Applied Mathematics, 2010, 1, 534-541
doi:10.4236/am.2010.16071 Published Online December 2010 (http://www.SciRP.org/journal/am)
Second-Order Duality for Continuous Programming
Containing Support Functions
Iqbal Husain1, Mashoob Masoodi2
1Department of Mat hem at ic s, Jaypee University of Technology, Guna, (M.P), India
2Department of Stat i st i c s, University of Kashmir, Hazratbal Sri na g ar , Kashmir, India
Received June 25, 2010; revised November 4, 2010; acce pte d November 9, 2010
Abstract
A second-order dual problem is formulated for a class of continuous programming problem in which both
objective and constrained functions contain support functions, hence it is nondifferentiable. Under second-
order invexity and second-order pseudoinvexity, weak, strong and converse duality theorems are established
for this pair of dual problems. Special cases are deduced and a pair of dual continuous problems with natural
boundary values is constructed. A close relationship between duality results of our problems and those of the
corresponding (static) nonlinear programming problem with support functions is briefly outlined.
Keywords: Continuous Programming, Second-Order Invexity, Second-Order Pseudoinvexity, Second-Order
Duality, Nonlinear Programming, Support Functions, Natural Boundary Values
1. Introduction
Second-order duality in mathematical programming has
been extensively investigated in the literature. A second-
order dual formulation for a non-linear programming
problem was introduced by Mangasarian [1]. Later Mond
[2] established various duality theorems under a condi-
tion which is called “Second order convexity”. This con-
dition is much simpler than that used by Mangasarian [1].
In [3], Mond and Weir reconstructed the second-order
and higher order dual models to derive usual duality re-
sults. It is remarked here that second-order dual to a ma-
thematical programming problem presents a tighter
bound and because of which it enjoys computational
advantage over a first order dual.
Duality and optimality for continuous programming
have been widely investigated by many authors in the
recent past, notably, Mond and Hanson [4], Bector,
Chandra and Husain [5], Mond and Husain [6] and Chen
[7] and other cited references in these expositions.
Chen [7] was the first to identify second-order dual
formulated for a constrained variational problem and
established various duality results under an involved in-
vexity- like assumptions. Husain et al. [8] have presented
Mond-Weir type duality for the problem of [7] and by
introducing continuous-time version of second-order
invexity and generalized second-order invexity, validated
various duality results. Recently, Husain and Masoodi [9]
have studied Wolfe type duality for a class of nondiffe-
rentiable continuous programming problem and estab-
lished relationship between these results and the duality
results of Husain et al. [10] for nonlinear programming
problems with support functions.
In this paper, we formulate a Wolfe type second-order
dual to a class of nondifferentiability continuous pro-
gramming containing support functions. The popularity
of this type of problems seems to originate from the fact
that, even though the objective function and or/constraint
functions are non-smooth, a simple representation of the
dual problem may be found. The theory of non-smooth
mathematical programming deals with more general type
of functions by means of generalized sub-differentials.
However, square root of positive semi-definite quadratic
form and support functions are amongst few cases of the
nondifferentiable functions for which one can write
down the sub-or quasi-differentials explicitly. Here, var-
ious duality theorems for this pair of Wolfe type dual
problems are validated under second-order invexity and
second-order pseudoinvexity conditions. The special
cases as in [1] are derived. A pair of Wolfe type dual
variational problems with natural boundary values rather
than fixed end points is presented and the proofs of its
duality results are indicated. It is also shown that our
second-order duality results can be considered as dy-
namic generalizations of corresponding (Static) second-
order duality results established for nonlinear program-

I. HUSAIN ET AL.
535
ming problem with support function, considered by Hu-
sain et al. [10].
2. Pre-requisites and Expression of the
Problem
Let


=,Iab
be a real interval, :nn
IR RR  
:nn m andI RRR   be twice continuously diffe-
rentiable functions. In order to consider
  
,,txt xt 
where :n
x
IRis differentiable with
derivative
x
, denoted by
x
x
and

, the first order of

with respect to




x
tandxt
, respectively, that is,
12 12
,,,,, ,,
TT
xx
nn
xx xxx x
  

   


  



