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Duality for a class of symmetric nondifferentiable multiobjective fractional variational problems with generalized (F,α,ρ,d) -convexity
Mathematical and Computer Modelling 57 (2013) 1453–1465
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Duality for a class of symmetric nondifferentiable multiobjective fractional variational problems with generalized (F , α, ρ, d)-convexity N. Kailey a , S.K. Gupta b,∗ a
Department of Applied Sciences, Gulzar Institute of Engineering and Technology, G.T. Road, Khanna-141 401, India
b
Department of Mathematics, Indian Institute of Technology Patna, Patna-800 013, India
article
info
Article history: Received 8 September 2011 Received in revised form 5 December 2012 Accepted 6 December 2012
abstract A pair of nondifferentiable symmetric dual problems for a class of multiobjective fractional variational programs over arbitrary cones is formulated. Weak, strong and converse duality relations are then proved under generalized (F , α, ρ, d)-convexity assumptions. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Symmetric duality Multiobjective nondifferentiable programming Fractional variational programming (F , α, ρ, d)-convexity Efficient solutions
1. Introduction Calculus of variation provides an excellent uniform analytical method to find that curve connecting two given points which either maximizes or minimizes some given integral; for example, to determine a curve which will generate the surface of revolution of smallest area when revolved about the x-axis. In general, we wish to find the curve x = x(t ) where x(a) = α1 and x(b) = α2 such that for some given known function f (t , x(t ), x˙ (t )) of variables t , x, x˙ , the integral b
f (t , x, x˙ )dt a
is either a maximum or minimum (also called extremum values). The curve which satisfies this property is said to be an extremal. The problem of finding a piecewise smooth extremal x = x(t ) for the above program is known as a variational problem. Hanson [1] surveyed and generalized the relationship between mathematical programming and classical calculus of variation. Optimality conditions and duality results were obtained for scalar valued variational problems by Mond and Hanson [2] under convexity. Bector et al. [3] extended symmetric duality to variational programming, giving continuous analogous of the results in [4,5]. The concept of (F , ρ)-convexity was introduced by Preda [6], an extension of F -convexity defined by Hanson and Mond [7] and ρ -convexity given by Vial [8]. The class of (F , ρ)-convex functions was extended to second order and duality results for Mangasarian, Mond–Weir and general Mond–Weir type multiobjective dual problems were obtained in [9].
∗
Corresponding author. Tel.: +91 612 2552025; fax: +91 612 2277384. E-mail addresses: [email protected] (N. Kailey), [email protected] (S.K. Gupta).
0895-7177/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2012.12.007
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N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
Multiple objective programming problems with the concept of weak minima are extended to multiple objective variational problems by Mukherjee and Mishra [10]. Mishra et al. [11] studied Mond–Weir type dual for a class of nondifferentiable multiobjective variational problems in which every component of the objective function contains a term involving the square root of a certain positive semidefinite quadratic form and usual duality theorems are developed for conditionally properly efficient solutions under V -invexity assumptions. Mukherjee and Rao [12] generalized the concept of mixed type duality to the class of multiobjective variational problems and proved duality results for ρ -convex functions. Ahmad and Gulati [13] considered mixed type dual for multiobjective variational problem and obtained duality relations by relating properly efficient solutions between the primal and mixed dual problems under generalized (F , ρ)-convexity assumptions. Ahmad and Husain [14] introduced second-order (F , α, ρ, d)-convex functions, their generalizations and developed usual duality theorems for second-order Mond–Weir type multiobjective primal–dual pair. Hachimi and Aghezzaf [15] extended the concepts of (F , α, ρ, d) type-I functions and established various sufficient optimality conditions and mixed duality results for multiobjective variational problems. Stancu-Minasian and Mititelu [16,17] considered multiobjective fractional variational problems with (ρ, b)quasiinvexity for which they established necessary conditions for normal efficient solutions. Based on this normal efficiency criteria, a Mond–Weir type dual is formulated and appropriate duality theorems are presented under (ρ, b)-quasiinvexity on the functions involved. The appropriate duality results are proved for a class of multiobjective fractional variational problems through a parametric approach by Mishra and Mukherjee [18]. Mishra et al. [19] formulated a symmetric dual pair for a class of nondifferentiable multiobjective fractional variational problems and duality theorems are obtained under invexity assumptions. Later on, Ahmad and Sharma [20] formulated a pair of multiobjective fractional variational symmetric dual problems over arbitrary cones and obtained usual duality results under generalized F -convexity assumptions. Recently, Mishra et al. [21] focused on symmetric duality for a class of nondifferentiable fractional variational problems. They introduced a symmetric dual pair for a class of nondifferentiable vector fractional variational problems and established appropriate duality results under certain invexity assumptions. The results obtained in this paper extend their work over arbitrary cones and assumptions as (F , α, ρ, d)-convexity/pseudoconvexity. This paper is organized as follows: In Section 2, we give some definitions and preliminaries. In Section 3, we formulate a pair of nondifferentiable symmetric dual programs for a class of multiobjective fractional variational problems over arbitrary cones and prove weak, strong and converse duality theorems under (F , α, ρ, d)-convexity/pseudoconvexity assumptions. 2. Notations and preliminaries Let I = [a, b] be a real interval and C1 ⊂ Rn , C2 ⊂ Rm , be closed convex cones with nonempty interiors having positive polars C1∗ and C2∗ , respectively. Let, for each i ∈ L = {1, 2, . . . , l}, f i (t , x(t ), x˙ (t ), y(t ), y˙ (t )) and g i (t , x(t ), x˙ (t ), y(t ), y˙ (t )), where x : I → Rn and y : I → Rm , with derivatives x˙ and y˙ , are twice continuously differentiable functions. For i ∈ L, the symbols fxi , fx˙i , fyi and fy˙i denote gradient vectors of the scalar function f i (t , x, x˙ , y, y˙ ) with respect to x, x˙ , y and y˙ , respectively. We have fxi
=
∂f i ∂f i , . . . , ∂ x1 ∂ xn
T
,
fx˙i
=
∂f i ∂f i , . . . , ∂ x˙ 1 ∂ x˙ n
T
.
Similarly, gxi , gx˙i , gyi and gy˙i denote the gradient vectors of g i (t , x, x˙ , y, y˙ ) with respect to x, x˙ , y and y˙ , respectively. The following observations are used for proving the strong duality theorem (Theorem 3): Dfy˙i = fy˙it + fy˙iy y˙ + fy˙iy˙ y¨ + fy˙ix x˙ + fy˙ix˙ x¨ . Consequently,
∂ i Df = Dfy˙iy , ∂ y y˙ ∂ i Df = Dfy˙ix , ∂ x y˙
∂ i Df = Dfy˙iy˙ + fy˙iy , ∂ y˙ y˙ ∂ i Df = Dfy˙ix˙ + fy˙ix , ∂ x˙ y˙
∂ i Df = fy˙iy˙ , ∂ y¨ y˙ ∂ i Df = fy˙ix˙ , i ∈ L. ∂ x¨ y˙
Similarly, Dgy˙i can be defined. Let X (I , Rn ) denotes the space of piecewise smooth functions x with the norm
∥ x ∥=∥ x ∥∞ + ∥ Dx ∥∞ , where the differentiation operator D is given by u = Dx ⇔ x(t ) = α1 +
t
u(s)ds, a
where α1 is a given boundary value. Therefore, dtd ≡ D except at discontinuities. Denote by Y (I , Rm ), the space of piecewise smooth functions y : I → Rm with the norm as that of space X (I , Rn ).
N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
1455
Consider the following multiobjective variational problem: (FP) Minimize b
φ(t , x(t ), x˙ (t ))dt =
b
φ (t , x(t ), x˙ (t ))dt , . . . , 1
φ (t , x(t ), x˙ (t ))dt l
a
a
a
b
subject to x(a) = 0 = x(b), x˙ (a) = 0 = x˙ (b), h(t , x(t ), x˙ (t )) 5 0,
t ∈ I,
where φ : I × R × R → R and h : I × Rn × Rn → Rm are differentiable functions. n
n
l
Let S be the set of all feasible solutions of (FP), i.e., S = {x ∈ X (I , Rn ) | x(a) = 0 = x(b), x˙ (a) = 0 = x˙ (b), h(t , x(t ), x˙ (t )) 5 0, t ∈ I }. Definition 2.1. A point x¯ ∈ S is an efficient (or Pareto optimal) solution of (FP) if there exist no other x ∈ S such that b
φ(t , x(t ), x˙ (t ))dt ≤
b
a
φ(t , x¯ (t ), x˙¯ (t ))dt . a
Definition 2.2. A point x¯ ∈ S is a weak efficient solution of (FP) if there exist no other x ∈ S such that b
φ(t , x(t ), x˙ (t ))dt <
b
a
φ(t , x¯ (t ), x˙¯ (t ))dt . a
Definition 2.3. A functional F : I × Rn × Rn × Rn × Rn × Rn → R is said to be sublinear with respect to its sixth argument, if for all x, x˙ , u, u˙ ∈ X (I , Rn ), (i) F (t , x, x˙ , u, u˙ ; a1 + a2 ) 5 F (t , x, x˙ , u, u˙ ; a1 ) + F (t , x, x˙ , u, u˙ ; a2 ), ∀a1 , a2 ∈ Rn , (ii) F (t , x, x˙ , u, u˙ ; δ a) = δ F (t , x, x˙ , u, u˙ ; a), for all δ ∈ R+ and a ∈ Rn . Definition 2.4. Let C be a compact convex set in Rn . The support function of C is defined by s(x | C ) = max{xT y : y ∈ C }. A support function, being convex and everywhere finite, has a subdifferential, that is, there exists z ∈ Rn such that s(y | C ) = s(x | C ) + z T (y − x) for all y ∈ C . The subdifferential of s(x | C ) is given by
∂ s(x | C ) = {z ∈ C : z T x = s(x | C )}. For any set M ⊂ Rn , the normal cone to M at a point x ∈ M is defined by NM (x) = {y ∈ Rn : yT (z − x) 5 0 for all z ∈ M }. Also, for a compact convex set C , y ∈ NC (x) if and only if s(y | C ) = xT y. Definition 2.5. Let D be a closed convex cone in Rn with nonempty interior. The positive polar cone D∗ of D is defined as D∗ = {z : xT z = 0, for all x ∈ D}. Let F and G be sublinear functionals with respect to sixth argument and d = (d1 , d2 ), where d1 = (d11 , d12 , . . . , d1l ) : I × R ×Rn ×Rn ×Rn → Rl , d2 = (d21 , d22 , . . . , d2l ) : I ×Rm ×Rm ×Rm ×Rm → Rl . Let f = (f 1 , f 2 , . . . , f l ) : I ×Rn ×Rn ×Rm ×Rm → Rl be a differentiable function, α = (α1 , α2 ), where α1 : Rn × Rn → R+ \ {0}, α2 : Rm × Rm → R+ \ {0} and ρ = (ρ 1 , ρ 2 ), where ρ 1 = (ρ11 , ρ21 , . . . , ρl1 ), ρ 2 = (ρ12 , ρ22 , . . . , ρl2 ) ∈ Rl . n
Definition 2.6. For each i ∈ L, b
f i (t , x, x˙ , y, y˙ )dt − a
b a
f i (t , x, x˙ , y, y˙ )dt is said to be (F , α1 , ρi1 , d1i )-convex in x and x˙ for fixed y and y˙ , if
b
f i (t , u, u˙ , y, y˙ )dt = a
b
F (t , x, x˙ , y, y˙ ; α1 (x, u)(fxi (t , u, u˙ , y, y˙ ) a
− Dfx˙i (t , u, u˙ , y, y˙ )))dt + ρi1
b
d1i (t , x, x˙ , u, u˙ )
a
for all x, u : I → Rn and for some arbitrary sublinear functional F .
