Convergence of exponential penalty function method for multiobjective fractional programming problems

ENGINEERING PHYSICS AND MATHEMATICS

Convergence of exponential penalty

function method for multiobjective fractional

programming problems

Anurag Jayswal, Sarita Choudhury

Department of Applied Mathematics, Indian School of Mines, Dhanbad 826 004, Jharkhand, India Received 20 January 2014; revised 6 May 2014; accepted 24 July 2014

Available online 23 August 2014

KEYWORDS

Exponential penalty function method;

Multiobjective fractional programming problem; Convergence

Abstract In this paper, we extend the application of exponential penalty function method for solv-ing multiobjective programmsolv-ing problem introduced by Liu and Feng (2010) to solve multiobjective fractional programming problem and analyze the relationship between weak efficient solutions of penalized problems and multiobjective fractional programming problem. Furthermore, we examine the convergence of this method for multiobjective fractional programming problems.

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1. Introduction

In various practical problems arising in decision theory, eco-nomics, game theory, portfolio selection, etc., it is required to optimize ratio of several linear or nonlinear functions to achieve the goal efficiently. These optimization problems are called multiobjective fractional programming problems. The study of multiobjective fractional programming problems

[3,13,14] has been of great interest in the recent past due to its diversified application. Dinkelbach [4] and Jagannathan

[7]used the concept of parametric approach for scalar frac-tional programming. Basically, the multiobjective fracfrac-tional programming problem is either parametrized or transformed by suitable methods to obtain an equivalent multiobjective programming problem.

One of the important methods to solve general nonlinear constrained optimization problems is the penalty function method. In this method, the original problem is replaced by a sequence of simpler unconstrained subproblems, referred to as the penalized problems, in which the constraints are repre-sented by terms added to the objective. The penalty function method is effective if the sequence of solutions of unstrained problems converge to the solution of related con-strained optimization problem.

White [15] initiated the application of penalty function method to multiobjective programming problem and studied the convergence and efficiency properties of the sequence of penalized unconstrained multiobjective programming prob-lems. Huang and Yang[5]introduced nonlinear penalty func-tions and formulated nonlinear penalty problems for multiobjective constrained optimization problems.

* Corresponding author. Tel.: +91 8539094504.

E-mail addresses: [email protected] (A. Jayswal),

[email protected](S. Choudhury).

q

The research of the first author is financially supported by the DST, New Delhi, India through Grant no.: SR/FTP/MS-007/2011. Peer review under responsibility of Ain Shams University.

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Thereafter, Huang et al.[6]studied convergence properties of a class of penalization methods for vector optimization problems with cone constraints in infinite dimensional space and established that any limit point of a sequence of approxi-mate efficient solutions to the penalty problems is a weakly efficient solution to the original cone constrained optimization problem.

The class of penalty functions has been studied by many researchers, see for instance,[1,2,8–12]. Murphy[11]developed a class of exponential penalty functions for nonlinear program and established its convergence rate. Recently, Liu and Feng

[8]constructed exponential penalty function method for multi-objective programming problems and proved its convergence. Motivated by the work of Liu and Feng[8], we present this paper. In this paper, we apply the exponential penalty function method to solve multiobjective fractional programming prob-lem. This paper is organized as follows: In Section 2, we recall some definitions and lemma to help us prove our results. In Section 3, an exponential penalty function is used to construct a sequence of unconstrained penalized problems for multiob-jective fractional programming problem. In Section 4, we examine the convergence of this method and investigate the relationship between the sequence of weak efficient solutions to the penalized problems and the weak efficient solution to the original multiobjective fractional programming problem. Finally, in Section 5, we conclude the paper.

2. Notations and preliminaries

Let Rn _{denote the n-dimensional Euclidean space and R}n þ

denote the non-negative orthant of Rn_{. Let x; y be arbitrary}

vectors in Rn_{. The following conventions will be adopted for}

equalities and inequalities throughout this paper.

