Short Communication

Technical note on symmetric duality in multiobjectiveprogramming: Some remarks on recent results

Suresh Chandra *, Abha

Department of Mathematics, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India

Received 30 September 1997; accepted 3 February 1999

Abstract

Certain shortcomings are pointed out in some recent works of Das and Nanda [European Journal of Operational

Research 97 (1997) 167±171] and Kim et al. [European Journal of Operational Research 107 (1998) 686±691] which

show that these studies are highly restricted. Ó 2000 Elsevier Science B.V. All rights reserved.

Keywords: Symmetric duality; Multiobjective programs; Pseudo-invex functions; E�cient solutions

1. Introduction

Das and Nanda [6] in a recent paper attemptedto study symmetric duality in multiobjective pro-gramming on the lines of the formulation givenearlier for scalar nonlinear programming problemsby Dorn [7], Dantzig et al. [5], Bazarra and Goode[1] and Mond [11]. Multiobjective programmingsymmetric duality has earlier been studied bymany authors. Mond and Weir [15] presented twomodels for symmetric duality in multiobjectiveprogramming, one motivated by Dantzig et al. [5]and Mond [11] and the other motivated by Mondand Weir [12] for scalar programming problems.Later Weir [14] studied its fractional analogue andChandra and Durga Prasad [2] introduced a pair

of symmetric dual multiobjective programs by as-sociating a vector valued in®nite game. Very re-cently Kim et al. [9] studied multiobjectivesymmetric duality with cone constraints.

The purpose of this note is to point out certainshortcomings in the formulation and the proofs ofDas and Nanda [6]. Further it is observed thatresults of Kim et al. [9] are highly restricted as theyare not valid even for the convex case. The mainreason for this is the fact that Kim et al. [9] haveused some earlier results of Nanda and Das [13]which are very much restricted as has been pointedout in Kumar et al. [10], Chandra and Kumar [4]and Chandra and Abha [3].

2. Preliminaries

Let Rn denote the n-dimensional Euclideanspace and Rn

� be its non-negative orthant. The

European Journal of Operational Research 124 (2000) 651±654www.elsevier.com/locate/dsw

* Corresponding author. Tel.: +91-11-6861977; fax: +91-11-

6862037.

E-mail address: chandra@maths.iitd.ernet.in (S. Chandra).

0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 1 8 6 - 1

following conventions for vectors in Rn will beused:

x > y () xi > yi �i � 1; 2; . . . ; n�;x=y () xi=yi �i � 1; 2; . . . ; n�;x P y () xi=yi �i � 1; 2; . . . ; n� but x 6� y

x� y is the negation of x P y:

Let f �x; y� be real valued twice di�erentiablefunction de®ned on Rn � Rm. Let rxf ��x; �y� andryf ��x; �y� denote the partial derivatives of f �x; y�with respect to x and y, respectively, evaluated at��x; �y�. Also let r2

xf ��x; �y� denote the Hessian matrixwith respect to x evaluated at ��x; �y�. The symbolsrxyf ��x; �y� and ryxf ��x; �y� and r2

y f ��x; �y� are de®nedsimilarly. Now consider the following vectorminimization problem (VP).

Problem (VP)

Min F �x� � �f1�x�; f2�x�; . . . ; fp�x��s:t: h�x�50;

where F : Rn ! Rp, h : Rn ! Rm. Let us denote thefeasible region of (VP) by X.

De®nition 1. A feasible solution x° is said to be anefficient solution for the problem (VP) if there ex-ists no other feasible point x such thatF �x�6 F �x°�.

De®nition 2. A feasible point x° is said to beproperly efficient solution for problem (VP) if it isefficient for (VP) and there exists a scalar M > 0such that, for each i, we have

fi�x°� ÿ fi�x�fj�x� ÿ fj�x°� 5M

for some j such that fj�x� > fj�x°� whenever x 2 Xand fi�x� < fi�x°�.

Let C1 and C2 be closed convex cones with non-empty interiors in Rn and Rm, respectively. Let for�j � 1; 2� C�j denote the polar of the cone Cj andfor �i � 1; 2; . . . ; p� Ki : C1 � C2 ! R.

