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

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 proble

OPTIMIZING OVER THE EFFICIENT SET OF A CONVEX

Assoc Prof Nguyen Thi Bach Kim1 and M.Sc.Tran Ngoc Thang2,

1,2

School of Applied Mathematics and Informatics

Hanoi University of Science and Technology

1

Email: kim.nguyenthibach@hust.edu.vn

2

Email: thang.tranngoc@hust.edu.vn

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 Rn Let f x i i( ), = 1, ,k, k 2,

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

When k = 2,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 Rn The central problem of interest in this paper is

the following problem

min ( ) s.t.h x xE X, (OP0) where E X is the efficient decision set for problem ( CMP ) and defined as follows:

= { | such that ( ) ( ) and ( ) ( )}

X

1 THIS RESEARCH IS FUNDED BY VIETNAM NATIONAL FOUNDATION FOR SCIENCE AND TECHNOLOGY

DEVELOPMENT (NAFOSTED) UNDER GRANT NUMBER "101.01-2013.19"

As usual, we use the notation y1  y2, where y y1, 2Rk, to indicate y1i  y i2 for all

= 1, ,

i k

It is well-known that, the set E X 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 f1, ,f k of f are linear and

X is a polyhedral convex set Therefore, problem (OP0) 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 (OP0) These special-case procedures require quite little computational effort in comparison to ones required by algorithms for the general problem (OP0)

2 Preliminaries

Let Q be a nonempty subset in Rk The set of all efficient points of Q is denoted by

ep

Q and given by

= { | such that and }

ep

It is clear that a point q0Q ep if Q(q0 Rk) = { }q0 , where

= { | 0, = 1, , }

i

Let

= { k | = ( ) for some }

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 f i i, = 1, ,k are continuous and X Rn is a nonempty, compact set, the outcome set Y is also compact set in Rk Therefore the efficient set Y ep is nonempty [7] The relationship between the efficient decision set E X and the efficient set Y ep

of the outcome set Y is described as follows

Proposition 2.1

i) For any y0Y ep , if x0 X satisfies f x( 0) y0, then x0E X

ii) For any x0E X , if y0 = f x( 0), then y0Y ep

Proof This fact follows from the definitions 

Let

= k = { k| for some }

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 T ep =Y ep

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

miny i s.t yT

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

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

k

y y y is called the ideal point of the set T Notice that the ideal

point y I need not belong to T

Proposition 2.3 If y IT then = { }I

ep

Proof This fact follows from the definitions and Proposition 2.2 

It is clear that the case of = { }I

ep

Y y is a special case of problem ( CMP ) Let

= { | ( ) , = 1, , }

By the definition, X id is a convex set The following corollary is immediate from Proposition 2.1 and Proposition 2.3

Proposition 2.4 If X id is not empty then y IT and = id

X

E X Otherwise, y I does not belong to T

The next discuss concerns with the case that ( CMP ) is a bicriteria convex programming problem, i.e k = 2, and the objective function h x ( ) of the problem ( OP ) is defined by

1 1 2 2

( ) = ( ) ( ) = , ( )

h x  f x  f x  f x 

where = ( , 1 2)TR2 The case could happen in certain common situations For instance, problem (OP0) represents the minimization of a criterion function f x i( ), i  {1, 2}, of problem ( CMP ) over the efficient decision set E X

Let

= { ( ) | }.

The set E Y is called the efficient outcome set for problem ( CMP ) The outcome-space reformulation of problem (OP0) can be given by

min ,y s.t.yE Y (OP1)

By the definition, it is easy to see that E Y =Y ep Combining this fact and Proposition 2.2, problem (OP1) can be rewritten as follows

min,y s.t.yT ep (OP2) Here, instead of solving problem (OP1), we solve problem (OP2)

Since T is a nonempty convex subset in R2, it is well know [9] that the efficient set

ep

T is homeomorphic to a nonempty closed interval of R1 By geometry, it is easily seen that the problem

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

min{y :yT y, = y I} (PE) has an unique optimal solution y E If y I T then y S  y E and the efficient set T ep is a curve

on the boundary of T with starting point y S and the end point y E

Direct computation shows that the equation of the line through y S and y E is

, =

  , where

a

It is clear that the vector a is strictly positive Now, let

*

= { | , } and = \ ( S, E),

where (y S,y E) = { =y ty S (1 t y) E| 0 < < 1}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

Consider following convex problem

min ,y s.t.yM, (OP3) that has the explicit information as follows

min , s.t ( ) 0

, , ,

y

a y





 



where vector a  R2 and the real number  is determined by (2)

The relationship between the optimal solutions to problem (OP0) and the optimal solutions to problem (OP3) is presented in the following proposition

Proposition 2.5 Suppose that ( ,x y* *) is an optimal solution of the problem (OP3) Then x* is an optimal solution of problem (OP0)

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 (OP2) Since

= =

T E Y , by definition, we have ,y*  ,y for all yE Y and y*Y ep Then

*

,y , ( ) ,f x x E X

Since ( ,x y* *) is a feasible solution of problem (OP3), we have f x( )*  y* By Proposition 2.1, x*E X Furthermore, we have f x( )* Y and y*Y ep 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 y I belongs to the outcome set Y and the objective function

( )

h x of problem (OP0) is convex

By Proposition 2.4, to detect whether the ideal point y I belongs to T and solve problem