(i) x 6 y() xi6yi; i¼ 1; . . . ; n, with strict inequality

holding for at least one i; (ii) x 5 y() xi6yi; i¼ 1; . . . ; n;

(iii) x¼ y () xi¼ yi; i¼ 1; . . . ; n;

(iv) x < y() xi< yi; i¼ 1; . . . ; n.

We consider the following multiobjective fractional pro-gramming problem: ðMFPÞ min FðxÞ ¼ f1ðxÞ g₁ðxÞ; f2ðxÞ g₂ðxÞ; . . . ; fmðxÞ g_mðxÞ   subject to hjðxÞ 6 0; j ¼ 1; 2; . . . ; p:

Let D denote the set of all feasible solutions to (MFP), i.e., D¼ fx 2 Rn_{: h}

jðxÞ 6 0; 1 6 j 6 pg:

The functions fi; gi: R n

# R; i¼ 1; 2; . . . ; m; hj: Rn# R;

j¼ 1; 2; . . . ; p are continuous on Rn _{and f}

iðxÞ P 0; giðxÞ > 0;

8x 2 D; i ¼ 1; 2; . . . ; m.

In multiobjective programming problems, there may not exist a point which simultaneously optimizes each objective. Therefore, the concept of efficient (Pareto optimal) solution is used.

Definition 2.1 [13]. A point x2 D is said to be a weakly efficient solution to (MFP) if there exists no x2 D such that

f_iðxÞ giðxÞ

<fiðxÞ giðxÞ

; 1 6 i 6 m:

We denote the set of weakly efficient solutions to (MFP) by X_.

Now, we consider the following nonfractional multiobjec-tive programming problem associated to (MFP) with a parameter v2 Rm

þ based on the parametric approach of Dinkelbach [4] and Jagannathan [7]for the scalar fractional programming problem.

ðMFPÞv minðf1ðxÞ  v1g1ðxÞ; . . . ; fmðxÞ  vmgmðxÞÞ

subject to

hjðxÞ 6 0; j ¼ 1; 2; . . . ; p:

Definition 2.2 [13]. A point x2 D is said to be a weakly effi-cient solution to (MFP)vif there exists no x2 D such that

fiðxÞ  vigiðxÞ < fiðxÞ  vigiðxÞ; 1 6 i 6 m:

The following lemma establishes the equivalence between weakly efficient solutions to (MFP) and (MFP)v.

Lemma 2.1 [13]. Let x be a weakly efficient solution for (MFP). Then there exists v2 Rm_{such that x is a weakly}

effi-cient solution for (MFP)v. Conversely, if x is a weakly efficient

solution for (MFP)vwhere v¼_gðfðxÞ_xÞthen x is also a weakly

effi-cient solution for (MFP).

3. The exponential penalty function method for multiobjective fractional programming problems

In this section, we use the exponential penalty function method to transform the constrained multiobjective fractional pro-gramming problem (MFP) to an unconstrained one. The expo-nential penalized problem (MFP)n associated with

multiobjective fractional programming problem (MFP) is con-structed as: ðMFPÞ_n min Fðx; qnÞ ¼ f1ðxÞ  v1g1ðxÞ þ 1 q_n Xp j¼1 ðeqnhjðxÞ_{ 1Þ; . . . ; f} mðxÞ  vmgmðxÞ þ 1 q_n Xp j¼1 ðeqnhjðxÞ_{ 1Þ} ! where v2 Rm

þ;qn>0 is the penalty parameter such that

limn!1qn¼ þ1.

Definition 3.1. A point x2 Rn_{is said to be a weakly efficient} solution to (MFP)nif there exists no x2 Rnsuch that

Fðx; qnÞ < Fðx;qnÞ:

Let X_ndenote the set of weakly efficient solutions to (MFP)n.