 
Denote by
x
x

the Hessian matrix of

, and
x

the mn matrix respectively, that is, with respect to


x
t, that is,
2
,,1,2, ,
xx ij ij n
xx








 ,
x

the
mn matrix
11 1
11
22 2
12
12
n
n
x
mm m
nmn
xxx
xx x
xx x
 
 

 


 

 


 

 



 


  


 

The symbols ,,
x
xx xxx
and
 
 
have analogous
representations.
Designate by X the space of piecewise smooth func-
tions :n
x
IR, with the norm
x
xDx

 ,
where the differentiation operator D is given
by
 
t
a
uDx xtusds 
,
Thus dD
dt  except at discontinuities.
We incorporate the following definitions which are
required in the subsequent analysis.
Definition 2.1 (Second-Order Invex): If there exists a
vector function


,, n txx R  
where :nn n IR RR   and with 0

 at t = a and t
= b, such that for a scalar function

,,txx 
, the func-
tional


,,
I
txxdt 
 where :nn
IR RR  
satisfies

 
 





1
,,,, 2
,, ,,,
T
II
T
TT
xx
I
txxdttxxptGptdt
txxDtxxGpt dt

 

 

 





then


,,
I
txxdt 
 is second-order invex with respect to

where2
2
x
xxxxx
GDD

  

, and


,n
p
CIR,
the space of n-dimensional continuous vector functions.
Definition 2.2 (Second-Order Pseudoinvex): The
functional


,,
I
txxdt 
 is said to be second-order
pseudoinvex with respect to

if
 





 
0
1
,, ,,.
2
T
TT
xx
I
T
II
DGptdt
txxdttxxptGptdt
 

 

 






Definition 2.3 (Second-Order Quasi-Invex): The
functional


,,
I
txxdt 
 is to be second-order qua-
si-invex with respect to

if




 
 

1
,,,, 2
0.
T
II
T
TT
xx
I
txxdttxxptGpt dt
DGtptdt

 

 

 




Consider the following nondifferentiable continuous
programming problem (CP) with support functions of
Husain and Jabeen [11]:
(CP): Minimize
  


,, |
I
f
txxS xtKdt

Subject to




0
x
axb (2.1)






,,|0,1,2,,,
jj
g
txxSxtCjmtI

 
 (2.2)
where f and g are continuously differentiable and each
Cj ,(j=1,2,…,m) is a compact convex set in Rn. In [11],
Husain and Zamrooda derived the following optimality
conditions for the problem (CP):
Lemma 2.1 (Fritz-John Neccesary Optimality Condi-
tions): If the problem (CP) attains a minimum at
x
xX

, there exist rR

and piecewise smooth func-
tion :m
yI Rwith
 


12
,,,
m
yty tytyt,
:n zI Rand :,1,2,,
jn
wI Rjm, such that


 





1
,, ,,
,,,, ,
mjjj
xj
T
xx
r ftxxztytgtxxwt
Drf txxytg txxtI


 
 

 

 







 
1
,, 0,
mT
jj j
j
ytgtxx xtwttI






   
|,
T
x
tzt SxtKtI





|,1,2,,,
Tjj
x
twt SxtCjmtI



I. HUSAIN ET AL.
536
 
,,1,2,,,
jj
ztKwtCjmtI 




,0,rytt I




,0,ryttI
The minimum
x
of (CP) may be described as normal
if 1rso that the Fritz John optimality conditions re-
duce to Karush-Kuhn-Tucker optimality conditions. It
suffices for 1rthat Slater’s condition holds at
x
.
Now we review some well known facts about a sup-
port function for easy reference.
Let be a compact set inn R, then the support func-
tion of is defined by





max: ,
T
Sxtxt vtvttI 
A support function, being convex everywhere finite,
has a subdifferential in the sense of convex analysis i.e.,
there exist


,
n
ztR tI
, such that


()()() ()T SytS xtytxtzt
From [12], the subdifferential of




Sxtis given
by
    


,such that.
T
SxtzttIxtztSxt 
For any setn
A
R, the normal cone to A at a point


x
tA is defined by
  





() ()()0,
n
A
NxtytRytztxtztA 
It can be verified that for a compact convex set B,