2
dt
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N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
Definition 2.7. For each i ∈ L,
b a
f i (t , x, x˙ , y, y˙ )dt is said to be (F , α1 , ρi1 , d1i )-pseudoconvex in x and x˙ for fixed y and y˙ , if
b
F (t , x, x˙ , y, y˙ ; α1 (x, u)(fxi (t , u, u˙ , y, y˙ ) − Dfx˙i (t , u, u˙ , y, y˙ )))dt = 0 a
⇒
b
f i (t , x, x˙ , y, y˙ )dt =
b
f i (t , u, u˙ , y, y˙ )dt + ρi1 a
a
b
2
d1i (t , x, x˙ , u, u˙ )
dt ,
a
for all x, u : I → Rn and for some arbitrary sublinear functional F . In the sequel, we will write F (t , x, u; ξ ) for F (t , x, x˙ , u, u˙ ; ξ ) and G(t , v, y; η) for G(t , v, v˙ , y, y˙ ; η). 3. Symmetric duality Duality in variational programming problems has been developed extensively in last few years. Mond and Hanson [2] presented a constrained variational problem as a mathematical programming problem and presented its Wolfe type dual for validating various duality results under convexity. Smart and Mond [22] extended the symmetric duality results to variational problems by using continuous version of invexity. Later, Bector et al. [3] studied duality theorems for variational programming problems with weaker convexity assumptions. Bector and Husain [23] extended the results appeared in [3,4] to multiobjective case and obtained duality relations for Wolfe and Mond–Weir type duals for multiobjective variational problems under convexity. Later on, Kim and Lee [24] formulated a pair of symmetric dual variational problems in the spirit of Mond and Weir [5] using pseudo-invexity. After that, Hachimi and Aghezzaf [15] proved mixed duality results for multiobjective variational programming problems using the concept of generalized (F , α, ρ, d)-type I functions. Recently, Ahmad and Sharma [20] formulated multiobjective fractional variational symmetric dual problems over arbitrary cones and established usual duality theorems under generalized F -convexity assumptions. Now, we consider the problem of finding functions x : I → Rn and y : I → Rm , where (˙x, y˙ ) is piecewise smooth on I, to solve the following pair of symmetric dual multiobjective nondifferentiable fractional variational problems over arbitrary cones: Primal problem (VP) Minimize
b a
{f (t , x, x˙ , y, y˙ ) + s(x | B) − yT z }dt
b
{g (t , x, x˙ , y, y˙ ) − s(x | E ) + yT r }dt b b l {f 1 (t , x, x˙ , y, y˙ ) + s(x | B1 ) − yT z 1 }dt {f (t , x, x˙ , y, y˙ ) + s(x | Bl ) − yT z l }dt a a = b ,..., b {g 1 (t , x, x˙ , y, y˙ ) − s(x | E1 ) + yT r 1 }dt {g l (t , x, x˙ , y, y˙ ) − s(x | El ) + yT r l }dt a a subject to x(a) = 0 = x(b), y(a) = 0 = y(b), x˙ (a) = 0 = x˙ (b), y˙ (a) = 0 = y˙ (b), l i F i (x, y) ∈ C2∗ , t ∈ I , − λi fy − Dfy˙i − z i − gyi − Dgy˙i + r i i G ( x , y ) i =1 l F i ( x, y ) yT λi fyi − Dfy˙i − z i − gyi − Dgy˙i + r i i = 0, t ∈ I , G (x, y) i=1 a
λ > 0, z i ∈ Di ,
x( t ) ∈ C 1 , r i ∈ Hi ,
t ∈ I, i = 1, 2, . . . , l.
Dual problem (VD) Maximize
b
{f (t , u, u˙ , v, v˙ ) − s(v | D) + uT w}dt a b {g (t , u, u˙ , v, v˙ ) + s(v | H ) − uT s}dt a b b l {f 1 (t , u, u˙ , v, v˙ ) − s(v | D1 ) + uT w 1 }dt {f (t , u, u˙ , v, v˙ ) − s(v | Dl ) + uT w l }dt a a = b ,..., b {g 1 (t , u, u˙ , v, v˙ ) + s(v | H1 ) − uT s1 }dt {g l (t , u, u˙ , v, v˙ ) + s(v | Hl ) − uT sl }dt a a subject to u(a) = 0 = u(b), v(a) = 0 = v(b), u˙ (a) = 0 = u˙ (b), v˙ (a) = 0 = v˙ (b),
N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465 l
fxi − Dfx˙i + w i − gxi − Dgx˙i − si
λi
M i (u, v)
i =1
u
l
T
λi
fxi
−
Dfx˙i
+w − i
gxi
−
Dgx˙i
−s
∈ C1∗ ,
Li (u, v)
i
M i (u, v)
5 0,
Li (u, v)
i=1
λ > 0, w i ∈ Bi ,
1457
t ∈ I, t ∈ I,
v(t ) ∈ C2 , t ∈ I , si ∈ Ei , i = 1, 2, . . . , l,
where f i : I × Rn × Rn × Rm × Rm → R and g i : I × Rn × Rn × Rm × Rm → R, i ∈ L, are continuously differentiable functions, and F i ( x, y ) =
b
f i (t , x, x˙ , y, y˙ ) + s(x | Bi ) − yT z i dt ;
a
Gi (x, y) =
b
g i (t , x, x˙ , y, y˙ ) − s(x | Ei ) + yT r i dt ;
a
M i (u, v) =
b
f i (t , u, u˙ , v, v˙ ) − s(v | Di ) + uT w i dt ;
a
and L (u, v) = i
b
g i (t , u, u˙ , v, v˙ ) + s(v | Hi ) − uT si dt .
a
Assume that Gi (x, y) > 0,
Li (u, v) > 0,
F i (x, y) = 0,
M i (u, v) = 0,
∀i ∈ L.
In order to simplify the notations, let pi =
F i (x, y) Gi (x, y)
b
a b a
=
f i (t , x, x˙ , y, y˙ ) + s(x | Bi ) − yT z i dt
g i (t , x, x˙ , y, y˙ ) − s(x | Ei ) + yT r i dt
and qi =
M i (u, v) Li (u, v)
b a
f i (t , u, u˙ , v, v˙ ) − s(v | Di ) + uT w i dt
a
g i (t , u, u˙ , v, v˙ ) + s(v | Hi ) − uT si dt
= b
and express problems (VP) and (VD) equivalently as follows: (FVP) Minimize p = (p1 , p2 , . . . , pl ) subject to x(a) = 0 = x(b),
y(a) = 0 = y(b),
x˙ (a) = 0 = x˙ (b),
y˙ (a) = 0 = y˙ (b),
(1) (2)
b
f i (t , x, x˙ , y, y˙ ) + s(x | Bi ) − yT z i dt − pi
a
−
b
g i (t , x, x˙ , y, y˙ ) − s(x | Ei ) + yT r i dt = 0,
i ∈ L,
(3)
a
l
λi
λi
fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i
∈ C2∗ ,
t ∈ I,
(4)
i =1
yT
l
fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i
= 0,
t ∈ I,
(5)
i=1
λ > 0,
x(t ) ∈ C1 ,
z ∈ Di ,
r ∈ Hi ,
i
i
(FVD) Maximize q = (q1 , q2 , . . . , ql )
t ∈ I, i = 1, 2, . . . , l.
(6) (7)
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N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
subject to u(a) = 0 = u(b),
v(a) = 0 = v(b), u˙ (a) = 0 = u˙ (b), v˙ (a) = 0 = v˙ (b), b i f (t , u, u˙ , v, v˙ ) − s(v | Di ) + uT w i dt − qi
(8) (9) b
g i (t , u, u˙ , v, v˙ ) + s(v | Hi ) − uT s dt = 0,
i
a
a
i ∈ L, l
λi
(10)
fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si
λi
∈ C1∗ ,
t ∈ I,
(11)
i =1 l
uT
fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si
5 0,
t ∈ I,
(12)
i =1
λ > 0,
v(t ) ∈ C2 ,
w ∈ Bi ,
s ∈ Ei ,
i
i
t ∈ I,
(13)
i = 1, 2, . . . , l.
(14)
In the above problems (FVP) and (FVD), it is to be noted that p and q are nonnegative. Let P and Q denote the sets of feasible solutions of (FVP) and (FVD), respectively. In the subsequent analysis, usual duality theorems are discussed in terms of (FVP) and (FVD), but equally applicable to (VP) and (VD). Now, we establish weak, strong and converse duality theorems for the problems (FVP) and (FVD). Theorem 3.1 (Weak Duality). Let (x, y, p, λ, z , r ) ∈ P and (u, v, q, λ, w, s) ∈ Q . Let the sublinear functionals F : I ×Rn ×Rn ×Rn → R and G : I × Rm × Rm × Rm → R satisfy the following conditions: F (t , x, u; a) + α1−1 aT u = 0,
for all a ∈ C1∗ , t ∈ I
G (t , v, y; b) + α2 b y = 0,
for all b ∈ C2 , t ∈ I .
−1 T
(A) (B)
∗
Suppose that
λi
b
λi
(iii) either ρ
1 b
(i)
l
(ii)
i=1 l i=1
a b a
a
f i (t , ·, ·, v, v˙ ) + (·)T w i − qi g i (t , ·, ·, v, v˙ ) − (·)T si
f i (t , x, x˙ , ·, ·) − (·)T z
dt is (F , α1 , ρ 1 , d1 )-pseudoconvex in x and x˙ ,
− pi g i (t , x, x˙ , ·, ·) + (·)T r dt is (G, α2 , ρ 2 , d2 )-pseudoconcave in y and y˙ and, b 2 2 d1 (t , x, x˙ , u, u˙ ) dt + ρ 2 a d2 (t , v, v˙ , y, y˙ ) dt = 0 or ρ 1 = 0 and ρ 2 = 0. i
i
Then p ̸≤ q. Proof. As (x, y, p, λ, z , r ) is feasible for the primal problem (FVP) and (u, v, q, λ, w, s) is feasible for the dual problem (FVD), α1 (x, u) > 0, by the dual constraint (11), the vector a = α1 (x, u) li=1 λi {[fxi − Dfx˙i + w i ] − qi [gxi − Dgx˙i − si ]} ∈ C1∗ , t ∈ I, and so from hypothesis (A), we obtain
t , x, u; α1 (x, u)
F
l
λi
fxi
−
Dfx˙i
+ w − qi i
gxi
−
Dgx˙i
i
−s
i =1
= −u T
l
λi
fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si
i=1
(by dual constraint (12)).
=0
By using sublinearity of F and λi > 0, i ∈ L, we get l
λi F t , x, u; α1 (x, u) fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si = 0,
i=1
which implies l i=1
λi
b
F t , x, u; α1 (x, u)
a
fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si
dt = 0.
(15)
N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
This, in view of (F , α1 , ρ 1 , d1 )-pseudoconvexity of x and x˙ , we have l
b
λi
1459
b i ˙ ) + (·)T w i − qi g i (t , ·, ·, v, v˙ ) − (·)T si }dt in i=1 λi a { f (t , ·, ·, v, v
l
f i (t , x, x˙ , v, v˙ ) + xT w i − qi g i (t , x, x˙ , v, v˙ ) − xT si
dt
a
i =1 l
=
b
λi
f i (t , u, u˙ , v, v˙ ) + uT w i − qi g i (t , u, u˙ , v, v˙ ) − uT si
dt + ρ 1
2
dt .