0

(OP) in this case, we solve the following convex programming problem

min ( ) s.t.h x xX id (CP

I) Namely, the procedure for this case is described as follows

Procedure 1

Step 1 For each i = 1, , k, find the optimal value y i I of problem ( )P i

Step 2 Solve the convex programming problem (CP I)

If problem (CP I) is not feasible Then STOP (the Case 1 does not apply)

Else Find any optimal solution x* to problem (CP I) STOP

(x* is an optimal solution to problem (OP0))

Below we present two numerical examples to illustrate Procedure 1

Example 1 Consider problem (OP0), where E X is the efficient solution set to the problem

Vmin ( ( ),f x f x( )) = ( ,x x )

s.t (x 2) (x 2) 4

1 2 2

1 2

2x x 

1 2 2

and h x( ) = min{0.5x10.25x220.2;2x124.6x25.8}

In the case  = 0

Step 1 Solving problem ( )P1 and ( )P2 , we obtain the ideal point

= (0.6667.0.6667)

I

Step 2 Solving problem (CP I), we find that it is not feasible and the algorithm is terminated It means that the ideal point y I does not belong to T

In the case  = 1

Step 1 Solving problem ( )P1 and ( )P2 , we obtain the ideal point y I = (1,1)

Step 2 Solving problem (CP I), we find an optimal solution x* = (1,1) Then x* is the optimal solution to problem (OP0) and the optimal value h x( ) = 0.9500.*

Example 2 Consider problem (OP0), with h x( ) = 10x12  (1 2x13 )x2 2 and E X is the efficient solution set to problem ( CMP ) where k = 2 and

1( ) = 3 1 2 2 3, 2( ) = 1 2 1,

= { |10 14 5 32 22 22 0,

1 4, 5 1 3 2 8, 4 1 3 2 4}

Step 1 Solving problem ( )P1 and ( )P2 , we obtain the ideal point

= (1.8000, 2.6000)

I

Step 2 Solving problem (CP I), we can find an optimal solution x* = (1.6000,0.0000)

Case 2 Problem ( CMP ) is a bicriteria convex programming problem and the objective function h x ( ) of the problem (OP0) 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 i= 1, 2, find the optimal value y i I of problem ( )P i

Step 2 Solve the convex programming problem (CP I)

If problem (CP I) is not feasible Then Go to Step 3 (the ideal point y I T)

Else Find any optimal solution x* to the problem (CP I), where k = 2 STOP

(x* is an optimal solution for problem (OP0))

Step 3 Solve the convex programming problems (P S) and (P E) to find the efficient point y S and y E, respectively

Step 4 Solve problem (OP3) to find any optimal solution ( ,x y* *) STOP

(x* is an optimal solution of problem (OP0))

We give below some examples to illustrate Procedure 2

Example 3 Consider the problem (OP0), where E X is the efficient solution set to problem ( CMP ) with k = 2 and

1( ) = ( 1 2) 1, 2( ) = ( 2 4) 1,

= | 25 4 100 0, 2 4 0 ,

and h x( ) =1 1f x( )2f x2( )

x y* h x( *)

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

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 ( )P1 and ( )P2 , we obtain the ideal point y I = (1.000, 2.3804)

Step 2 Solving problem (CP I), we can find that it is not feasible Then go to Step 3

Step 3 Solve two problems (P S) and (P E) to obtain the points

= (14.3236, 2.3804)

S

Step 4 For each  = ( , 1 2)R2, solve problem (OP3) to find the optimal solution

* *

( ,x y ) Then x* is an optimal solution to problem (OP0) and h x( *) is the optimal value of 0

(OP) The computational results are shown in Table 1

Example 4 Consider the problem (OP0), where h x( ) =1 1f x( )2f x2( ) and E X is the efficient solution set to the following problem

Vmin ( ) = ( ), ( ) s.t , 0, ( ) 0,

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



( ) = 0.5( 1) 1.4( 0.5) 1.1,

2 2

1( ) = 1 2 0.4 1 4 ,2

f x x x  x  x and

2( ) = max (0.5 1 0.25 2 0.2); 2 1 4.6 2 5.8

x y* h x( *)

1 (1.0,0.0) (0.2724,1.2724) ( 3.2875, 0.4916)    3.2875

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 ( )P1 and ( )P2 , we obtain the ideal point

= ( 3.2875, 1.2000)

I

Step 2 Solving problem (CP I), we can find that it is not feasible Then go to Step 3

Step 3 Solve two problems (P S) and (P E) to obtain the points

= ( 1.3365, 1.2000)

S

y   and y E = ( 3.2875, 0.4916) 

Step 4 For each  = ( , 1 2)R2, solve problem (OP3) to find the optimal solution

* *

( ,x y ) Then x* is an optimal solution to problem (OP0) and h x( *) is the optimal value of 0

(OP) The computational results are shown in Table 2

4 Conclusion

Problem (OP0) 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(OP0) in two special cases: i) The ideal point y I belongs to the outcome set Y and the objective function

( )

h x of problem (OP0) is convex; ii) Problem ( CMP ) is a bicriteria convex programming problem and the objective function h x ( ) of the problem (OP0) has the form as (1) These

procedures require quite little computational effort in comparison to that required to solve the general problem (OP0) Therefore, when solving problem (OP0) , 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

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