This document summarizes key duality theorems for optimization problems involving n-set functions: 1) The weak duality theorem shows that the objective value of any feasible dual problem solution provides a lower bound on the objective of any primal problem solution. 2) The strong duality theorem establishes that under suitable convexity conditions, the optimal objective values of the primal and dual problems are equal. 3) Mangasarian's strict converse duality theorem proves that if a solution satisfies the Kuhn-Tucker conditions and strong duality holds, then the objective function is strictly convex at that solution.

1. 79
International Journal of Engineering Issues
Vol. 2015, no. 2, pp. 79-84
ISSN: 2458-651X
Copyright © Infinity Sciences
Weak and Strong Duality on Optimization
Problems with n-set Functions
Wasantha Daundasekera
Department of Mathematics, Faculty of Science, University of Peradeniya, Sri Lanka
Email: wbd@pdn.ac.lk.
Abstract - In the present paper we develop some duality results for mathematical programming problems with
differentiable convex n - set functions. Our main objective is to prove, the (i) weak duality theorem, (ii) strong duality
theorem, and (iii) Mangasarian’s strict converse duality theorem for a minimization problem with differentiable convex
n - set functions and its dual problem.
Keywords: Duality, Differentiable, Convex, n – set functions, Objective, Minimization
I. INTRODUCTION
Our main focus in this paper is to develop the duality theory for nonlinear programming involving n - set
functions. Some early work has been done on duality theory by Corley in [1] who developed the general theory of n -
set functions. Corely obtained saddle point optimality conditions and also Lagrangian duality results for the problem
(MP) given below. Ref. [2], Zalmai presented several duality results under generalized  -convexity assumptions for the
same (MP) problem.
Ref. [3], Bector, Bhatia, and Pandey considered a class of multiobjective programming problems with
differentiable n - set functions and established duality results and later in [4], Bector, Bhatia, and Pandey obtained
duality results for a nonlinear miltiobjective fractional programming problem.
In this paper our aim is to develop weak duality, strong duality, and Mangasarian strict converse duality results for
the minimization problem (MP) with differentiable convex n-set functions.
Let n
A be a family and let F and G be, respectively, convex n - set functions and convex m - dimensional n -
set functions, both defined on n
A .
The (primal) minimization problem (MP) is defined as follows:
minimize F ),...( 1 nRR
subject to (MP)
,
),...,( 1
n
n SRR  ,
where
},...,1,0),...(:),...,{( 11 mjRRGARRS nj
n
n
n
 .

2. W. Daundasekera / International Journal of Engineering Issues
80
The (dual) maximization problem (DP) of the (MP) is defined as follows:
maximize F 

m
j
njjn SSGuSS
1
11 ),...(),...( ,
subject to


m
j
iSiRsj
i
js
i
guf
1
,   0, for all ,ARi  ,,...1 ni  (DP)
),...( 1 nSS  n
A ,
ju  0, mj ,...,1 ,
s
i
f and jS
i
g respectively the i th partial derivatives of F and jG at ),...( 1 nSS . Denote by  the set of all
feasible solutions ),...;,...( 11 mn uuSS which satisfy the constraints of (DP).
II. RESULTS AND DISCUSSION
Theorem 1 below, referred as the weak duality theorem, shows that the objective function value of any feasible
solution to the dual problem yields a lower bound on the objective function value of any feasible solution to the primal
problem.
Weak duality theorem for n-set functions:
Theorem 1. IfAonabledifferentibeGandFletandsubfamilyabeALet nn
.
thenRRatconvexareGandFanduuRRSSS nmn
n
n ),,...(,),...,;,...,(,),...( 1111 


m
j
njjnn RRGuRRFSSF
1
111 ).,...,(),...,(),...,(
Proof. Since F is convex and differentiable, by Theorem 4.5 in [1], we have


n
i
RiSR
i
nn i
fRRFSSF
1
11 ,),...(),...,( 
  

n
i
m
j
RiSjR
i
jn i
guRRF
1 1
1 ,),...( 
( since ),...;,...,( 11 mn uuRR )


m
j
njnjjn SSGRRGuRRF
1
111 )),...,(),...,((),...,(
( by Theorem 4.5 in [1] )


m
j
njjn RRGuRRF
1
11 ).,...,(),...,(
(since ),...,1,0),...,( 1 mjSSGu njj  .
This concludes the proof of the theorem.