2
dt .
d1 (t , x, x˙ , u, u˙ )
d1 (t , x, x˙ , u, u˙ )
(16)
a
a
i=1
b
Using xT w i 5 s(x | Bi ), w i ∈ Bi , i = 1, 2, . . . , l, (16) can be written as l
λi
b
f i (t , x, x˙ , v, v˙ ) + s(x | Bi ) − qi g i (t , x, x˙ , v, v˙ ) − xT si
dt
f i (t , u, u˙ , v, v˙ ) + uT w i − qi g i (t , u, u˙ , v, v˙ ) − uT si
a
i =1
l
=
b
λi
dt + ρ 1
a
a
i=1
b
Using dual constraint (10) and xT si 5 s(x | Ei ), si ∈ Ei , v T r i 5 s(v | Hi ), r i ∈ Hi , qi = 0, i = 1, 2, . . . , l in the above inequality, we obtain l
λi
b
f i (t , x, x˙ , v, v˙ ) + s(x | Bi ) − s(v | Di ) − qi g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i
dt
a
i =1
= ρ1
b
2
dt .
d1 (t , x, x˙ , u, u˙ )
(17)
a
By the primal constraint (4), α2 (v, y) > 0, taking vector b = −α2 (v, y) t ∈ I , and using hypothesis (B), we obtain
G t , v, y; −α2 (v, y)
l
λi
fyi
−
Dfy˙i
i
− z − pi
gyi
−
Dgy˙i
+r
i
λi fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i ∈ C2∗ ,
l
i =1
i =1
= yT
l
λi
fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i
i =1
(by primal constraint (5)).
=0
Using sublinearity of G and λi > 0, i ∈ L, we get l
λi G t , v, y; −α2 (v, y) fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i = 0,
i=1
which shows that l
λi
b
G t , v, y; −α2 (v, y)
fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i
dt = 0.
(18)
a
i=1
This further from hypothesis (ii) yields l
λi
f i (t , x, x˙ , v, v˙ ) − v T z i − pi g i (t , x, x˙ , v, v˙ ) + v T r i
dt
a
i=1
5
b
l
b
λi
f i (t , x, x˙ , y, y˙ ) − yT z i − pi g i (t , x, x˙ , y, y˙ ) + yT r i
a
i =1
dt − ρ 2
b
2
d2 (t , v, v˙ , y, y˙ )
dt .
(19)
a
From (3) and v T z i 5 s(v | Di ), z i ∈ Di , i = 1, 2, . . . , l, (19) becomes l i=1
λi
b
f i (t , x, x˙ , v, v˙ ) + s(x | Bi ) − s(v | Di ) − pi g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i
dt
a
5 −ρ 2
b
2
d2 (t , v, v˙ , y, y˙ )
a
dt .
(20)
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N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
Using hypothesis (iii) in the addition of (17) and (20), we obtain l
λi (pi − qi )
b
g i (t , x(t ), x˙ (t ), v(t ), v˙ (t )) − s(x | Ei ) + v T r i dt = 0.
(21)
b
g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i dt > 0, i = 1, 2,
a
i=1
If, for all i ∈ L, pi 5 qi and pj < qj , for some j ∈ L, then from λ > 0 and . . . , l, implies that l
λi (pi − qi )
a
b
g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i dt < 0
a
i=1
which contradicts (21). Hence p ̸≤ q.
Theorem 3.2 (Weak Duality). Let (x, y, p, λ, z , r ) ∈ P and (u, v, q, λ, w, s) ∈ Q . Let the sublinear functionals F : I ×Rn ×Rn ×Rn → R and G : I × Rm × Rm × Rm → R satisfy the following conditions: F (t , x, u; a) + α1−1 aT u = 0,
for all a ∈ C1∗ , t ∈ I
G (t , v, y; b) + α2 b y = 0,
for all b ∈ C2 , t ∈ I .
−1 T
(A) (B)
∗
Suppose that (i) (ii)
b
f i (t , ·, ·, v, v˙ ) + (·)T w i − qi g i (t , ·, ·, v, v˙ ) − (·)T si
ab
f (t , x, x˙ , ·, ·) − (·) z
a
dt is (F , α1 , ρi1 , d1i )-convex in x and x˙ ,
− pi g (t , x, x˙ , ·, ·) + (·) r dt is (G, α2 , ρi2 , d2i )-concave in y and y˙ , and b 2 2 l d1i (t , x, x˙ , u, u˙ ) dt + i=1 λi ρi2 a d2i (t , v, v˙ , y, y˙ ) dt = 0 or ρi1 = 0 and ρi2 = 0, i ∈ L.
i
T
i
T
i
1 b
l
(iii) either
i
i =1
λi ρi
a
Then p ̸≤ q.
(22)
Proof. Assume by contradiction that (22) is not true, that is, p ≤ q, i.e., pi < qi , for some i ∈ L and pj 5 qj , for all j ∈ L, then from λ > 0 and l
b a
g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i dt > 0, i = 1, 2, . . . , l, we have
λi (pi − qi )
b
g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i dt < 0.
(23)
a
i=1
Since (x, y, p, λ, z , r ) is feasible for the primal problem (FVP) and (u, v, q, λ, w, s) is feasible (FVD), for the dual problem α1 (x, u) > 0, by the dual constraint (11), the vector a = α1 (x, u) li=1 λi fxi − Dfx˙i + wi − qi gxi − Dgx˙i − si ∈ C1∗ , t ∈ I , and so from hypothesis (A), we obtain
t , x, u; α1 (x, u)
F
l
λi
fxi − Dfx˙i + w
i
− qi gxi − Dgx˙i − s
i
i=1
= −u T
l
λi
fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si
i=1
=0
(by dual constraint (12)).
By using sublinearity of F , we get l
λi F t , x, u; α1 (x, u) fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si = 0,
i=1
which further implies l i=1
λi
b
F t , x, u; α1 (x, u)
a
By hypothesis (i), we have
fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si
dt = 0.
(24)
N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
1461
b
f i (t , x, x˙ , v, v˙ ) + xT w i − qi g i (t , x, x˙ , v, v˙ ) − xT si
dt
a b
−
f i (t , u, u˙ , v, v˙ ) + uT w i − qi g i (t , u, u˙ , v, v˙ ) − uT si
dt
a b
F t , x, u; α1 (x, u)
=
fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si
dt
a b
+ ρi1
2
dt .
d1i (t , x, x˙ , u, u˙ )
(25)
a
Using (13) and (24) in (25), we obtain l
b
λi
f i (t , x, x˙ , v, v˙ ) + xT w i − qi g i (t , x, x˙ , v, v˙ ) − xT si
dt
a
i =1
l
−
λi
l
f i (t , u, u˙ , v, v˙ ) + uT w i − qi g i (t , u, u˙ , v, v˙ ) − uT si
dt
a
i =1
=
b
λi ρi1
b
2
d1i (t , x, x˙ , u, u˙ )
dt .
(26)
a
i=1
Using (10) and xT w i 5 s(x | Bi ), w i ∈ Bi , i = 1, 2, . . . , l in the above inequality, we obtain l
λi
b
f i (t , x, x˙ , v, v˙ ) + s(x | Bi ) − s(v | Di ) − qi g i (t , x, x˙ , v, v˙ ) + s(v | Hi ) − xT si
dt
a
i =1
l
=
λi ρi1
b
2
d1i (t , x, x˙ , u, u˙ )
dt .
a
i=1
It follows from xT si 5 s(x | Ei ), si ∈ Ei , v T r i ≤ s(v | Hi ), r i ∈ Hi , qi = 0, i = 1, 2, . . . , l that l
λi
b
f i (t , x, x˙ , v, v˙ ) + s(x | Bi ) − s(v | Di ) − qi g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i
dt
a
i =1 l
=
λi ρi1
b
2
d1i (t , x, x˙ , u, u˙ )
dt .
(27)
a
i=1
By the primal constraint (4), α2 (v, y) > 0, taking vector b = −α2 (v, y)
l
i=1
λi
fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i
∈
C2 , t ∈ I, and using hypothesis (B), we obtain ∗
G t , v, y; −α2 (v, y)
l
λi
fyi
−
Dfy˙i
i
− z − pi
gyi
−
Dgy˙i
+r
i
i =1
= yT
l
λi
fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i
i =1
=0
(by primal constraint (5)).
Using sublinearity of G, we get l
λi G t , v, y; −α2 (v, y) fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i = 0,
i=1
which shows that l i=1
λi
b
G t , v, y; −α2 (v, y)
a
fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i
dt = 0.
(28)
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N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
The (G, α2 , ρi2 , d2i )-concavity of
b i { f (t , x, x˙ , ·, ·) − (·)T z i − pi g i (t , x, x˙ , ·, ·) + (·)T r i }dt in y and y˙ yields a
b
f i (t , x, x˙ , v, v˙ ) − v T z i − pi g i (t , x, x˙ , v, v˙ ) + v T r i
dt
a b
G t , v, y; −α2 (v, y)
f (t , x, x˙ , y, y˙ ) − y z
+
fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i
dt
a b
5
i
T i
− pi g (t , x, x˙ , y, y˙ ) + y r
i
T i
dt − ρ
2 i
b
2
d2i (t , v, v˙ , y, y˙ )
dt .
(29)
a
a
From (6), (28) and (29), we have l
λi
f i (t , x, x˙ , v, v˙ ) − v T z i − pi g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i
dt
a
i =1
5
b
l
f i (t , x, x˙ , y, y˙ ) − yT z i − pi g i (t , x, x˙ , y, y˙ ) − s(x | Ei ) + yT r i
dt
a
i=1 l
−
b
λi
λi ρi2
b
d2i (t , v, v˙ , y, y˙ )
2
dt .
(30)
a
i=1
Since v T z i 5 s(v | Di ), z i ∈ Di , i = 1, 2, . . . , l, (30) becomes l
b
λi
f i (t , x, x˙ , v, v˙ ) + s(x | Bi ) − s(v | Di ) − pi g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i
dt
a
i=1
5−
l
λi ρi2
b
d2i (t , v, v˙ , y, y˙ )
2
dt .
(31)
a
i =1
Finally, using hypothesis (i) in the addition of (27) and (30), we obtain l
λi (pi − qi )
b
g i (t , x(t ), x˙ (t ), v(t ), v˙ (t )) − s(x | Ei ) + v T r i dt = 0,
(32)
a
i=1
This contradicts (23). Hence the result.
¯ will be denoted by (FVD)λ¯ . Any problem, say (FVD), in which λ is fixed to be λ ¯ z¯ , r¯ ) be a weakly efficient solution for (FVP). Assume that Theorem 3.3 (Strong Duality). Let (¯x, y¯ , p¯ , λ, i ¯ i [(fyyi − p¯ i gyy (i) [σ (t )T { i=1 λ )− D(fy˙iy − p¯ i gy˙i y )]}+ D[σ (t )T 0 implies σ (t ) = 0, ∀t ∈ I, and
l
(ii) the set of vectors
l
fyi − z¯ i − p¯ i gyi + r¯ i − D fy˙i − p¯ i gy˙i
i=1
λ¯ i {D(fy˙iy˙ − p¯ i gy˙i y˙ )}]+ D2 {−σ (t )T
l
l
i=1
λ¯ i (fy˙iy˙ − p¯ i gy˙i y˙ )}]σ (t ) =
is linearly independent.
i=1
Then there exist w ¯ i ∈ Rn , s¯i ∈ Rn , i = 1, 2, . . . , l, such that (¯x, y¯ , p¯ , w, ¯ s¯) is feasible for (FVD)λ¯ , and the objective values of (FVP) and (FVD)λ¯ are equal. Furthermore, if the hypotheses of a weak duality theorem are satisfied for all feasible solutions of (FVP)λ¯ and (FVD)λ¯ , then (¯x, y¯ , p¯ , w, ¯ s¯) is an efficient solution of (FVD)λ¯ .