Throughout this paper, let An Rk; n2 N ¼ f1; 2; . . .g. We denote

limn!1An¼ fx 2 Rk: x2 An for infinitely many n2 Ng;

limn!1An¼ fx 2 Rk: x2 An for all but finitely many n2 Ng:

lim

n!1An¼ limn!1An¼ limn!1An:

The following lemma shows that the feasibility of a solution to the original multiobjective fractional programming problem determines the limit point of the exponential penalty function with respect to the penalty parameter q. This lemma is very important as it is used to prove some of the main results in the sequel of this paper.

Lemma 3.1. Let D be the set of feasible solutions to the multi-objective fractional programming problem (MFP).

(I) If x2 D, then limq!þ11q

Pp j¼1ðe

qhjðxÞ_{ 1Þ ¼ 0.}

(II) If x R D, then limq!þ11q

Pp j¼1ðe

qhjðxÞ_{ 1Þ ¼ þ1.}

Proof.

(I) Let x2 D. Then, hjðxÞ 6 0; j ¼ 1; 2; . . . ; p. If hjðxÞ ¼ 0;

8j ¼ 1; 2; . . . ; p, then it is obvious that lim q!þ1 1 q Xp j¼1 ðeqhjðxÞ_{ 1Þ ¼ 0:}

Otherwise, suppose hjðxÞ ¼ rjðxÞ, where rjðxÞ P 0; j ¼

1; 2; . . . ; p. Thus lim q!þ1 1 q Xp j¼1 ðeqhjðxÞ_{ 1Þ ¼ lim} q!þ1 1 q Xp j¼1 ðeqrjðxÞ_{ 1Þ ¼}X p j¼1 lim q!þ1 1 eqrjðxÞ qeqrjðxÞ ¼X p j¼1 lim q!þ1 rjðxÞeqrjðxÞ eqrjðxÞ_{þ qrjðxÞe}qrjðxÞ¼ Xp j¼1 lim q!þ1 rjðxÞ 1þ qrjðxÞ ¼ 0:

(II) If x R D, then hjðxÞ > 0, for some j 2 f1; 2; . . . ; pg. Let

hj1ðxÞ > 0, where j12 f1; 2; . . . ; pg. Thus, lim q!þ1 1 q Xp j¼1 ðeqhjðxÞ_{ 1Þ ¼ lim} q!þ1 1 q Xp j¼ 1 j–j₁ ðeqhjðxÞ_{ 1Þ þ lim} q!þ1 1 q  ðeqh_j1ðxÞ_{ 1Þ} ¼ lim q!þ1 1 q Xp j¼ 1 j–j1 ðeqhjðxÞ_{ 1Þ} þ lim q!þ1hj1ðxÞe qh_j1ðxÞ ¼ þ1: 

UsingLemma 3.1, we derive the necessary condition for a point to be a feasible solution of multiobjective fractional pro-gramming problem in the following lemma.

Lemma 3.2. Let q_n>0 and limn!1q_n¼ þ1. If x_{2 lim} n!1 X_n, then x_{2 D.}

Proof. Since x_{2 lim}

n!1Xn, there exists a subsequencefnkg of

fNg such that x_{2 X}

nk; k¼ 1; 2; . . . Therefore, by the

defini-tion of weakly efficient soludefini-tion to ðMFPÞ_n

k, there exists no  x2 Rn _{such that} Fðx;qnkÞ < Fðx _;q nkÞ i:e: f_iðxÞ  vigiðxÞ þ 1 qnk Xp j¼1 ðeq_nkhjðxÞ_{ 1Þ < f} iðxÞ  vigiðxÞ þ 1 qnk Xp j¼1 ðeq_nkhjðxÞ_{ 1Þ i¼ 1; 2; . . . ; m; k ¼ 1; 2; . . . ð1Þ}

Suppose contrary to the result that x_{R D. Then by Lemma}

3.1(II), we have lim k!1 Xp j¼1 ðeq_nkhjðxÞ_{ 1Þ} qnk ¼ þ1:

