem
()CMP
where
=2k
and

1 1 2 2 1 2
( ) = 3 2 3, ( ) = 1,f x x x f x x x   

2 2 2
1 1 2 2 1 2
={ |10 14 5 32 22 22 0,X x x x x x x x      R

1 1 2 1 2
4, 5 3 8, 4 3 4}x x x x x      

Step 1. Solving problem
1
()P
and
2
()P
, we obtain the ideal point
= (1.8000,2.6000)
I
y
.
Step 2. Solving problem

()
I
CP
, we can find an optimal solution
*
= (1.6000,0.0000)x
. Then
*
x
is the optimal solution to problem
0
()OP
and the optimal value
*
( ) = 43.2400.hx

Kỷ yếu công trình khoa học 2014 – Phần I
Trường Đại học Thăng Long 32

Case 2. Problem
()CMP
is a bicriteria convex programming problem and the
objective function
()hx
of the problem
0
()OP
has the form as
(1)
.

In this case, the procedure for solving problem is established basing on Proposition 2.4
and Proposition 2.5.

Procedure 2.
Step 1. For each
=1,2i
, find the optimal value
I
i
y
of problem
()
i
P
.
Step 2. Solve the convex programming problem
()
I
CP
.
If problem
()
I
CP
is not feasible Then Go to Step 3 (the ideal point
I
yT
).
Else Find any optimal solution
*

x
to the problem
()
I
CP
, where
=2k
. STOP
(
*
x
is an optimal solution for problem
0
()OP
).
Step 3. Solve the convex programming problems
()
S
P
and
()
E
P
to find the efficient
point
S
y
and
E
y

, respectively.
Step 4. Solve problem
3
()OP
to find any optimal solution
**
( , )xy
. STOP
(
*
x
is an optimal solution of problem
0
()OP
)
We give below some examples to illustrate Procedure 2.
Example 3. Consider the problem
0
()OP
, where
X
E
is the efficient solution set to
problem
()CMP
with
=2k
and

22

1 1 2 2
( ) = ( 2) 1, ( ) = ( 4) 1,f x x f x x   


 
2 2 2
1 2 1 2
= | 25 4 100 0, 2 4 0 ,X x x x x x      R

and
1 1 2 2
( ) = ( ) ( )h x f x f x


.
Case


*
x

*
y


*
()hx

1

(1.0,0.0)


(1.9931,0.4128)


(1.0000,13.8730)


1.0000

2

(0.8,0.2)


(1.6471,1.1765)


(1.1246,8.9723)


2.6941

3

(0.5,0.5)


(0.8000,1.6000)

(2.4400,6.7600)


4.6000

4

(0.2,0.8)


( 1.0000,2.5000)


(10.0000,3.2500)


4.6000

Kỷ yếu công trình khoa học 2014 – Phần I
Trường Đại học Thăng Long 33
5

(0.0,1.0)


( 1.6502,2.8251)

(14.3236,2.3804)


2.3804

6

( 0.2,0.8)


( 1.6502,2.8251)


(14.3236,2.3804)


0.9604

7

(0.8, 0.2)


(1.9932,0.4121)


(1.0000,13.8780)


1.9756

8

( 0.5, 0.5)


( 1.6502,2.8251)


(14.3236,2.3804)


8.3520

Table 1: Computational results of Example 3

Step 1. Solving problem
1
()P
and
2
()P
, we obtain the ideal point
= (1.000,2.3804)
I
y
.
Step 2. Solving problem
()
I

CP
, we can find that it is not feasible. Then go to Step 3.
Step 3. Solve two problems
()
S
P
and
()
E
P
to obtain the points
= (14.3236,2.3804)
S
y
and
= (1.0000,13.8781)
E
y
.
Step 4. For each
2
12
= ( , )
  
R
, solve problem
3
()OP
to find the optimal solution
**

( , )xy
. Then
*
x
is an optimal solution to problem
0
()OP
and
*
()hx
is the optimal value of
0
()OP
. The computational results are shown in Table 1.
Example 4. Consider the problem
0
()OP
, where
1 1 2 2
( ) = ( ) ( )h x f x f x


and
X
E
is
the efficient solution set to the following problem

 
12

Vmin ( ) = ( ), ( )
s.t. , 0, ( ) 0,
f x f x f x
Ax b x g x  

where
1.0 2.0 1.0
1.0 1.0 1.0
= , = ,
2.0 1.0 4.0
2.0 5.0 10.0
1.0 1.0 1.5
Ab

   
   

   
   
   
   
   
  
   

22
12
( ) = 0.5( 1) 1.4( 0.5) 1.1,g x x x   

22

1 1 2 1 2
( ) = 0.4 4 ,f x x x x x  
and
 
2 1 2 1 2
( ) = max (0.5 0.25 0.2); 2 4.6 5.8 .f x x x x x     

Case




*
x


*
y


*
()hx

1

(1.0,0.0)


(0.2724,1.2724)

( 3.2875, 0.4916)