¯ z¯ , r¯ ) be a weakly efficient solution of (FVP), by Fritz John optimality conditions [25], there exist Proof. Since (¯x, y¯ , p¯ , λ, α ∈ Rl , β ∈ Rl , piecewise smooth γ (t ) : I → C2 , ξ (t ) : I → R and δ ∈ Rl such that ˜ = M
l i =1
αi pi +
l
βi
f i + s(¯x | Bi ) − y¯ T z¯ i − p¯ i g i − s(¯x | Ei ) + y¯ T r¯ i
i=1
l i i i T i i i + γ − ξ y¯ λi fy − z¯ − p¯ i gy + r¯ − D fy˙ − p¯ i gy˙ − δ T λ¯ i =1
¯ z¯ , r¯ ): satisfies the following conditions at (¯x, y¯ , p¯ , λ,
˜ x − DM ˜ x˙ + D2 M ˜ x¨ {x(t ) − x¯ (t )} = 0, M
∀x(t ) ∈ C1 , t ∈ I ,
(33)
N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
˜ y − DM ˜ y˙ + D2 M ˜ y¨ = 0, M ˜ λ = 0, M
1463
t ∈ I,
(34)
t ∈ I,
(35)
˜ p = 0, t ∈ I , M b βi f i + s(¯x | Bi ) − y¯ T z¯ i − p¯ i (g i − s(¯x | Ei ) + y¯ T r¯ i ) dt = 0,
(36) i ∈ L, t ∈ I ,
(37)
a
γT
l
λ¯ i
fyi − z¯ i − p¯ i gyi + r¯ i − D fy˙i − p¯ i gy˙i
= 0,
t ∈ I,
(38)
i=1
ξ y¯ T
l
λi
fyi − z¯ i − p¯ i gyi + r¯ i − D fy˙i − p¯ i gy˙i
= 0,
t ∈ I,
(39)
i =1
δ T λ¯ = 0,
(40)
s(¯x | Bi ) = x¯ T ηi ,
ηi ∈ Bi , i = 1, 2, . . . , l,
(41)
θi ∈ Ei , i = 1, 2, . . . , l, ¯ βi y¯ + [γ − ξ y¯ ] λi ∈ NDi (¯z i ), ¯ i ∈ NHi (¯r i ), p¯ i βi y¯ + (γ − ξ y¯ ) λ
(42)
(α, β, γ , ξ , δ) ̸= 0,
(45)
s(¯x | Ei ) = x¯ θi , T
T
(43) (44)
t ∈ I.
¯ z¯ , r¯ ), where (33) and (34) hold for The above relation hold throughout the interval I, except at the corners of (¯x, y¯ , p¯ , λ, unique right and left hand limits. The piecewise smooth functions γ and ξ are continuously differentiable except possibly ¯ z¯ , r¯ ). of the corners of (¯x, y¯ , p¯ , λ, Using the observations on Dfy˙i and Dgy˙i , i = 1, 2, . . . , l, from Section 2, Eqs. (33)–(36) become
l
βi
fxi + ηi − p¯ i gxi − θi
− D fx˙i − p¯ i gx˙i
i =1
+ [γ − ξ y¯ ]
T
l
λ¯ i
i fyx
i pi gyx
−¯
−D
fy˙ix
pi gy˙i x
−¯
i=1
− D (γ − ξ y¯ )
T
l
λ¯ i
fyix˙
pi gyi x˙
−¯
−D
fy˙ix˙
pi gy˙i x˙
−¯
−
fy˙ix
pi gy˙i x
−¯
i =1
2
+ D
l
− [γ − ξ y¯ ]
T
fy˙ix˙
pi gy˙i x˙
−¯
{x(t ) − x¯ (t )} = 0,
∀x(t ) ∈ C1 , t ∈ I ,
(46)
i=1 l βi − ξ λ¯ i fyi − z¯ i − p¯ i gyi + r¯ i − D fy˙i − p¯ i gy˙i i=1
+ [γ − ξ y¯ ]
T
l
λ¯ i
i fyy
i pi gyy
−¯
−D
fy˙iy
pi gy˙i y
−¯
+ D (γ − ξ y¯ )
T
i =1
l
λ¯ i D fy˙iy˙ − p¯ i gy˙i y˙
i=1
+ D2 − (γ − ξ y¯ )T
l
λ¯ i fy˙iy˙ − p¯ i gy˙i y˙
= 0,
t ∈ I,
(47)
i=1
fyi − z¯ i − p¯ i gyi + r¯ i − D fy˙i − p¯ i gy˙i
[γ − ξ y¯ ]T − δi = 0,
i = 1, 2, . . . , l, t ∈ I ,
(48)
and
αi − βi g i − θi + y¯ T r¯ i − [γ − ξ y¯ ]T gyi + r¯ i − Dgy˙i = 0,
i = 1, 2, . . . , l, t ∈ I .
(49)
¯ > 0, (40) yields δ = 0. Consequently (48) becomes Since δ = 0 and λ
fyi − z¯ i − p¯ i gyi + r¯ i − D fy˙i − p¯ i gy˙i
[γ − ξ y¯ ]T = 0,
i = 1, 2, . . . , l, t ∈ I .
(50)
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N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
Multiplying (47) by [γ − ξ y¯ ] and using (50), we have [γ − ξ y¯ ]T
l
λ¯ i
i i fyy − p¯ i gyy − D fy˙iy − p¯ i gy˙i y
[γ − ξ y¯ ]
i=1
+ D (γ − ξ y¯ )
T
l
λ¯ i D
fy˙iy˙
pi gy˙i y˙
−¯
[γ − ξ y¯ ]
i =1
− (γ − ξ y¯ )
T
2
+D
l
¯λi fy˙iy˙ − p¯ i gy˙i y˙ [γ − ξ y¯ ] = 0,
t ∈ I.
(51)
i=1
This, in view of hypothesis (i), yields
γ − ξ y¯ = 0,
t ∈ I.
(52)
From (47), we have l βi − ξ λ¯ i fyi − z¯ i − p¯ i gyi + r¯ i − D fy˙i − p¯ i gy˙i = 0, i=1
which, because of hypothesis (ii), implies
βi − ξ λ¯ i = 0,
i = 1, 2, . . . , l,
or
¯ β = ξ λ.
(53)
If, for t ∈ I, ξ = 0, then from (52) and (53), we get γ = 0 and β = 0, respectively. Also, (49) gives α = 0. Hence (α, β, γ , ξ , δ) = 0, contradicting (45). Thus ξ > 0, t ∈ I. From (52), we have
γ ∈ C2 , ξ
y¯ =
t ∈ I.
(54)
Now, (46) along with (52) and (53), and with ξ > 0, gives l
λ¯ i
fxi + ηi − p¯ i gxi − θi
− D fx˙i − p¯ i gx˙i (x(t ) − x¯ (t )) = 0,
t ∈ I.
(55)
i=1
Let x(t ) ∈ C1 . Then x(t ) + x¯ (t ) ∈ C1 , t ∈ I and so (55) show that for every x(t ) ∈ C1 , l
λ¯ i
fxi + ηi − p¯ i gxi − θi
− D fx˙i − p¯ i gx˙i x(t ) = 0,
t ∈ I.
i=1
Therefore, l
λ¯ i
fxi + ηi − p¯ i gxi − θi
− D fx˙i − p¯ i gx˙i ∈ C1∗ ,
t ∈ I.
(56)
i=1
Also, by letting x(t ) = 0 and x(t ) = 2x¯ (t ) simultaneously in (55), we obtain x¯ (t )T
l
λ¯ i
fxi + ηi − p¯ i gxi − θi
− D fx˙i − p¯ i gx˙i = 0,
t ∈ I.
i=1
Thus, from (54), (56) and (57), it follows that (¯x, y¯ , p¯ , w ¯ = η, s¯ = θ ) is a feasible solution for the dual problem (FVD)λ¯ . Further, from (43), (52) and (53) and ξ > 0, t ∈ I , we have for i = 1, 2, . . . , l,
λ¯ i y¯ ∈ NDi (¯z i ) ¯ i > 0, i = 1, 2, . . . , l. or y¯ ∈ NDi (¯z i ), usingλ Since Di is a compact convex set in Rm , y¯ T z¯ i = s(¯y | Di ), i = 1, 2, . . . , l. Also, from (44), (52) and (53) and ξ > 0, t ∈ I, we have for i = 1, 2, . . . , l,
¯ i y¯ ∈ NHi (¯r i ) p¯ i λ ¯ i > 0, i = 1, 2, . . . , l. or y¯ ∈ NHi (¯r i ), using λ
(57)
N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
1465
Since Hi is a compact convex set in Rm , y¯ T r¯ i = s(¯y | Hi ), i = 1, 2, . . . , l. Thus (FVP) and (FVD)λ¯ have equal objective function values (i.e., p¯ = q¯ ).
If (¯x, y¯ , p¯ , w, ¯ s¯) is not an efficient solution of (FVD)λ¯ , then there exists (¯u, v¯ , q¯ , w, ¯ s¯) feasible for (FVD)λ¯ such that p¯ ≤ q¯ ,
which contradicts weak duality theorem (Theorem 3.1 or 3.2). Thus (¯x, y¯ , p¯ , w, ¯ s¯) is an efficient solution of (FVD)λ¯ . Hence the result.
¯ w, Theorem 3.4 (Converse Duality). Let (¯u, v¯ , q¯ , λ, ¯ s¯) be a weakly efficient solution for (FVD). Assume that i ¯ i [(fxxi − q¯ i gxx (i) [ψ(t )T { i=1 λ ) − D(fx˙ix − q¯ i gx˙i x )]} + D[ψ(t )T ψ(t )dt = 0 implies ψ(t ) = 0, ∀t ∈ I, and
l
(ii) the set of vectors
fxi + w ¯ i − q¯ i gxi − s¯i − D fx˙i − q¯ i gx˙i
l
l i=1
i=1
λ¯ i {D(fx˙ix˙ − q¯ i gx˙i x˙ )}] + D2 {−ψ(t )T
l
i =1
λ¯ i (fx˙ix˙ − q¯ i gx˙i x˙ )}]
is linearly independent.
Then there exist z¯ i ∈ Rm , r¯ i ∈ Rm , i = 1, 2, . . . , l, such that (¯u, v¯ , q¯ , z¯ , r¯ ) is feasible for (FVP)λ¯ , and the objective values of (FVP)λ¯ and (FVD) are equal. Furthermore, if the hypotheses of a weak duality theorem are satisfied for all feasible solutions of (FVP)λ¯ and (FVD)λ¯ , then (¯u, v¯ , q¯ , z¯ , r¯ ) is an efficient solution of (FVP)λ¯ . Proof. Follows on the lines of Theorem 3.3.