() ()
B
ytN xtif and only if


() (),
T
SytBxtyt t I
3. Second Order Duality
The following problem is formulated as Wolfe type dual
for the problem (CP):
(CD): Maximize
 

1
,, ,,
m
TT T
jj j
j
I
f
tuuu tztytgtuuutwt

 



 
1
2
T
ptHt ptdt



Subject to




0uaub (3.1)
 



 
1
,, ,,
,, ,,0,
u
mT
jj j
uj
T
uu
ftuuztytgtuu wt
DftuuytgtuuH tpttI

 






(3.2)




,,,1,2,,
jj
ztKw tCtIjm (3.3)


0,ytt I (3.4)
where
 





2
,, ,,
2,, ,,
,, ,,
T
uuu u
T
uuu u
T
uuu u
Htftuuytg tuu
D ftuuytgtuu
Dftuuytgtuu



















If


0,pttI

, the above dual becomes the dual of
the problem studied in [11].
Theorem 3.1 (Weak duality): Let


x
tX

be a
feasible solution of (CP) and








1
,,, ,utytztw t
 




2,..., , m
wt wtpt be
feasible solution for (CD). If
for all feasible









 




12
,,,,,,,,
m
x
tutytztwtwt wtpt
and with respect to

=


,,txu 



( ),.,.T
I
ift ztdt

and





1
,.,. .
mjj j
jI
ytgtwtdt


second-order invex .
Or


1
(),.,..,.,. .
m
TT
jj j
j
I
iiftz tytgtwtdt



 





is second-order pseudoinvex then.
inf (CP) ≥ sup (CD).
Proof:
  

 

 



1
1
,, |,,,,2
m
TTT T
jj j
j
II
f
txxS xtKdtftuuutztytgtuuutwtptHtptdt


 



 
 


 





 
1
1
,,,,,, 2
m
TT TT
jj j
j
II II
f
txxxtztdtftuu utztdtytgtuuutwtdtptHtptdt

  

 
 
 


 

 
1
,, ,,,,
1,
2
m
TTTT
TT jjj
uu j
III
T
I
f
tuuztDftuuFt ptdtptFt ptdtytgtuuutwtdt
ptHt ptdt



 







 

I. HUSAIN ET AL. 537
(using


2
2
x
xxx xx
F
tf DfDf 

and the second-order invexity of


,.,..)
T
I
f
tztdt

     


 
1
,, ,,,,,,
11
22
m
tb TT
TTjjj
uu u
ta j
I I
TT
II
f
tuuztDftuuF tptdtftuuytgtuuutwtdt
ptFtptdtptHtptdt











 
 

 

 

  
1 1
,, ,,,,
11
22
u
m m
TT TT
Tjj jjj j
u
j j
II I
TT
II
y
tgtuuwt DytgtuuGtptdtytgtuuutwtdt
ptFt ptdtptHtptdt

 

 





 


 
 

  

  
1 1
,, ,,
11
22
u
m m
TTTTT
Tjj jjjj
u
j j
II I
TT
II
ytgtuuwtDytgGt ptGt ptdtytgtuuutwtdt
ptFt ptdtptHt ptdt

 

 




 



 

   
1
111
,, 222
mTT TTT
jj j
jIIII
ytgtxxutwtdtptGt ptdtptFtptdtptHt ptdt

 


  


1
,,| 0
mT
jj j
jI
ytgtxx SxtCdt

 

This implies,
  

 

 



1
1
,, |,,,,2
m
TTT
jj j
j
II
f
txxSxtKdtftuuutztytgtxxutwtptHtptdt






 
yielding,
inf (CP) ≥ sup (CD).
(ii) From (3.2), we have
   





  
1
0,,,,,,,,
mTT T
Tjjj
uuu
j
I
f
tuuztytgtuuutwtDftuuytgtuuH tptdt 

 
 



 
   



 


1
,, ,,
,, ,,
u
mT
Tjjj
uj
I
tb
TT
TT
uu uu
ta
f tuuztytgtuuwt
DftuuytgtuuH tptdtfyg














 