For any point x2 D, byLemma 3.1(I), it follows that lim k!1 Xp j¼1 ðeq_nkhjðxÞ_{ 1Þ} q_n k ¼ 0:

Thus for sufficiently large k, i.e. k > k0(say), we conclude that

fiðxÞ  vigiðxÞ þ 1 qnk Xp j¼1 ðeq_nkhjðxÞ_{ 1Þ < f} iðxÞ  vigiðxÞ þ 1 q_n k Xp j¼1 ðeq_nkhjðxÞ_{ 1Þ i¼ 1; 2; . . . ; m; k > k} 0;

which contradicts(1). This completes the proof. h

4. Convergence of the exponential penalty function method for multiobjective fractional programming problems

In this section, we shall prove that the sequence of set of weakly efficient solutions to the unconstrained exponential penalized multiobjective programming problem (MFP)n

con-verge to the set of weakly efficient solutions of the original multiobjective fractional programming problem (MFP). Theorem 4.1. limn!1 Xnn X



 

is an empty set.

Proof. Suppose contrary to the result, x2 limn!1 Xnn X

 

. Therefore, there exists a number n0>0 such that x2 Xnn X



for n P n0. Firstly, let us assume that x2 D. Since x R X, then

byLemma 2.1, there exists x2 D such that

fiðxÞ  vigiðxÞ < fiðxÞ  vigiðxÞ; ð2Þ

where vi¼ fiðxÞ

giðxÞ; i¼ 1; 2; . . . ; m.

Further, x2 X

nimplies that the inequality

fiðxÞ  vigiðxÞ þ 1 qn Xp j¼1 ðeqnhjðxÞ_{ 1Þ} < fiðxÞ  vigiðxÞ þ 1 q_n Xp j¼1 ðeqnhjðxÞ_{ 1Þ ð3Þ}

does not hold for i¼ 1; 2; . . . ; m and n > n0.

Taking limit as n! 1 in the above inequality and using

Lemma 3.1(I), it follows that

fiðxÞ  vigiðxÞ < fiðxÞ  vigiðxÞ; i ¼ 1; 2; . . . ; m;

does not hold. This contradicts(2).

Now, we assume that x R D. Then byLemma 3.1(II), we have

lim n!1 1 qn Xp j¼1 ðeqnhjðxÞ_{ 1Þ ¼ þ1:}

Let x2 D. ByLemma 3.1(I), it follows that lim n!1 1 qn Xp j¼1 ðeqnhjðxÞ_{ 1Þ ¼ 0:}

Therefore for sufficiently large n, we conclude that f_iðxÞ  vigiðxÞ þ 1 qn Xp j¼1 ðeqnhjðxÞ_{ 1Þ} < fiðxÞ  vigiðxÞ þ 1 qn Xp j¼1 ðeqnhjðxÞ_{ 1Þ;}

which contradicts(3). This completes the proof. h Theorem 4.2. limn!1 Xnn X



 

is an empty set.

Proof. Suppose contrary to the result, x2 limn!1 Xnn X

 

. Therefore, there exists a subsequence fnkg of N such that

x2 X nkn X

_{. Let x2 D. Since x R X}_{, then by Lemma 2.1,}

there exists x2 D such that

fiðxÞ  vigiðxÞ < fiðxÞ  vigiðxÞ; ð4Þ

where vi¼ fiðxÞ

giðxÞ; i¼ 1; 2; . . . ; m.