Acknowledgments The authors wish to thank the referees for their valuable and constructive suggestions which have considerably improved the presentation of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
M.A. Hanson, Bounds for functionally convex optimal control problems, J. Math. Anal. Appl. 8 (1964) 84–89. B. Mond, M.A. Hanson, Duality for variational problems, J. Math. Anal. Appl. 18 (1967) 355–364. C.R. Bector, S. Chandra, I. Husain, Generalized concavity and duality in continuous programming, Util. Math. 25 (1984) 171–190. G.B. Dantzig, E. Eisenberg, R.W. Cottle, Symmetric dual nonlinear programs, Pacific J. Math. 15 (1965) 809–812. B. Mond, T. Weir, Generalized concavity and duality, in: S. Schaible, W.T. Ziemba (Eds.), Generalized Concavity in Optimization and Economics, Academic Press, New York, 1981, pp. 263–279. V. Preda, On efficiency and duality for multiobjective programs, J. Math. Anal. Appl. 166 (1992) 365–377. M.A. Hanson, B. Mond, Further generalizations of convexity in mathematical programming, J. Inform. Optim. Sci. 3 (1982) 25–32. J.P. Vial, Strong and weak convexity of sets and functions, Math. Oper. Res. 8 (1983) 231–259. J. Zhang, B. Mond, Second order duality for multiobjective nonlinear programming involving generalized convexity, in: B.M. Glower, B.D. Craven, D. Ralph (Eds.), Proceedings of the Optimization Miniconference III, University of Ballarat, 1997, pp. 79–95. R.N. Mukherjee, S.K. Mishra, Generalized invexity and duality in multiple objective variational problems, J. Math. Anal. Appl. 195 (1995) 307–322. S.K. Mishra, Generalized proper efficiency and duality for a class of nondifferentiable multiobjective variational problems with V -invexity, J. Math. Anal. Appl. 202 (1996) 53–71. R.N. Mukherjee, C.P. Rao, Mixed type duality for multiobjective variational problems, J. Math. Anal. Appl. 252 (2000) 571–586. I. Ahmad, T.R. Gulati, Mixed type duality for multiobjective variational problems with generalized (F , ρ)-convexity, J. Math. Anal. Appl. 306 (2005) 669–683. I. Ahmad, Z. Husain, Second-order (F , α, ρ, d)-convexity and duality in multiobjective programming, Inform. Sci. 176 (2006) 3094–3103. M. Hachimi, B. Aghezzaf, Sufficiency and duality in multiobjective variational problems with generalized type I functions, J. Global Optim. 34 (2006) 191–218. I.M. Stancu-Minasian, St. Mititelu, Multiobjective fractional variational problems with (ρ, b)-quasiinvexity, Proc. Rom. Acad. Ser. A 9 (2008) 5–11. St. Mititelu, I.M. Stancu-Minasian, Efficiency and duality for multiobjective fractional variational problems with (ρ, b)-quasiinvexity, Yugosl. J. Oper. Res. 19 (1) (2009) 85–99. S.K. Mishra, R.N. Mukherjee, Duality for multiobjective fractional variational problems, J. Math. Anal. Appl. 186 (1994) 711–725. S.K. Mishra, S.Y. Wang, K.K. Lai, Symmetric duality for a class of nondifferentiable multi-objective fractional variational problems, J. Math. Anal. Appl. 333 (2007) 1093–1110. I. Ahmad, S. Sharma, Symmetric duality for multiobjective fractional variational problems involving cones, European J. Oper. Res. 188 (2008) 695–704. S.K. Mishra, S.Y. Wang, K.K. Lai, Generalized convexity and vector optimization, in: Nonconvex Optimization and its Applications Volume 90, 2009, pp. 199–253. I. Smart, B. Mond, Symmetric duality with invexity in variational problems, J. Math. Anal. Appl. 152 (1990) 536–545. C.R. Bector, I. Husain, Duality for multiobjective variational problems, J. Math. Anal. Appl. 166 (1992) 214–229. D.S. Kim, G.M. Lee, Symmetric duality with pseudo-invexity in variational problems, Optimization 28 (1993) 9–16. S.K. Suneja, S. Aggarwal, S. Davar, Multiobjective symmetric duality involving cones, European J. Oper. Res. 141 (2002) 471–479.
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Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Duality for a class of symmetric nondifferentiable multiobjective fractional variational problems with generalized (F , α, ρ, d)-convexity N. Kailey a , S.K. Gupta b,∗ a
Department of Applied Sciences, Gulzar Institute of Engineering and Technology, G.T. Road, Khanna-141 401, India
b
Department of Mathematics, Indian Institute of Technology Patna, Patna-800 013, India
article
info
Article history: Received 8 September 2011 Received in revised form 5 December 2012 Accepted 6 December 2012
abstract A pair of nondifferentiable symmetric dual problems for a class of multiobjective fractional variational programs over arbitrary cones is formulated. Weak, strong and converse duality relations are then proved under generalized (F , α, ρ, d)-convexity assumptions. © 2012 Elsevier Ltd. All rights reserved.
Keywords: Symmetric duality Multiobjective nondifferentiable programming Fractional variational programming (F , α, ρ, d)-convexity Efficient solutions
1. Introduction Calculus of variation provides an excellent uniform analytical method to find that curve connecting two given points which either maximizes or minimizes some given integral; for example, to determine a curve which will generate the surface of revolution of smallest area when revolved about the x-axis. In general, we wish to find the curve x = x(t ) where x(a) = α1 and x(b) = α2 such that for some given known function f (t , x(t ), x˙ (t )) of variables t , x, x˙ , the integral b
f (t , x, x˙ )dt a
is either a maximum or minimum (also called extremum values). The curve which satisfies this property is said to be an extremal. The problem of finding a piecewise smooth extremal x = x(t ) for the above program is known as a variational problem. Hanson [1] surveyed and generalized the relationship between mathematical programming and classical calculus of variation. Optimality conditions and duality results were obtained for scalar valued variational problems by Mond and Hanson [2] under convexity. Bector et al. [3] extended symmetric duality to variational programming, giving continuous analogous of the results in [4,5]. The concept of (F , ρ)-convexity was introduced by Preda [6], an extension of F -convexity defined by Hanson and Mond [7] and ρ -convexity given by Vial [8]. The class of (F , ρ)-convex functions was extended to second order and duality results for Mangasarian, Mond–Weir and general Mond–Weir type multiobjective dual problems were obtained in [9].
∗
Corresponding author. Tel.: +91 612 2552025; fax: +91 612 2277384. E-mail addresses: [email protected] (N. Kailey), [email protected] (S.K. Gupta).
0895-7177/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2012.12.007
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N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
Multiple objective programming problems with the concept of weak minima are extended to multiple objective variational problems by Mukherjee and Mishra [10]. Mishra et al. [11] studied Mond–Weir type dual for a class of nondifferentiable multiobjective variational problems in which every component of the objective function contains a term involving the square root of a certain positive semidefinite quadratic form and usual duality theorems are developed for conditionally properly efficient solutions under V -invexity assumptions. Mukherjee and Rao [12] generalized the concept of mixed type duality to the class of multiobjective variational problems and proved duality results for ρ -convex functions. Ahmad and Gulati [13] considered mixed type dual for multiobjective variational problem and obtained duality relations by relating properly efficient solutions between the primal and mixed dual problems under generalized (F , ρ)-convexity assumptions. Ahmad and Husain [14] introduced second-order (F , α, ρ, d)-convex functions, their generalizations and developed usual duality theorems for second-order Mond–Weir type multiobjective primal–dual pair. Hachimi and Aghezzaf [15] extended the concepts of (F , α, ρ, d) type-I functions and established various sufficient optimality conditions and mixed duality results for multiobjective variational problems. Stancu-Minasian and Mititelu [16,17] considered multiobjective fractional variational problems with (ρ, b)quasiinvexity for which they established necessary conditions for normal efficient solutions. Based on this normal efficiency criteria, a Mond–Weir type dual is formulated and appropriate duality theorems are presented under (ρ, b)-quasiinvexity on the functions involved. The appropriate duality results are proved for a class of multiobjective fractional variational problems through a parametric approach by Mishra and Mukherjee [18]. Mishra et al. [19] formulated a symmetric dual pair for a class of nondifferentiable multiobjective fractional variational problems and duality theorems are obtained under invexity assumptions. Later on, Ahmad and Sharma [20] formulated a pair of multiobjective fractional variational symmetric dual problems over arbitrary cones and obtained usual duality results under generalized F -convexity assumptions. Recently, Mishra et al. [21] focused on symmetric duality for a class of nondifferentiable fractional variational problems. They introduced a symmetric dual pair for a class of nondifferentiable vector fractional variational problems and established appropriate duality results under certain invexity assumptions. The results obtained in this paper extend their work over arbitrary cones and assumptions as (F , α, ρ, d)-convexity/pseudoconvexity. This paper is organized as follows: In Section 2, we give some definitions and preliminaries. In Section 3, we formulate a pair of nondifferentiable symmetric dual programs for a class of multiobjective fractional variational problems over arbitrary cones and prove weak, strong and converse duality theorems under (F , α, ρ, d)-convexity/pseudoconvexity assumptions. 2. Notations and preliminaries Let I = [a, b] be a real interval and C1 ⊂ Rn , C2 ⊂ Rm , be closed convex cones with nonempty interiors having positive polars C1∗ and C2∗ , respectively. Let, for each i ∈ L = {1, 2, . . . , l}, f i (t , x(t ), x˙ (t ), y(t ), y˙ (t )) and g i (t , x(t ), x˙ (t ), y(t ), y˙ (t )), where x : I → Rn and y : I → Rm , with derivatives x˙ and y˙ , are twice continuously differentiable functions. For i ∈ L, the symbols fxi , fx˙i , fyi and fy˙i denote gradient vectors of the scalar function f i (t , x, x˙ , y, y˙ ) with respect to x, x˙ , y and y˙ , respectively. We have fxi
=
∂f i ∂f i , . . . , ∂ x1 ∂ xn
T
,
fx˙i
=
∂f i ∂f i , . . . , ∂ x˙ 1 ∂ x˙ n
T
.
Similarly, gxi , gx˙i , gyi and gy˙i denote the gradient vectors of g i (t , x, x˙ , y, y˙ ) with respect to x, x˙ , y and y˙ , respectively. The following observations are used for proving the strong duality theorem (Theorem 3): Dfy˙i = fy˙it + fy˙iy y˙ + fy˙iy˙ y¨ + fy˙ix x˙ + fy˙ix˙ x¨ . Consequently,
∂ i Df = Dfy˙iy , ∂ y y˙ ∂ i Df = Dfy˙ix , ∂ x y˙
∂ i Df = Dfy˙iy˙ + fy˙iy , ∂ y˙ y˙ ∂ i Df = Dfy˙ix˙ + fy˙ix , ∂ x˙ y˙
∂ i Df = fy˙iy˙ , ∂ y¨ y˙ ∂ i Df = fy˙ix˙ , i ∈ L. ∂ x¨ y˙
Similarly, Dgy˙i can be defined. Let X (I , Rn ) denotes the space of piecewise smooth functions x with the norm
∥ x ∥=∥ x ∥∞ + ∥ Dx ∥∞ , where the differentiation operator D is given by u = Dx ⇔ x(t ) = α1 +
t
u(s)ds, a
where α1 is a given boundary value. Therefore, dtd ≡ D except at discontinuities. Denote by Y (I , Rm ), the space of piecewise smooth functions y : I → Rm with the norm as that of space X (I , Rn ).
N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
1455
Consider the following multiobjective variational problem: (FP) Minimize b
φ(t , x(t ), x˙ (t ))dt =
b
φ (t , x(t ), x˙ (t ))dt , . . . , 1
φ (t , x(t ), x˙ (t ))dt l
a
a
a
b
subject to x(a) = 0 = x(b), x˙ (a) = 0 = x˙ (b), h(t , x(t ), x˙ (t )) 5 0,
t ∈ I,
where φ : I × R × R → R and h : I × Rn × Rn → Rm are differentiable functions. n
n
l
Let S be the set of all feasible solutions of (FP), i.e., S = {x ∈ X (I , Rn ) | x(a) = 0 = x(b), x˙ (a) = 0 = x˙ (b), h(t , x(t ), x˙ (t )) 5 0, t ∈ I }. Definition 2.1. A point x¯ ∈ S is an efficient (or Pareto optimal) solution of (FP) if there exist no other x ∈ S such that b
φ(t , x(t ), x˙ (t ))dt ≤
b
a
φ(t , x¯ (t ), x˙¯ (t ))dt . a
Definition 2.2. A point x¯ ∈ S is a weak efficient solution of (FP) if there exist no other x ∈ S such that b
φ(t , x(t ), x˙ (t ))dt <
b
a
φ(t , x¯ (t ), x˙¯ (t ))dt . a
Definition 2.3. A functional F : I × Rn × Rn × Rn × Rn × Rn → R is said to be sublinear with respect to its sixth argument, if for all x, x˙ , u, u˙ ∈ X (I , Rn ), (i) F (t , x, x˙ , u, u˙ ; a1 + a2 ) 5 F (t , x, x˙ , u, u˙ ; a1 ) + F (t , x, x˙ , u, u˙ ; a2 ), ∀a1 , a2 ∈ Rn , (ii) F (t , x, x˙ , u, u˙ ; δ a) = δ F (t , x, x˙ , u, u˙ ; a), for all δ ∈ R+ and a ∈ Rn . Definition 2.4. Let C be a compact convex set in Rn . The support function of C is defined by s(x | C ) = max{xT y : y ∈ C }. A support function, being convex and everywhere finite, has a subdifferential, that is, there exists z ∈ Rn such that s(y | C ) = s(x | C ) + z T (y − x) for all y ∈ C . The subdifferential of s(x | C ) is given by
∂ s(x | C ) = {z ∈ C : z T x = s(x | C )}. For any set M ⊂ Rn , the normal cone to M at a point x ∈ M is defined by NM (x) = {y ∈ Rn : yT (z − x) 5 0 for all z ∈ M }. Also, for a compact convex set C , y ∈ NC (x) if and only if s(y | C ) = xT y. Definition 2.5. Let D be a closed convex cone in Rn with nonempty interior. The positive polar cone D∗ of D is defined as D∗ = {z : xT z = 0, for all x ∈ D}. Let F and G be sublinear functionals with respect to sixth argument and d = (d1 , d2 ), where d1 = (d11 , d12 , . . . , d1l ) : I × R ×Rn ×Rn ×Rn → Rl , d2 = (d21 , d22 , . . . , d2l ) : I ×Rm ×Rm ×Rm ×Rm → Rl . Let f = (f 1 , f 2 , . . . , f l ) : I ×Rn ×Rn ×Rm ×Rm → Rl be a differentiable function, α = (α1 , α2 ), where α1 : Rn × Rn → R+ \ {0}, α2 : Rm × Rm → R+ \ {0} and ρ = (ρ 1 , ρ 2 ), where ρ 1 = (ρ11 , ρ21 , . . . , ρl1 ), ρ 2 = (ρ12 , ρ22 , . . . , ρl2 ) ∈ Rl . n
Definition 2.6. For each i ∈ L, b
f i (t , x, x˙ , y, y˙ )dt − a
b a
f i (t , x, x˙ , y, y˙ )dt is said to be (F , α1 , ρi1 , d1i )-convex in x and x˙ for fixed y and y˙ , if
b
f i (t , u, u˙ , y, y˙ )dt = a
b
F (t , x, x˙ , y, y˙ ; α1 (x, u)(fxi (t , u, u˙ , y, y˙ ) a
− Dfx˙i (t , u, u˙ , y, y˙ )))dt + ρi1
b
d1i (t , x, x˙ , u, u˙ )
a
for all x, u : I → Rn and for some arbitrary sublinear functional F .