(by integrating by parts) Using boundary conditions (2.1) and (3.1), we have
   






1
,,,,,, ,,0
u
mTT
Tjjj T
uuu
j
I
f tuuztytgtuuwtDf tuuytg tuuHtptdt 


 
 

 
This, in view of second-order pseudo-invexity of
  

1
,.,., ,
m
TT
jj j
j
I
f
tztytgtwtdt


 




yields,
  

  

 
1
1
,, ,,
1
,,,, 2
m
TT
jj j
j
I
m
TTT
jj j
j
I
f txxxtztytgtxxxtwtdt
f
tuuutztytgtuuutwtptHtptdt



 













I. HUSAIN ET AL.
538
    
 
  


1
1
,, |,, |
1
,,,, 2
mjj j
j
I
m
TTT T
jj j
j
I
ftxxS xtKytgtxxS xtCdt
f
tuuutztytgtuuutwtptHtptdt



 


 
 
 
 






Using (2.2) and (3.4) together with
 


|
T
x
tzt SxtK
and
 


|, ,1,2,,
Tjj
x
twtSxtC tIjm
This gives,
 



 

 
1
,, |
1
,,,, 2
I
m
TTT
jj j
j
I
ftxxSxt Kdt
f
tuuutztytgtuuutwtptH tptdt











That is, inf (CP) ≥ sup (CD).
Theorem 3.2 (Strong Duality): If


x
tXis a local
(or global) optimal solution of (CP) and is also normal,
then there exist piece wise smooth factor:m yI R,
:n zI Rand :(1,2,,)
jn
wI Rjm such that
 

12
,,, ,,,,0
m
xtyt zt wt wtwtpt is a
feasible solution of (CD) and the two objective values
are equal. Furthermore, If the hypotheses of Theorem 3.1
hold, the














12
,,, ,,,,()
m
x
tytztwtwt wtpt
is an optimal solution of (CD).
Proof: From Lemma 2.1, there exist piecewise smooth
functions :m yI R,:n zI R
and :(1,2,,)
jn
wI Rjmsatisfying

 
 




,, ,,
,,,, 0,
mT
jj j
xx
ji
T
xx
f
txxztytg txxw
D ftxxytgtxxtI

 
 





 
 
  

,, ,,
,,,, 0,
mT
jj j
xx
ji
T
xx
f
txxztytg txxw
D ftxxytgtxxtI

 
 




 

,, 0,
mT
jj j
x
ji
ytgtxx wtI



 


|,
T
x
tzt SxtKtI


  


|,1,2,,,
Tjj
x
twt SxtCjmtI






,,1,2,,,
jj
ztKw tCjmtI

 


0,ytt I
Hence












12
,,, ,,,,0
m
xtyt zt wt wtwtpt

 sa-
tisfies the constraints of (CD) and

  

 



 

1
1
,,,, 2
,,
m
TTT T
jj j
j
I
I
f
txxutztytgtxxxtwtptHtptdt
ftxxSxt Kdt

 

 
 






(3.5)
That is, the objective values are equal. Furthermore, for every feasible solution, we have

  

 


1
1
,,,, 2
m
TTT T
jj j
j
I
f
txxutztytgtxxxtwtptHtptdt

 

 
 


  

 
1
1
,,,, 2
m
TTT T
jj j
j
I
f
tuuutztytgtuuutwtptH tptdt

 
 
 
 


So,














12
,,,,,,
m
x
tytztwtwtwt is op-
timal for the problem (CD).
Theorem 3.3 (Converse duality): Let f and g are
thrice continuously differentiable
and




12
,,,,,,,
m
x
tytztwtwt wtpt be
an optimal solution of (CD). If the following conditions
hold:
(A1): The Hessian matrix H(t) is non-singular, and
(A2):



 

TT
x
x
tHt tDtHt t   

I. HUSAIN ET AL.
539
 

20,
x
tDHttt I  



0,ttI  
Then


x
t is feasible solution of (CP), and





1
,, 0,
mT
jj j
j
ytgtxx xtwttI


. In addi-
tion, if the hypotheses in Theorem 3.1 hold, then


x
t is
an optimal solution of the problem (CP).
Proof: Since







 