3. The symmetric dual formulation of Das and

Nanda

Das and Nanda [6] considered the followingpair of symmetric dual programs called (SPV) and(SDV):

Problem (SPV)

Min F1�x; y� � �K1�x; y� ÿ yTryK1�x; y�; . . . ;

Kp�x; y� ÿ yTryKp�x; y��s:t: ryKi�x; y� 2 C�2 �i � 1; 2; . . . ; p�;

�x; y� 2 C1 � C2:

Problem (SDV)

Max G1�u; v� � �K1�u; v� ÿ uTrxK1�u; v�; . . . ;

Kp�u; v� ÿ uTrxKp�u; v��s:t: ÿrxKi�u; v� 2 C�1 �i � 1; 2; . . . ; p�;

�u; v� 2 C1 � C2:

They also considered the following parametricprograms (SPVr) and (SDVr) corresponding to�SPV � and �SDV �, respectively.

Problem (SPVr)

Min rTF1�x; y�s:t: ryKi�x; y� 2 C�2 �i � 1; 2; . . . ; p�;

r > 0;Xp

i�1

ri � 1;

r 2 Rp; �x; y� 2 C1 � C2:

Problem (SDVr)

Max rTG1�u; v�s:t: ÿrxKi�u; v� 2 C�1 �i � 1; 2; . . . ; p�;

r > 0;Xp

i�1

ri � 1;

r 2 Rp; �u; v� 2 C1 � C2:

There are the following shortcomings in the con-struction of these pairs of symmetric dual pro-grams and the proofs of various duality theorems:1. For proving various results in [6], Lemma 2 is

very crucial. It has been remarked that Lemma2 of [6] follows from Theorem 2 of [8], but thisdoes not seem to be correct because Theorem 2

652 S. Chandra, Abha / European Journal of Operational Research 124 (2000) 651±654

of [8] connects an e�cient solution of (SPV)with the corresponding parametric problem�SPVr�. What the authors probably mean isCorollary 1 of the comprehensive theorem of[8]. But even this does not seem to be correct be-cause for using this result (and hence to provethe main duality theorem) the constraints in(SPV) should be the subset of K-K-T systemof the problem (SDV) and conversely, i.e. in-stead of ryKi�x; y�50 for all i � 1; 2; . . . ; p in�SPV � one should have ry

Ppi�1 kiKi�x; y�50.

Similar comments hold for (SDV) as well. Thisseemingly incorrect formulation of �SPV � and�SDV � makes Lemma 2, Theorems 3 and 4 of[6] some what invalid.

2. In Theorem 2, ��xÿ �u� 2 C1, on taking C1 � Rn�,

this will mean ��xÿ �u�=0 for all �x; �u 2 Rn�,

which is not always true. Also the vector��vÿ �y� cannot be written as ���x; �v� ÿ ��x; �y�� be-cause the former is in Rm where as the later isin Rn � Rm.

4. The symmetric dual formulation of Kim et al.

Kim et al. [9] recently studied multiobjectivesymmetric duality with cone constraints by for-mulating the following pair of problems:

(MSP)

Min f �x; y� ÿ �yTry�kTf ��x; y��es:t: ry�kTf ��x; y� 2 C�2 ;

�x; y� 2 C1 � C2;

k P 0; kTe � 1:

(MSD)

Max f �u; v� ÿ �uTrx�kTf ��u; v��es:t: ÿrx�kTf ��u; v� 2 C�1 ;

�u; v� 2 C1 � C2;

k P 0; kTe � 1:

Here f : C1 � C2 ! Rp is twice di�erentiablefunction, k 2 Rp and e � �1; 1; . . . ; 1�T 2 Rp. Alsorx�kf ��x; y� and ry�kf ��x; y� are gradients of�kTf ��x; y� with respect to x and y, respectively.

Unlike Das and Nanda [6] there is no di�cultywith this formulation and all constraints in (MSP)and (MSD) are correctly speci®ed. However, dueto certain assumptions made in the proofs ofvarious theorems, these results seem to be highlyrestricted. This is mainly because of the assump-tion on g : C2 � C2 ! C2. For the convex caseg�x; y� � �xÿ y� and therefore by taking C2 � Rn

�it implies that x=0, y=0 mean �xÿ y�=0, some-thing not always true. Because of this, even theconvex case does not follow as a special case of thisstudy.

This point has already been noted in [3,4,10],and these modi®cations can be incorpated in theresults of Kim et al. [9] and that will make thework of Das and Nanda [6] and Kim et al. [9]complete on all counts.

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654 S. Chandra, Abha / European Journal of Operational Research 124 (2000) 651±654