2
dt
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N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
Definition 2.7. For each i ∈ L,
b a
f i (t , x, x˙ , y, y˙ )dt is said to be (F , α1 , ρi1 , d1i )-pseudoconvex in x and x˙ for fixed y and y˙ , if
b
F (t , x, x˙ , y, y˙ ; α1 (x, u)(fxi (t , u, u˙ , y, y˙ ) − Dfx˙i (t , u, u˙ , y, y˙ )))dt = 0 a
⇒
b
f i (t , x, x˙ , y, y˙ )dt =
b
f i (t , u, u˙ , y, y˙ )dt + ρi1 a
a
b
2
d1i (t , x, x˙ , u, u˙ )
dt ,
a
for all x, u : I → Rn and for some arbitrary sublinear functional F . In the sequel, we will write F (t , x, u; ξ ) for F (t , x, x˙ , u, u˙ ; ξ ) and G(t , v, y; η) for G(t , v, v˙ , y, y˙ ; η). 3. Symmetric duality Duality in variational programming problems has been developed extensively in last few years. Mond and Hanson [2] presented a constrained variational problem as a mathematical programming problem and presented its Wolfe type dual for validating various duality results under convexity. Smart and Mond [22] extended the symmetric duality results to variational problems by using continuous version of invexity. Later, Bector et al. [3] studied duality theorems for variational programming problems with weaker convexity assumptions. Bector and Husain [23] extended the results appeared in [3,4] to multiobjective case and obtained duality relations for Wolfe and Mond–Weir type duals for multiobjective variational problems under convexity. Later on, Kim and Lee [24] formulated a pair of symmetric dual variational problems in the spirit of Mond and Weir [5] using pseudo-invexity. After that, Hachimi and Aghezzaf [15] proved mixed duality results for multiobjective variational programming problems using the concept of generalized (F , α, ρ, d)-type I functions. Recently, Ahmad and Sharma [20] formulated multiobjective fractional variational symmetric dual problems over arbitrary cones and established usual duality theorems under generalized F -convexity assumptions. Now, we consider the problem of finding functions x : I → Rn and y : I → Rm , where (˙x, y˙ ) is piecewise smooth on I, to solve the following pair of symmetric dual multiobjective nondifferentiable fractional variational problems over arbitrary cones: Primal problem (VP) Minimize
b a
{f (t , x, x˙ , y, y˙ ) + s(x | B) − yT z }dt
b
{g (t , x, x˙ , y, y˙ ) − s(x | E ) + yT r }dt b b l {f 1 (t , x, x˙ , y, y˙ ) + s(x | B1 ) − yT z 1 }dt {f (t , x, x˙ , y, y˙ ) + s(x | Bl ) − yT z l }dt a a = b ,..., b {g 1 (t , x, x˙ , y, y˙ ) − s(x | E1 ) + yT r 1 }dt {g l (t , x, x˙ , y, y˙ ) − s(x | El ) + yT r l }dt a a subject to x(a) = 0 = x(b), y(a) = 0 = y(b), x˙ (a) = 0 = x˙ (b), y˙ (a) = 0 = y˙ (b), l i F i (x, y) ∈ C2∗ , t ∈ I , − λi fy − Dfy˙i − z i − gyi − Dgy˙i + r i i G ( x , y ) i =1 l F i ( x, y ) yT λi fyi − Dfy˙i − z i − gyi − Dgy˙i + r i i = 0, t ∈ I , G (x, y) i=1 a
λ > 0, z i ∈ Di ,
x( t ) ∈ C 1 , r i ∈ Hi ,
t ∈ I, i = 1, 2, . . . , l.
Dual problem (VD) Maximize
b
{f (t , u, u˙ , v, v˙ ) − s(v | D) + uT w}dt a b {g (t , u, u˙ , v, v˙ ) + s(v | H ) − uT s}dt a b b l {f 1 (t , u, u˙ , v, v˙ ) − s(v | D1 ) + uT w 1 }dt {f (t , u, u˙ , v, v˙ ) − s(v | Dl ) + uT w l }dt a a = b ,..., b {g 1 (t , u, u˙ , v, v˙ ) + s(v | H1 ) − uT s1 }dt {g l (t , u, u˙ , v, v˙ ) + s(v | Hl ) − uT sl }dt a a subject to u(a) = 0 = u(b), v(a) = 0 = v(b), u˙ (a) = 0 = u˙ (b), v˙ (a) = 0 = v˙ (b),
N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465 l
fxi − Dfx˙i + w i − gxi − Dgx˙i − si
λi
M i (u, v)
i =1
u
l
T
λi
fxi
−
Dfx˙i
+w − i
gxi
−
Dgx˙i
−s
∈ C1∗ ,
Li (u, v)
i
M i (u, v)
5 0,
Li (u, v)
i=1
λ > 0, w i ∈ Bi ,
1457
t ∈ I, t ∈ I,
v(t ) ∈ C2 , t ∈ I , si ∈ Ei , i = 1, 2, . . . , l,
where f i : I × Rn × Rn × Rm × Rm → R and g i : I × Rn × Rn × Rm × Rm → R, i ∈ L, are continuously differentiable functions, and F i ( x, y ) =
b
f i (t , x, x˙ , y, y˙ ) + s(x | Bi ) − yT z i dt ;
a
Gi (x, y) =
b
g i (t , x, x˙ , y, y˙ ) − s(x | Ei ) + yT r i dt ;
a
M i (u, v) =
b
f i (t , u, u˙ , v, v˙ ) − s(v | Di ) + uT w i dt ;
a
and L (u, v) = i
b
g i (t , u, u˙ , v, v˙ ) + s(v | Hi ) − uT si dt .
a
Assume that Gi (x, y) > 0,
Li (u, v) > 0,
F i (x, y) = 0,
M i (u, v) = 0,
∀i ∈ L.
In order to simplify the notations, let pi =
F i (x, y) Gi (x, y)
b
a b a
=
f i (t , x, x˙ , y, y˙ ) + s(x | Bi ) − yT z i dt
g i (t , x, x˙ , y, y˙ ) − s(x | Ei ) + yT r i dt
and qi =
M i (u, v) Li (u, v)
b a
f i (t , u, u˙ , v, v˙ ) − s(v | Di ) + uT w i dt
a
g i (t , u, u˙ , v, v˙ ) + s(v | Hi ) − uT si dt
= b
and express problems (VP) and (VD) equivalently as follows: (FVP) Minimize p = (p1 , p2 , . . . , pl ) subject to x(a) = 0 = x(b),
y(a) = 0 = y(b),
x˙ (a) = 0 = x˙ (b),
y˙ (a) = 0 = y˙ (b),
(1) (2)
b
f i (t , x, x˙ , y, y˙ ) + s(x | Bi ) − yT z i dt − pi
a
−
b
g i (t , x, x˙ , y, y˙ ) − s(x | Ei ) + yT r i dt = 0,
i ∈ L,
(3)
a
l
λi
λi
fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i
∈ C2∗ ,
t ∈ I,
(4)
i =1
yT
l
fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i
= 0,
t ∈ I,
(5)
i=1
λ > 0,
x(t ) ∈ C1 ,
z ∈ Di ,
r ∈ Hi ,
i
i
(FVD) Maximize q = (q1 , q2 , . . . , ql )
t ∈ I, i = 1, 2, . . . , l.
(6) (7)
1458
N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
subject to u(a) = 0 = u(b),
v(a) = 0 = v(b), u˙ (a) = 0 = u˙ (b), v˙ (a) = 0 = v˙ (b), b i f (t , u, u˙ , v, v˙ ) − s(v | Di ) + uT w i dt − qi
(8) (9) b
g i (t , u, u˙ , v, v˙ ) + s(v | Hi ) − uT s dt = 0,
i
a
a
i ∈ L, l
λi
(10)
fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si
λi
∈ C1∗ ,
t ∈ I,
(11)
i =1 l
uT
fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si
5 0,
t ∈ I,
(12)
i =1
λ > 0,
v(t ) ∈ C2 ,
w ∈ Bi ,
s ∈ Ei ,
i
i
t ∈ I,
(13)
i = 1, 2, . . . , l.
(14)
In the above problems (FVP) and (FVD), it is to be noted that p and q are nonnegative. Let P and Q denote the sets of feasible solutions of (FVP) and (FVD), respectively. In the subsequent analysis, usual duality theorems are discussed in terms of (FVP) and (FVD), but equally applicable to (VP) and (VD). Now, we establish weak, strong and converse duality theorems for the problems (FVP) and (FVD). Theorem 3.1 (Weak Duality). Let (x, y, p, λ, z , r ) ∈ P and (u, v, q, λ, w, s) ∈ Q . Let the sublinear functionals F : I ×Rn ×Rn ×Rn → R and G : I × Rm × Rm × Rm → R satisfy the following conditions: F (t , x, u; a) + α1−1 aT u = 0,
for all a ∈ C1∗ , t ∈ I
G (t , v, y; b) + α2 b y = 0,
for all b ∈ C2 , t ∈ I .
−1 T
(A) (B)
∗
Suppose that
λi
b
λi
(iii) either ρ
1 b
(i)
l
(ii)
i=1 l i=1
a b a
a
f i (t , ·, ·, v, v˙ ) + (·)T w i − qi g i (t , ·, ·, v, v˙ ) − (·)T si
f i (t , x, x˙ , ·, ·) − (·)T z
dt is (F , α1 , ρ 1 , d1 )-pseudoconvex in x and x˙ ,
− pi g i (t , x, x˙ , ·, ·) + (·)T r dt is (G, α2 , ρ 2 , d2 )-pseudoconcave in y and y˙ and, b 2 2 d1 (t , x, x˙ , u, u˙ ) dt + ρ 2 a d2 (t , v, v˙ , y, y˙ ) dt = 0 or ρ 1 = 0 and ρ 2 = 0. i
i
Then p ̸≤ q. Proof. As (x, y, p, λ, z , r ) is feasible for the primal problem (FVP) and (u, v, q, λ, w, s) is feasible for the dual problem (FVD), α1 (x, u) > 0, by the dual constraint (11), the vector a = α1 (x, u) li=1 λi {[fxi − Dfx˙i + w i ] − qi [gxi − Dgx˙i − si ]} ∈ C1∗ , t ∈ I, and so from hypothesis (A), we obtain
t , x, u; α1 (x, u)
F
l
λi
fxi
−
Dfx˙i
+ w − qi i
gxi
−
Dgx˙i
i
−s
i =1
= −u T
l
λi
fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si
i=1
(by dual constraint (12)).