12
,,,,,,,
m
x
tytztwtwt wtpt
is an optimal solution for (CD), then there exist piece
wise smooth :n
I
R   and:m
IR  such that fol-
lowing Fritz John type optimality conditions [7] are sa-
tisfied:

 



 




 










 











1
1
,,,, 2
1
,, ,,,,
2
,, ,,
,, 0,
mT
jj
xx x
j
TTT T
xxx x
x
x
TT
xx xxx x
x
xx
T
xxx x
x
ftxxztytgtxxwtptHtpt
Dftxxyt gtxxptHtpttftxxyt g
Dftxxyt gHtptDftxxyt g
DftxxytgHtptt I




 





 




 
 


 
 


 


(3.6)

 

 

 
2
1
,,20,,1,2,,
2xxxxxx xx
TT
jjjjjjj
g
txxxtwtptg pttgDgDgptttIIm 

 



(3.7)

 


 



 

1
,,,,,, ,,0,
mT
jj j
xxx
j
f
txxztytgtxxwtD ftxxytgtxxHt pttI


 




(3.8)
 


T
K
x
ttNzt   (3.9)
 

,1,2,,
j
Tjj j
C
x
tyttyt Nwt jm   
(3.10)
   
0,
T
tptHt tI  
(3.11)
 
0,
T
tyt tI  
(3.12)

,() 0,ttI   (3.13)


,(),() 0tt tI 

 (3.14)
By the singularity of H(t), (3.11) implies,
()()0,tpt tI



 (3.15)
If 0


, then


0,ttI  
and so


0,t 

tI

. This contradicts (3.14),
Hence 0


 

 

 


 




 








 


1
11
22
,, 0,
x
mTTT
jjj
xxx
x
x
j
TT T
xxx xxxx
xx
TT
xxx xxxx
xx
fztytgwtptHtptDfytgptBtpt
tf ytgDfytgHtpt
D fytgD ftxxytgHt pttI



 


 
  
 





(3.16)
Using the expression of H (t) and (3.16), this gives
 


  
20,
TT
x
T
x
p
tHt ptDptHt pt
ptDHt pttI




This, in view of the hypothesis (A2) implies, yields,


0,ptt I (3.17)
The relations (3.9) and (3.10) imply
 


T
K
x
tNzt
 

,1,2,,
j
Tj
C
and xtNwtjm
which respectively yields,
  


|,
T
x
tzt SxtK
tI

and
  


|,1,2,,,
Tjj
x
twt SxtCjmtI


The relation (3.7) with


0,ptt I and (3.12)
gives


 


1
,, 0,
mjj j
j
ytgtxx xtwttI



 (3.11)
The relation (3.7) with

0, ,pttI


0,
jt
 

I. HUSAIN ET AL.
540
tI and
  
|,,
T
x
tzt SxtKtI
yields



,,|0,1,2,, ,
jj
g
txxS xtCjmtI

That is,
x
is feasible to (CP).
Now, in view of  


|,
T
x
tzt SxtKtI
and
(3.11) and

 

 






1
1
,,,, 2
,, |
m
TTT T
jj j
j
I
I
f
txxxtztytgtxxxtwtptHtptdt
ftxxSxt Kdt

 

 
 






This, along with the hypotheses of Theorem 3.1, yields
that


x
tis an optimal solution of (CP).
4. Special Cases
Let for


,tIBt positive semi-definite matrices and
continuous on I. Then
 

 
12|,
T
x
tBtxt SxtKtI
where
 


1,
T
K
Btzt ztBtzttI
Replacing




|Sxt K by
 

12
T
xt Btxt and
suppressing each




|j Sxt C, j=1,2,…,m from the
constraints of (CD), we have following problems treated
by Husain and Masoodi [9]
(CP2): Minimize
 


12
,, T
I
f
txxxtBtxtdt







Subject to




0
x
axb


,, 0,
g
txxt I

(CD): Maximize


  
1
,,,, 2
TT T
I
f
tuuut Btztyt gtuuptHtptdt

 