3.2875

Kỷ yếu công trình khoa học 2014 – Phần I
Trường Đại học Thăng Long 34
2

(0.8,0.2)


(0.2500,1.2500)


( 3.2750, 0.5500)


2.7300

3

(0.5,0.5)


(0.3829,1.2731)


( 3.1639, 0.7021)

1.9330

4

(0.2,0.8)


(0.8623,1.3826)


( 2.5304, 0.9768)


1.2875

5

(0.0,1.0)


(1.3170,1.3659)


( 1.3365, 1.2000)


1.2000

6

( 0.2,0.8)


(1.3170,1.3659)


( 1.3365, 1.2000)


0.6927

7

(0.8, 0.2)


(0.2724,1.2724)


( 3.2875, 0.4916)


2.5316

8

( 0.5, 0.5)

(1.3170,1.3659)


( 1.3365, 1.2000)


1.2683

Table 2: Computational results of Example 4
Step 1. Solving problem
1
()P
and
2
()P
, we obtain the ideal point
= ( 3.2875, 1.2000)
I
y 
.
Step 2. Solving problem
()
I
CP
, we can find that it is not feasible. Then go to Step 3.
Step 3. Solve two problems
()
S
P

and
()
E
P
to obtain the points
= ( 1.3365, 1.2000)
S
y 
and
= ( 3.2875, 0.4916)
E
y 
.
Step 4. For each
2
12
= ( , )
  
R
, solve problem
3
()OP
to find the optimal solution
**
( , )xy
. Then
*
x
is an optimal solution to problem
0

()OP
and
*
()hx
is the optimal value of
0
()OP
. The computational results are shown in Table 2.
4. Conclusion
Problem
0
()OP
is a very hard and interesting task in multiple objective optimization
and has some important applications in finance, economics, engineering, and other fields. In
this paper, we propose simple convex programming procedures for solving problem
0
()OP
in
two special cases: i) The ideal point
I
y
belongs to the outcome set
Y
and the objective function
()hx
of problem
0
()OP
is convex; ii) Problem
()CMP

is a bicriteria convex programming
problem and the objective function
()hx
of the problem
0
()OP
has the form as
(1)
. These
procedures require quite little computational effort in comparison to that required to solve the
general problem
0
()OP
. Therefore, when solving problem
0
()OP
, they can be used as
screening devices to detect and solve this two special cases.
Acknowledgements. The authors wish to thank Prof. Tran Vu Thieu for his help.
5. References
[1] An, L.T.H., Tao, P. D., Muu, L. D. (1996), “Numerical solution for optimization
over the efficient set by d.c. optimization algorithm”, Oper. Res. Lett., 19, 117-128.
Kỷ yếu công trình khoa học 2014 – Phần I
Trường Đại học Thăng Long 35
[2] Benson, H. P., Lee, D. (1996), “Outcome-based algorithm for optimizing over the
efficient set of a bicriteria linear programming problem”, J. Optim. Theory Appl., 88, 77-105.
[3] Fulop, J., Muu, L.D. (2000), “Branch-and-bound variant of an outcome-based
algorithm for optimizing over the efficient set of a bicriteria linear programming problem”, J.
Optim. Theory Appl., 105, 37-54.
[4] Horst, R., Thoai, N.V. (2007), “Yamamoto, Y., Zenke, D.: On optimization over the

efficient set in linear multicriteria programming”, J. Optim. Theory Appl., 134, 433-443.
[5] Kim, N.T.B., Muu, L.D. (2002), “On the projection of the efficient set and potential
application”, Optim., 51, 401-421.
[6] Kim, N.T.B., Thang, T.N. (2013) “Optimization over the Efficient Set of a Bicriteria
Convex Programming Problem”, Pacific J. Optim., 9, 103-115.
[7] Luc, D.T. (1989), Theory of Vector Optimization, Springer-Verlag, Berlin,
Germany.
[8] Luc L. T., Muu L. D. (1997), “Global optimization approach to optimization over
the efficient set”, The Proceedings of the 8th French-German Conference on Optimiza- tion,
Lecture Notes in Economics and Mathematical System, 452 , Springer-Verlag, Berlin, 213-221.
[9] Philip, J. (1972), “Algorithms for the vector maximization problem”, Math.
Program., 2, 207-229.
[10] Phu, H. X. (2005), “On efficient sets in
2
R
”, Vietnam J. Math., 33, 463-468.

[11] Schandle B., Klamroth K., Wiecek M. M. (2001), “Norm-Based Approximation in
Bicriteria Programming”, Computational Optimization and Applications, 20, pp 23-42.
[12] Thoai, N. V. (2010), “Reverse convex programming approach in the space of
extreme criteria for optimization over efficient sets”, J. Optim. Theory and Appl., 147, 263-
277.
[13] Tuy H. (1998), Convex Analysis and Global Optimization, Academic Kluwer
Publisher.
[14] Yamamoto, Y. (2002), “Optimization over the efficient set: overview”, J. Global
Optim., 22, 285-317.
[15] Yu, P. L. (1985), Multiple-Criteria Decision Making, Plenum Press, New York and
London.