=0
By using sublinearity of F and λi > 0, i ∈ L, we get l
λi F t , x, u; α1 (x, u) fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si = 0,
i=1
which implies l i=1
λi
b
F t , x, u; α1 (x, u)
a
fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si
dt = 0.
(15)
N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
This, in view of (F , α1 , ρ 1 , d1 )-pseudoconvexity of x and x˙ , we have l
b
λi
1459
b i ˙ ) + (·)T w i − qi g i (t , ·, ·, v, v˙ ) − (·)T si }dt in i=1 λi a { f (t , ·, ·, v, v
l
f i (t , x, x˙ , v, v˙ ) + xT w i − qi g i (t , x, x˙ , v, v˙ ) − xT si
dt
a
i =1 l
=
b
λi
f i (t , u, u˙ , v, v˙ ) + uT w i − qi g i (t , u, u˙ , v, v˙ ) − uT si
dt + ρ 1
2
dt .
2
dt .
d1 (t , x, x˙ , u, u˙ )
d1 (t , x, x˙ , u, u˙ )
(16)
a
a
i=1
b
Using xT w i 5 s(x | Bi ), w i ∈ Bi , i = 1, 2, . . . , l, (16) can be written as l
λi
b
f i (t , x, x˙ , v, v˙ ) + s(x | Bi ) − qi g i (t , x, x˙ , v, v˙ ) − xT si
dt
f i (t , u, u˙ , v, v˙ ) + uT w i − qi g i (t , u, u˙ , v, v˙ ) − uT si
a
i =1
l
=
b
λi
dt + ρ 1
a
a
i=1
b
Using dual constraint (10) and xT si 5 s(x | Ei ), si ∈ Ei , v T r i 5 s(v | Hi ), r i ∈ Hi , qi = 0, i = 1, 2, . . . , l in the above inequality, we obtain l
λi
b
f i (t , x, x˙ , v, v˙ ) + s(x | Bi ) − s(v | Di ) − qi g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i
dt
a
i =1
= ρ1
b
2
dt .
d1 (t , x, x˙ , u, u˙ )
(17)
a
By the primal constraint (4), α2 (v, y) > 0, taking vector b = −α2 (v, y) t ∈ I , and using hypothesis (B), we obtain
G t , v, y; −α2 (v, y)
l
λi
fyi
−
Dfy˙i
i
− z − pi
gyi
−
Dgy˙i
+r
i
λi fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i ∈ C2∗ ,
l
i =1
i =1
= yT
l
λi
fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i
i =1
(by primal constraint (5)).
=0
Using sublinearity of G and λi > 0, i ∈ L, we get l
λi G t , v, y; −α2 (v, y) fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i = 0,
i=1
which shows that l
λi
b
G t , v, y; −α2 (v, y)
fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i
dt = 0.
(18)
a
i=1
This further from hypothesis (ii) yields l
λi
f i (t , x, x˙ , v, v˙ ) − v T z i − pi g i (t , x, x˙ , v, v˙ ) + v T r i
dt
a
i=1
5
b
l
b
λi
f i (t , x, x˙ , y, y˙ ) − yT z i − pi g i (t , x, x˙ , y, y˙ ) + yT r i
a
i =1
dt − ρ 2
b
2
d2 (t , v, v˙ , y, y˙ )
dt .
(19)
a
From (3) and v T z i 5 s(v | Di ), z i ∈ Di , i = 1, 2, . . . , l, (19) becomes l i=1
λi
b
f i (t , x, x˙ , v, v˙ ) + s(x | Bi ) − s(v | Di ) − pi g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i
dt
a
5 −ρ 2
b
2
d2 (t , v, v˙ , y, y˙ )
a
dt .
(20)
1460
N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
Using hypothesis (iii) in the addition of (17) and (20), we obtain l
λi (pi − qi )
b
g i (t , x(t ), x˙ (t ), v(t ), v˙ (t )) − s(x | Ei ) + v T r i dt = 0.
(21)
b
g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i dt > 0, i = 1, 2,
a
i=1
If, for all i ∈ L, pi 5 qi and pj < qj , for some j ∈ L, then from λ > 0 and . . . , l, implies that l
λi (pi − qi )
a
b
g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i dt < 0
a
i=1
which contradicts (21). Hence p ̸≤ q.
Theorem 3.2 (Weak Duality). Let (x, y, p, λ, z , r ) ∈ P and (u, v, q, λ, w, s) ∈ Q . Let the sublinear functionals F : I ×Rn ×Rn ×Rn → R and G : I × Rm × Rm × Rm → R satisfy the following conditions: F (t , x, u; a) + α1−1 aT u = 0,
for all a ∈ C1∗ , t ∈ I
G (t , v, y; b) + α2 b y = 0,
for all b ∈ C2 , t ∈ I .
−1 T
(A) (B)
∗
Suppose that (i) (ii)
b
f i (t , ·, ·, v, v˙ ) + (·)T w i − qi g i (t , ·, ·, v, v˙ ) − (·)T si
ab
f (t , x, x˙ , ·, ·) − (·) z
a
dt is (F , α1 , ρi1 , d1i )-convex in x and x˙ ,
− pi g (t , x, x˙ , ·, ·) + (·) r dt is (G, α2 , ρi2 , d2i )-concave in y and y˙ , and b 2 2 l d1i (t , x, x˙ , u, u˙ ) dt + i=1 λi ρi2 a d2i (t , v, v˙ , y, y˙ ) dt = 0 or ρi1 = 0 and ρi2 = 0, i ∈ L.
i
T
i
T
i
1 b
l
(iii) either
i
i =1
λi ρi
a
Then p ̸≤ q.
(22)
Proof. Assume by contradiction that (22) is not true, that is, p ≤ q, i.e., pi < qi , for some i ∈ L and pj 5 qj , for all j ∈ L, then from λ > 0 and l
b a
g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i dt > 0, i = 1, 2, . . . , l, we have
λi (pi − qi )
b
g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i dt < 0.
(23)
a
i=1
Since (x, y, p, λ, z , r ) is feasible for the primal problem (FVP) and (u, v, q, λ, w, s) is feasible (FVD), for the dual problem α1 (x, u) > 0, by the dual constraint (11), the vector a = α1 (x, u) li=1 λi fxi − Dfx˙i + wi − qi gxi − Dgx˙i − si ∈ C1∗ , t ∈ I , and so from hypothesis (A), we obtain
t , x, u; α1 (x, u)
F
l
λi
fxi − Dfx˙i + w
i
− qi gxi − Dgx˙i − s
i
i=1
= −u T
l
λi
fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si
i=1
=0
(by dual constraint (12)).
By using sublinearity of F , we get l
λi F t , x, u; α1 (x, u) fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si = 0,
i=1
which further implies l i=1
λi
b
F t , x, u; α1 (x, u)
a
By hypothesis (i), we have
fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si
dt = 0.
(24)
N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
1461
b
f i (t , x, x˙ , v, v˙ ) + xT w i − qi g i (t , x, x˙ , v, v˙ ) − xT si
dt
a b
−
f i (t , u, u˙ , v, v˙ ) + uT w i − qi g i (t , u, u˙ , v, v˙ ) − uT si
dt
a b
F t , x, u; α1 (x, u)
=
fxi − Dfx˙i + w i − qi gxi − Dgx˙i − si
dt
a b
+ ρi1
2
dt .
d1i (t , x, x˙ , u, u˙ )
(25)
a
Using (13) and (24) in (25), we obtain l
b
λi
f i (t , x, x˙ , v, v˙ ) + xT w i − qi g i (t , x, x˙ , v, v˙ ) − xT si
dt
a
i =1
l
−
λi
l
f i (t , u, u˙ , v, v˙ ) + uT w i − qi g i (t , u, u˙ , v, v˙ ) − uT si
dt
a
i =1
=
b
λi ρi1
b
2
d1i (t , x, x˙ , u, u˙ )
dt .
(26)
a
i=1
Using (10) and xT w i 5 s(x | Bi ), w i ∈ Bi , i = 1, 2, . . . , l in the above inequality, we obtain l
λi
b
f i (t , x, x˙ , v, v˙ ) + s(x | Bi ) − s(v | Di ) − qi g i (t , x, x˙ , v, v˙ ) + s(v | Hi ) − xT si
dt
a
i =1
l
=
λi ρi1
b
2
d1i (t , x, x˙ , u, u˙ )
dt .
a
i=1
It follows from xT si 5 s(x | Ei ), si ∈ Ei , v T r i ≤ s(v | Hi ), r i ∈ Hi , qi = 0, i = 1, 2, . . . , l that l
λi
b
f i (t , x, x˙ , v, v˙ ) + s(x | Bi ) − s(v | Di ) − qi g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i
dt
a
i =1 l
=
λi ρi1
b
2
d1i (t , x, x˙ , u, u˙ )
dt .
(27)
a
i=1
By the primal constraint (4), α2 (v, y) > 0, taking vector b = −α2 (v, y)
l
i=1
λi
fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i
∈
C2 , t ∈ I, and using hypothesis (B), we obtain ∗
G t , v, y; −α2 (v, y)
l
λi
fyi
−
Dfy˙i
i
− z − pi
gyi
−
Dgy˙i
+r
i
i =1
= yT
l
λi
fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i
i =1
=0
(by primal constraint (5)).
Using sublinearity of G, we get l
λi G t , v, y; −α2 (v, y) fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i = 0,
i=1
which shows that l i=1
λi
b
G t , v, y; −α2 (v, y)
a
fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i
dt = 0.
(28)
1462
N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
The (G, α2 , ρi2 , d2i )-concavity of
b i { f (t , x, x˙ , ·, ·) − (·)T z i − pi g i (t , x, x˙ , ·, ·) + (·)T r i }dt in y and y˙ yields a
b
f i (t , x, x˙ , v, v˙ ) − v T z i − pi g i (t , x, x˙ , v, v˙ ) + v T r i
dt
a b
G t , v, y; −α2 (v, y)
f (t , x, x˙ , y, y˙ ) − y z
+
fyi − Dfy˙i − z i − pi gyi − Dgy˙i + r i
dt
a b
5
i
T i
− pi g (t , x, x˙ , y, y˙ ) + y r
i
T i
dt − ρ
2 i
b
2
d2i (t , v, v˙ , y, y˙ )
dt .
(29)
a
a
From (6), (28) and (29), we have l
λi
f i (t , x, x˙ , v, v˙ ) − v T z i − pi g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i
dt
a
i =1
5
b
l
f i (t , x, x˙ , y, y˙ ) − yT z i − pi g i (t , x, x˙ , y, y˙ ) − s(x | Ei ) + yT r i
dt
a
i=1 l
−
b
λi
λi ρi2
b
d2i (t , v, v˙ , y, y˙ )
2
dt .
(30)
a
i=1
Since v T z i 5 s(v | Di ), z i ∈ Di , i = 1, 2, . . . , l, (30) becomes l
b
λi
f i (t , x, x˙ , v, v˙ ) + s(x | Bi ) − s(v | Di ) − pi g i (t , x, x˙ , v, v˙ ) − s(x | Ei ) + v T r i
dt
a
i=1
5−
l
λi ρi2
b
d2i (t , v, v˙ , y, y˙ )
2
dt .