Also fromLemma 3.1(I), we have

lim k!1 1 q_n k Xp j¼1 ðeqnkhjðxÞ_{ 1Þ ¼ 0} and lim k!1 1 qnk Xp j¼1 ðeq_nkhjðxÞ_{ 1Þ ¼ 0:}

Thus from(4), it follows that f_iðxÞ  vigiðxÞ þ 1 q_n k Xp j¼1 ðeqnkhjðxÞ_{ 1Þ} < fiðxÞ  vigiðxÞ þ 1 qnk Xp j¼1 ðeq_nkhjðxÞ_{ 1Þ; ð5Þ}

for i¼ 1; 2; . . . ; m and sufficiently large k. Further, x2 X

nk implies that the inequality

f_iðxÞ  vigiðxÞ þ 1 qnk Xp j¼1 ðeq_nkhjðxÞ_{ 1Þ} < fiðxÞ  vigiðxÞ þ 1 qnk Xp j¼1 ðeq_nkhjðxÞ_{ 1Þ ð6Þ}

does not hold for i¼ 1; 2; . . . ; m and k ¼ 1; 2; . . ., which con-tradicts(5).

Now, let x R D. Then byLemma 3.1(II), we have

lim nk!1 1 q_n k Xp j¼1 ðeq_nkhjðxÞ_{ 1Þ ¼ þ1:}

Let x2 D. ByLemma 3.1(I), it follows that

lim nk!1 1 qnk Xp j¼1 ðeq_nkhjðxÞ_{ 1Þ ¼ 0:}

Therefore, for sufficiently large nk, we conclude that

f_iðxÞ  vigiðxÞ þ 1 qnk Xp j¼1 ðeq_nkhjðxÞ_{ 1Þ} < fiðxÞ  vigiðxÞ þ 1 qnk Xp j¼1 ðeq_nkhjðxÞ_{ 1Þ;}

which contradicts(6). This completes the proof. h Theorem 4.3. limn!1 Xnn X



 

is an empty set.

Proof. It follows directly from Theorems 4.2 and 4.1. h The following theorem shows that if a subsequence of the sequence of weakly efficient solutions of the unconstrained exponential penalized multiobjective programming problem (MFP)nconverges to a feasible solution of the original

multi-objective fractional programming problem (MFP), then the limit point of the penalty function at each element of the sub-sequence is zero.

Theorem 4.4. Let x_nbe a weakly efficient solution to (MFP)n,

n¼ 1; 2; . . . If fx

nkg is a convergent subsequence of fx

 ng and limk!1xnk is a feasible solution to (MFP), then

lim k!1 1 qnk Xm j¼1 ðeq_nkhjðx_nkÞ_{ 1Þ ¼ 0:}

Proof. Suppose contrary to the result that limk!1_q1

nk

Pm j¼1ðe

q_nkhjðx_nkÞ_{ 1Þ–0. Then, there exists a}

subse-quencefx nksg of fx  nkg such that lim k!1 1 q_n ks Xm j¼1 ðeqnkshjðx_nksÞ_{ 1Þ > ;} s¼ 1; 2; . . . ; ð7Þ where  is a positive number.

Since x

nks is the weakly efficient solution to the exponential

penalized problem ðMFPÞ_n

ks associated with multiobjective

fractional programming problem (MFP), therefore there exists no x2 Rn_{such that} f_iðxÞ  vigiðxÞ þ 1 qnks Xp j¼1 ðeqnkshjðxÞ_{ 1Þ} < fiðxnksÞ  vigiðx  nksÞ þ 1 q_n ks Xp j¼1 ðeqnkshjðx nksÞ 1Þ; i ¼ 1; 2; . . . ; m; s ¼ 1; 2; . . . ð8Þ

Clearly, there exists no x2 D such that (8) holds. Let lim_k!1x

nk¼ x

_{. Since x}_{2 D, inequality (8) does not hold}

for x_{, i.e.,} fiðxÞ  vigiðxÞ þ 1 q_n ks Xp j¼1 ðeqnkshjðxÞ  1Þ < fiðxn_ksÞ  vigiðxn_ksÞ þ 1 qnks Xp j¼1 ðeqnkshjðx_nksÞ  1Þ; does not hold for i¼ 1; 2; . . . ; m; s ¼ 1; 2; . . .