3. W. Daundasekera / International Journal of Engineering Issues
81
Theorem 2 below, referred as the strong duality theorem, shows that under suitable convexity assumptions, the
optimal objective function values of the primal and dual problems are equal. This theorem is considered to be one of the
more important duality theorems of nonlinear programming.
Strong duality theorem for n-set functions:
Theorem 2. functionssetnconvexabledifferentiGandFfamilyabeALet n
,
ker).(),...,(, 1 TucKuhnfollowingthesatisfiesGIfMPsolveSSletandAon n
n


thatsuchuuenonnegativexistthereisthatSSatconditions mn

,...,,),,...,( 11
niARallforguf i
iS
iRsj
i
m
j
js
i
,...,1,0,
1
 




0),...( 1
1




 nj
m
j
j SSGu
mjSSG nj ,...,1,0),...,( 1 

andDPsolvesuuSSthenSS
potheatGandFofsderivativepartialthithelyrespectivearegandfwhere
mnn
jsj
i
s
i
)(),...;,...,(),,...,(
int
111


.),...,(),...,(),...,(
1
111 


m
j
njjnn SSGuSSFSSF
Proof. ),...,(),...,(),...,( 1
1
11 n
m
j
njjn SSFSSGuSSF



 
(since 0),...,( 1
1




 n
m
j
jj SSGu )


m
j
njn RRGRRF
1
11 ),...,(),...,(
(by Theorem 1)
for all ),...,;,...,( 11 mn uuRR .
Hence the proof is complete.
Before we state the next theorem, we define the strict convexity of an n -set function.
Definition 3. aonconvexstrictlybetosaidisRSFfunctionsetnAn n
 :
 ,1,0),,...,(
),...,(,),...,(),,...,(
1
2111


andSS
RRSSSRReachforifAofSsubfamilyconvex
n
n
nn
nn
),...,()1(),...,())(),...,((suplim 111 nnn
ll
l
SSFRRFVVF  

.,...,1,,)}({ nieachforSRwithassociatedSVsequenceMorrisanyfor iii
l
 
Before we represent the next duality theorem, we state the following lemma, which we need to prove the theorem.
Lemma 4. FIfAofSsubfamilyconvexaonabledifferentibeRSFLet nnn
.: 

4. W. Daundasekera / International Journal of Engineering Issues
82
n
nn SSSRRallforthenconvexstrictlyis ),...,(),,...,(, 11
,,),...,(),...,(
1
11 *

n
i
SR
i
nn ii
fSSFRRF 
).,...( 1* n
i
SSatFofderivativepartialthitheisfwhere
This lemma can be proved by using Definition 4.3 and the proof of Theorem 4.5 in [1].
Another important duality theorem is the converse of the strong duality theorem. In order to obtain such a theorem
we have to modify the hypothesis of the strong duality theorem. Theorem 5 below is such a theorem referred as a strict
converse duality theorem. The theorem was originally introduced and proved by [5] for real functions defined on n
R .
The following is an extension of that theorem for n -set functions.
Mangasarian’s strict converse duality theorem:
Theorem 5. LetAonconvexandabledifferentibeGandFfamilyabeALet nn
.,
ifandDPofsolutionaisuuSSIfSSatconditions
TucKuhnthesatisfiesGthatassumeandMPofsolutionabeSS
mnn
n
)(),...,;,...,().,...(
ker)(),...,(
111
1





m
j
njjn RRGuRRF
1
11 ),..,(),...(
andMPsolvesalso
SSisthatSSSSthenSSatconvexstrictlyis nnnn
),(
),...,(,),,...,(),...,(),,...,( 1111