Subject to




0ua ub
 



,,,,,,,,0,
TTT
uu u
f
tuuutBtztytgtuuDf tuuytg tuuHtpttI 


 
1,0 .
T
ztBtztt Iyt


5. Problems with Natural Boundary
Conditions
In this section, we formulate a pair of nondifferentiable
dual variational problems with natural boundary values
rather than fixed end points. The proofs for duality theo-
rems for this pair of dual problems is omitted as they
follow immediately on the basis of analysis of the pre-
ceding section and, of course, slight modifications are
needed on the lines of [12]. The problems are:
(CP0): Minimize
  



,, |
I
f
txxSxtKdt

Subject to
 

,,|0,,1,2, ,
j
g
txxSxtCtIjm

(CD0): Maximize
   

1
,, ,,
m
TT T
jjj
j
I
f
txxxtztytgtxxxtw t

 



 
1
2
T
ptHt ptdt



Subject to




  
1
,, ,,
,, ,,0,
mjj j
xx
j
T
xx
ftxxztytgtxxwt
D ftxxytgtxxHt pttI

 










,,1,2,,,
jj
ztKwtCjmtI

 


0,ytt I

,,,, 0,
T
xx
ta
ftxxyt gtxx





,,,,0,
T
xx
tb
ftxxyt gtxx




6. Nonlinear Programming Problems
If all functions in the problems (CP0) and (CD0) are in-
dependent of t, then these problems will reduce to the

I. HUSAIN ET AL.
541
following nonlinear programming problems studied by
Husain et al. [10].
(CP1): Minimize
 


|
f
xSxtK
Subject to
 

|0,1,2,,
jj
g
xSxtCjm
(CD1): Maximize
 

1
1
2
mT
TjjTjT
j
f
uuztyt guuwtpHp

 

Subject to
 

1
0'
,,1,2,,.,
mT
jj j
u
j
jj
fuzty tguw tHp
zKw Cjm

 



where




.
T
uu uu
Hfuygu
7. References
[1] O. L. Mangasarian, “Second and Higher Order Duality in
Non linear Programming,” Journal of Mathematical
Analysis and Applications, Vol. 51, 1979, pp. 605-620.
[2] B. Mond,“Second Order Duality in Non-Linear Pro-
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[3] B. Mond and T. Weir, “Generalized Convexity and
Higher Order Duality,” Journal of Mathematical Analysis
and Applications, Vol. 46, 1974, pp. 169-174.
[4] B. Mond and M. A. Hanson, “Duality for Variational Pro-
blems,” Journal of Mathematical Analysis and Applica-
cations, Vol. 11, 1965, pp. 355-364.
[5] C. R. Bector, S. Chandra and I. Husain, “Generalized
Concavity and Duality in Continuous Programming,”
Utilitas Mathematica, Vol. 25, 1984.
[6] B. Mond and I. Husain, “Sufficient Optimality Criteria
and Duality for Avariational Problem with Generalized
Invexity,” The Journal of the Australian Mathematical
Society. Series B. Applied Mathematics, Vol. 31, No. 1,
1989, pp. 101-121.
[7] X. Chen, “Second-Order Duality for the Variational
Problems,” Journal of Mathematical Analysis and Appli-
cations, Vol. 216, 2003, pp. 261-270.
[8] I. Husain, A. Ahmed and M. Masoodi, “Second Order
Duality for Variational Problems,” European Journal of
Pure and Applied Mathematics, Vol. 2, No. 2, 2009, pp.
271- 295.
[9] I. Husain and M. Masoodi, “Second-Order Duality for a
Class of Nondifferentiable Continuous Programming
Problems,” European Journal of Pure and Applied Ma-
thematics, 2010, (In Press).
[10] I. Husain, A. Ahmed and M. Masoodi, “Second-Order
Duality in Mathematical Programming with Support
Functions,” Journal of Informatics and Mathematical
Sciences, Vol. 1, No. 2-3, 2009, pp. 165-182.
[11] I. Husain and Z. Jabeen, “Continuous Programming Con-
taining Support Functions,” Journal of Applied Mathe-
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75-106.
[12] B. Mond and M. Schechter, “Nondifferentiable Symme-
tric Duality,” Bulletin of the Australian Mathematical So-
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