This implies that for each point x nks, there exists is2 f1; 2; . . . ; mg such that fisðx _{Þ  v} isgisðx _{Þ þ} 1 qn_ks Xp j¼1 ðeqnkshjðxÞ_{ 1Þ} P fisðx  nksÞ  visgisðx  nksÞ þ 1 qn_ks Xp j¼1 ðeqnkshjðx nksÞ 1Þ; s¼ 1; 2; . . .

Thus we obtain an infinite sequence fisg 2 f1; 2; . . . ; mg. But

the setf1; 2; . . . ; mg is finite. Hence the infinite sequence fisg

contains infinite number of terms having the same value. With-out loss of generality, let the value of infinite number of terms infisg be 1. Therefore, it follows that

f1ðxÞ  v1g1ðxÞ þ 1 q_n ks Xp j¼1 ðeqnkshjðxÞ  1Þ P f₁ðx nksÞ  v1g1ðx  nksÞ þ 1 qnks Xp j¼1 ðeqnkshjðx_nksÞ  1Þ; s ¼ 1; 2; . . .

Taking limit as s! 1 and using(7)in the above inequality, we have f₁ðx_{Þ  v} 1g1ðxÞ þ lim_s!1 1 qnks Xp j¼1 ðeqnkshjðxÞ  1Þ P lim s!1ff1ðx  nksÞ  v1g1ðx  nksÞg þ :

Since f1 is continuous and limk!1xnk¼ x

_{, the above}

inequal-ity yields lim s!1 1 qnks Xp j¼1 ðeqnkshjðxÞ_{ 1Þ P : ð9Þ}

Again x_{2 D implies that}

lim s!1 1 qnks Xp j¼1 ðeqnkshjðxÞ_{ 1Þ ¼ 0:}

Thus from(9), we conclude that 0 P ;

which contradicts the fact that  > 0. This completes the proof. h

Theorem 4.5. limn!1Xn#limn!1Xn# D.

Proof. It directly follows fromLemma 3.2. h

In the following theorem, we prove that under certain con-ditions, a weakly efficient solution to the original multiobjec-tive fractional programming problem can be approached by a convergent subsequence of the sequence of weakly efficient solutions of the unconstrained penalized multiobjective pro-gramming problem.

Theorem 4.6. Let x

nbe a weakly efficient solution to (MFP)n,

n¼ 1; 2; . . . If fx

nkg is a convergent subsequence of fx

 ng and limk!1x

nkis a feasible solution to (MFP), then the limit point

of x

nkis a weakly efficient solution to (MFP).

Proof. Let limk!1xnk¼ x

_{. Since, x} nk2 X  nk, there exists no  x2 D such that f_iðxÞ  vigiðxÞ þ 1 q_n k Xp j¼1 ðeq_nkhjðxÞ_{ 1Þ} < fiðxnkÞ  vigiðx  nkÞ þ 1 qnk Xp j¼1 ðeq_nkhjðx_nkÞ_{ 1Þ;} i ¼ 1; 2; . . . ; m; k ¼ 1; 2; . . . : ð10Þ Further x2 D and limk!1xnk2 D. Thus fromLemma 3.1(I),

it follows that lim k!1 1 qnk Xp j¼1 ðeq_nkhjðxÞ_{ 1Þ ¼ 0} and lim k!1 1 qnk Xp j¼1 ðeq_nkhjðx_nkÞ_{ 1Þ ¼ 0:}

Taking limit as k! 1 and using the above two values in(10), we conclude that there exists no x2 D such that

fiðxÞ  vigiðxÞ < fiðxÞ  vigiðxÞ; i¼ 1; ;2; . . . ; m:

Thus x _{is a weak efficient solution to (MFP)}

v. Hence by

Lemma 2.1, it follows that x_{is a weakly efficient solution to}

(MFP). This completes the proof. h

Now, we give an example to show that exponential penalty function method provides an efficient way to solve constrained multiobjective fractional programming problem using well-known algorithms for unconstrained problems.