m
j
njjnn SSGuSSFSSF
1
111 ).,...,(),...(),...,(
Definition 6. ),...,(),...,( 111 njj
m
jn RRGuRRF

 is strictly convex at ),...,( 1 nSS

if either F is strictly
convex at ),...,( 1 nSS

or if for some j , 0

ju and jG is strictly convex at ),...,( 1 nSS

.
Proof. For simplicity we again let


m
j
njjnmn RRGuRRFuuRR
1
1111 ),...,(),...,(),...,;,...,( .
We shall assume that ),...,(),...,( 11 nn SSSS 

and exhibit a contradiction.
Since ),...,( 1 nSS is a solution of (MP), and G satisfies Kuhn-tucker conditions
at ),...,( 1 nSS , it follows from the previous theorem that there exists a m
m Ruu ),...,( 1
such that ),...,,,...,( 11 mn uuSS

solves (DP).
Hence, ),,...,;,...,(max),...,;,...,(),...,,,...,( 111111 mnmnmn uuRRuuSSuuSS  

over ),...,;,...,( 11 mn uuRR and 

),...,,,...,( 11 mn uuSS .
Let us define  


m
j
Sj
i
jS
i
uS
i
guf 1
, , the th
i partial derivative at

5. W. Daundasekera / International Journal of Engineering Issues
83


),...,;,...,( 11 mn uuSS .
Because 

),...,;,...,( 11 mn uuSS , we have that,
0,
1
,
 

n
i SS
uS
i
i
i
 for .,..,1, niAS i 
Hence, by the strict convexity of ),...,;,...,( 11 mn uuRR

 at ),...,( 1 nSS

and by the Lemma 4 it follows that
0,),...,;,...,(),...,,,...,(
1
1111
,
 


n
i S
i
mnmn
i
iSuS
uuSSuuSS  .
As a consequence
),...,;,...,(),...,;,...,(),...,,,...,( 111111 mnmnmn uuSSuuSSuuSS

  .
That is,   


m
j
m
j
njjnjj SSGuSSGu
1 1
1 ),...,1(),...,( .
But 0),...,(1
1 

m
j
njj SSGu (Kuhn-Tucker condition), hence
0),...,(
1
1 
m
j
njj SSGu .
This contradicts the facts that 0

ju and 0),...,( 1 nj SSG for each .,...,1 mj 
Hence, ),...,(),...,( 11 nn SSSS 

.
It is also the case that
).,...,;,...,(
),...,,,...,(
),,...(),...,(),...,(
11
11
1
111
mn
mn
n
m
j
jjnn
uuSS
uuSS
SSGuSSFSSF




 


Therefore, 


m
j
njjnn SSGuSSFSSF
1
111 ).,...,(),...(),...,(
The proof of the theorem is now complete.
III. CONCLUSION
In this paper, we concentrated on duality theorems for n-set functions. We considered a minimization problem and
its dual problem and obtained weak duality, strong duality, and Mangasarian strict converse duality results. Here, we
assumed that n-set functions are convex and differentiable.
REFERENCES
[1] Corely, H.W., “Optimization Theory for n-set Functions,” J. of Optim. Theory and Appl., V.127 , 1987, pp 193-205.
[2] Salami G.J., “Optimality Conditions and Duality for Multiobjective Measurable Subset Selection Problems,”
Optimization, V.22 No.2, 1981, pp 221-238.

6. W. Daundasekera / International Journal of Engineering Issues
84
[3] Bector C.R., Bathia D., and Pandey S., “Efficiency and Dualtiy for Nonlinear Multiobjective Programming Involving
n- Set Functions,” J. of Math Anal. and Appl. V.182, 1994, pp 486-500.
[4] Bector C.R., Bathia D., and Pandey S., “Dualtiy for Multiobjective Fractional Programming Involving n- Set
Functions,” J. of Math Anal. and Appl. V.186, 1994, pp 747-768.
[5] Mangasarian O.L., “Dualtiy in Nonlinear Programming,” Quarterly of Applied Mathematics, V.20, 1962, pp 300-
302.