Example 4.1. Let us consider the multiobjective fractional programming problem: ðMFPÞ min FðxÞ ¼ f1ðxÞ g1ðxÞ ;f2ðxÞ g2ðxÞ   subject to h1ðxÞ 6 0; h2ðxÞ 6 0;

where f_i; g_i; hi: R2#R; i¼ 1; 2 are defined as

f1ðxÞ ¼ 4x21þ 2x2þ 5 f2ðxÞ ¼ x21þ 2x2þ 6 g₁ðxÞ ¼ x2 1þ x2þ 2:5 g2ðxÞ ¼ x 2 1þ 2 h1ðxÞ ¼ x21 1 h2ðxÞ ¼ x22 1:

The set of all feasible solutions to (MFP) is given by D¼ fðx1; x2Þ 2 R2:1 6 x161^ 1 6 x261g:

Clearly, f_iðxÞ P 0; giðxÞ > 0; 8x 2 D and i ¼ 1; 2.

We transform the multiobjective fractional programming problem (MFP) to the following equivalent nonfractional form using the parametric approach.

ðMFPÞ_v minð4x2 1þ 2x2þ 5  v1ðx21þ x2þ 2:5Þ; x2 1þ 2x2þ 6  v2ðx21þ 2ÞÞ subject to x2 1 1 6 0; x2 2 1 6 0;

where v¼ ðv1; v2Þ 2 R2þ.

Now, we construct the unconstrained exponential penalized multiobjective programming problem associated to (MFP) as follows: ðMFPÞ_n min Fðx; qnÞ ¼ 4x 2 1þ 2x2þ 5  v1ðx21þ x2þ 2:5Þ  þ 1 q_n½ðe qnðx211Þ 1Þþeqnðx221Þ 1Þ; x2 1þ 2x2þ 6  v2ðx21þ 2Þ þ 1 q_n½ðe qnðx211Þ 1Þ þ eqnðx221Þ 1Þ 

We take v1¼ v2¼ 2 and qn¼ n. Then limn!1qn¼ þ1 and

(MFP)nreduces to min Fðx;qnÞ ¼ 2x21þ 1 n½ðe nðx2 11Þ 1Þ  þ enðx2 21Þ 1Þ; x2 1þ 2x2þ 2 þ 1 n½ðe nðx2 11Þ 1Þ þ enðx 2 21Þ 1Þ 

Using MATLAB1routine function fminunc for unconstrained optimization, we observe thatð0; 1Þ is a weakly efficient solu-tion for each of the unconstrained problem (MFP)n,

n¼ 1; 2; . . .. Thus, we obtain a convergent subsequence fð0; 1Þg of the weakly efficient solutions of unconstrained exponential penalized multiobjective programming problem (MFP)n with limit point ð0; 1Þ which is also feasible to

(MFP). Hence, byTheorem 4.6,ð0; 1Þ is also a weakly effi-cient solution to (MFP), which can be easily verified.

5. Conclusion

In this paper, we employed the exponential penalty function method for multiobjective fractional programming problem and established its convergence. Thus, we conclude that the method of exponential penalty function, used by Liu and Feng

[8]for multiobjective programming problems, is also valid for multiobjective fractional programming problems. Hence the results obtained in this paper provide a new method to solve multiobjective fractional programming problem which does not require convexity or differentiability of the functions involved. In this way, using the computational methods for unconstrained optimization, we can also solve complicated multiobjective fractional programming problems subject to multiple constraints, arising in different practical vector opti-mization problems, more effectively.

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Anurag Jayswal received his M.A. and Ph.D. from Banaras Hindu University, Varanasi-221005, U.P., India. He is currently an Assistant Professor at Indian School of Mines, Dhanbad, India since 2010. His research interests include mathematical programming,

nonsmooth analysis and generalized

convexity.

Sarita Choudhury received her M.Sc. in Mathematics in 2012 from Kolhan University, Chaibasa, Jharkhand, India. She is currently a Ph.D. student at Indian School of Mines, Dhanbad, India under the guidance of Dr. Anurag Jayswal.

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