(31)
a
i =1
Finally, using hypothesis (i) in the addition of (27) and (30), we obtain l
λi (pi − qi )
b
g i (t , x(t ), x˙ (t ), v(t ), v˙ (t )) − s(x | Ei ) + v T r i dt = 0,
(32)
a
i=1
This contradicts (23). Hence the result.
¯ will be denoted by (FVD)λ¯ . Any problem, say (FVD), in which λ is fixed to be λ ¯ z¯ , r¯ ) be a weakly efficient solution for (FVP). Assume that Theorem 3.3 (Strong Duality). Let (¯x, y¯ , p¯ , λ, i ¯ i [(fyyi − p¯ i gyy (i) [σ (t )T { i=1 λ )− D(fy˙iy − p¯ i gy˙i y )]}+ D[σ (t )T 0 implies σ (t ) = 0, ∀t ∈ I, and
l
(ii) the set of vectors
l
fyi − z¯ i − p¯ i gyi + r¯ i − D fy˙i − p¯ i gy˙i
i=1
λ¯ i {D(fy˙iy˙ − p¯ i gy˙i y˙ )}]+ D2 {−σ (t )T
l
l
i=1
λ¯ i (fy˙iy˙ − p¯ i gy˙i y˙ )}]σ (t ) =
is linearly independent.
i=1
Then there exist w ¯ i ∈ Rn , s¯i ∈ Rn , i = 1, 2, . . . , l, such that (¯x, y¯ , p¯ , w, ¯ s¯) is feasible for (FVD)λ¯ , and the objective values of (FVP) and (FVD)λ¯ are equal. Furthermore, if the hypotheses of a weak duality theorem are satisfied for all feasible solutions of (FVP)λ¯ and (FVD)λ¯ , then (¯x, y¯ , p¯ , w, ¯ s¯) is an efficient solution of (FVD)λ¯ .
¯ z¯ , r¯ ) be a weakly efficient solution of (FVP), by Fritz John optimality conditions [25], there exist Proof. Since (¯x, y¯ , p¯ , λ, α ∈ Rl , β ∈ Rl , piecewise smooth γ (t ) : I → C2 , ξ (t ) : I → R and δ ∈ Rl such that ˜ = M
l i =1
αi pi +
l
βi
f i + s(¯x | Bi ) − y¯ T z¯ i − p¯ i g i − s(¯x | Ei ) + y¯ T r¯ i
i=1
l i i i T i i i + γ − ξ y¯ λi fy − z¯ − p¯ i gy + r¯ − D fy˙ − p¯ i gy˙ − δ T λ¯ i =1
¯ z¯ , r¯ ): satisfies the following conditions at (¯x, y¯ , p¯ , λ,
˜ x − DM ˜ x˙ + D2 M ˜ x¨ {x(t ) − x¯ (t )} = 0, M
∀x(t ) ∈ C1 , t ∈ I ,
(33)
N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
˜ y − DM ˜ y˙ + D2 M ˜ y¨ = 0, M ˜ λ = 0, M
1463
t ∈ I,
(34)
t ∈ I,
(35)
˜ p = 0, t ∈ I , M b βi f i + s(¯x | Bi ) − y¯ T z¯ i − p¯ i (g i − s(¯x | Ei ) + y¯ T r¯ i ) dt = 0,
(36) i ∈ L, t ∈ I ,
(37)
a
γT
l
λ¯ i
fyi − z¯ i − p¯ i gyi + r¯ i − D fy˙i − p¯ i gy˙i
= 0,
t ∈ I,
(38)
i=1
ξ y¯ T
l
λi
fyi − z¯ i − p¯ i gyi + r¯ i − D fy˙i − p¯ i gy˙i
= 0,
t ∈ I,
(39)
i =1
δ T λ¯ = 0,
(40)
s(¯x | Bi ) = x¯ T ηi ,
ηi ∈ Bi , i = 1, 2, . . . , l,
(41)
θi ∈ Ei , i = 1, 2, . . . , l, ¯ βi y¯ + [γ − ξ y¯ ] λi ∈ NDi (¯z i ), ¯ i ∈ NHi (¯r i ), p¯ i βi y¯ + (γ − ξ y¯ ) λ
(42)
(α, β, γ , ξ , δ) ̸= 0,
(45)
s(¯x | Ei ) = x¯ θi , T
T
(43) (44)
t ∈ I.
¯ z¯ , r¯ ), where (33) and (34) hold for The above relation hold throughout the interval I, except at the corners of (¯x, y¯ , p¯ , λ, unique right and left hand limits. The piecewise smooth functions γ and ξ are continuously differentiable except possibly ¯ z¯ , r¯ ). of the corners of (¯x, y¯ , p¯ , λ, Using the observations on Dfy˙i and Dgy˙i , i = 1, 2, . . . , l, from Section 2, Eqs. (33)–(36) become
l
βi
fxi + ηi − p¯ i gxi − θi
− D fx˙i − p¯ i gx˙i
i =1
+ [γ − ξ y¯ ]
T
l
λ¯ i
i fyx
i pi gyx
−¯
−D
fy˙ix
pi gy˙i x
−¯
i=1
− D (γ − ξ y¯ )
T
l
λ¯ i
fyix˙
pi gyi x˙
−¯
−D
fy˙ix˙
pi gy˙i x˙
−¯
−
fy˙ix
pi gy˙i x
−¯
i =1
2
+ D
l
− [γ − ξ y¯ ]
T
fy˙ix˙
pi gy˙i x˙
−¯
{x(t ) − x¯ (t )} = 0,
∀x(t ) ∈ C1 , t ∈ I ,
(46)
i=1 l βi − ξ λ¯ i fyi − z¯ i − p¯ i gyi + r¯ i − D fy˙i − p¯ i gy˙i i=1
+ [γ − ξ y¯ ]
T
l
λ¯ i
i fyy
i pi gyy
−¯
−D
fy˙iy
pi gy˙i y
−¯
+ D (γ − ξ y¯ )
T
i =1
l
λ¯ i D fy˙iy˙ − p¯ i gy˙i y˙
i=1
+ D2 − (γ − ξ y¯ )T
l
λ¯ i fy˙iy˙ − p¯ i gy˙i y˙
= 0,
t ∈ I,
(47)
i=1
fyi − z¯ i − p¯ i gyi + r¯ i − D fy˙i − p¯ i gy˙i
[γ − ξ y¯ ]T − δi = 0,
i = 1, 2, . . . , l, t ∈ I ,
(48)
and
αi − βi g i − θi + y¯ T r¯ i − [γ − ξ y¯ ]T gyi + r¯ i − Dgy˙i = 0,
i = 1, 2, . . . , l, t ∈ I .
(49)
¯ > 0, (40) yields δ = 0. Consequently (48) becomes Since δ = 0 and λ
fyi − z¯ i − p¯ i gyi + r¯ i − D fy˙i − p¯ i gy˙i
[γ − ξ y¯ ]T = 0,
i = 1, 2, . . . , l, t ∈ I .
(50)
1464
N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
Multiplying (47) by [γ − ξ y¯ ] and using (50), we have [γ − ξ y¯ ]T
l
λ¯ i
i i fyy − p¯ i gyy − D fy˙iy − p¯ i gy˙i y
[γ − ξ y¯ ]
i=1
+ D (γ − ξ y¯ )
T
l
λ¯ i D
fy˙iy˙
pi gy˙i y˙
−¯
[γ − ξ y¯ ]
i =1
− (γ − ξ y¯ )
T
2
+D
l
¯λi fy˙iy˙ − p¯ i gy˙i y˙ [γ − ξ y¯ ] = 0,
t ∈ I.
(51)
i=1
This, in view of hypothesis (i), yields
γ − ξ y¯ = 0,
t ∈ I.
(52)
From (47), we have l βi − ξ λ¯ i fyi − z¯ i − p¯ i gyi + r¯ i − D fy˙i − p¯ i gy˙i = 0, i=1
which, because of hypothesis (ii), implies
βi − ξ λ¯ i = 0,
i = 1, 2, . . . , l,
or
¯ β = ξ λ.
(53)
If, for t ∈ I, ξ = 0, then from (52) and (53), we get γ = 0 and β = 0, respectively. Also, (49) gives α = 0. Hence (α, β, γ , ξ , δ) = 0, contradicting (45). Thus ξ > 0, t ∈ I. From (52), we have
γ ∈ C2 , ξ
y¯ =
t ∈ I.
(54)
Now, (46) along with (52) and (53), and with ξ > 0, gives l
λ¯ i
fxi + ηi − p¯ i gxi − θi
− D fx˙i − p¯ i gx˙i (x(t ) − x¯ (t )) = 0,
t ∈ I.
(55)
i=1
Let x(t ) ∈ C1 . Then x(t ) + x¯ (t ) ∈ C1 , t ∈ I and so (55) show that for every x(t ) ∈ C1 , l
λ¯ i
fxi + ηi − p¯ i gxi − θi
− D fx˙i − p¯ i gx˙i x(t ) = 0,
t ∈ I.
i=1
Therefore, l
λ¯ i
fxi + ηi − p¯ i gxi − θi
− D fx˙i − p¯ i gx˙i ∈ C1∗ ,
t ∈ I.
(56)
i=1
Also, by letting x(t ) = 0 and x(t ) = 2x¯ (t ) simultaneously in (55), we obtain x¯ (t )T
l
λ¯ i
fxi + ηi − p¯ i gxi − θi
− D fx˙i − p¯ i gx˙i = 0,
t ∈ I.
i=1
Thus, from (54), (56) and (57), it follows that (¯x, y¯ , p¯ , w ¯ = η, s¯ = θ ) is a feasible solution for the dual problem (FVD)λ¯ . Further, from (43), (52) and (53) and ξ > 0, t ∈ I , we have for i = 1, 2, . . . , l,
λ¯ i y¯ ∈ NDi (¯z i ) ¯ i > 0, i = 1, 2, . . . , l. or y¯ ∈ NDi (¯z i ), usingλ Since Di is a compact convex set in Rm , y¯ T z¯ i = s(¯y | Di ), i = 1, 2, . . . , l. Also, from (44), (52) and (53) and ξ > 0, t ∈ I, we have for i = 1, 2, . . . , l,
¯ i y¯ ∈ NHi (¯r i ) p¯ i λ ¯ i > 0, i = 1, 2, . . . , l. or y¯ ∈ NHi (¯r i ), using λ
(57)
N. Kailey, S.K. Gupta / Mathematical and Computer Modelling 57 (2013) 1453–1465
1465
Since Hi is a compact convex set in Rm , y¯ T r¯ i = s(¯y | Hi ), i = 1, 2, . . . , l. Thus (FVP) and (FVD)λ¯ have equal objective function values (i.e., p¯ = q¯ ).
If (¯x, y¯ , p¯ , w, ¯ s¯) is not an efficient solution of (FVD)λ¯ , then there exists (¯u, v¯ , q¯ , w, ¯ s¯) feasible for (FVD)λ¯ such that p¯ ≤ q¯ ,
which contradicts weak duality theorem (Theorem 3.1 or 3.2). Thus (¯x, y¯ , p¯ , w, ¯ s¯) is an efficient solution of (FVD)λ¯ . Hence the result.
¯ w, Theorem 3.4 (Converse Duality). Let (¯u, v¯ , q¯ , λ, ¯ s¯) be a weakly efficient solution for (FVD). Assume that i ¯ i [(fxxi − q¯ i gxx (i) [ψ(t )T { i=1 λ ) − D(fx˙ix − q¯ i gx˙i x )]} + D[ψ(t )T ψ(t )dt = 0 implies ψ(t ) = 0, ∀t ∈ I, and
l
(ii) the set of vectors
fxi + w ¯ i − q¯ i gxi − s¯i − D fx˙i − q¯ i gx˙i
l
l i=1
i=1
λ¯ i {D(fx˙ix˙ − q¯ i gx˙i x˙ )}] + D2 {−ψ(t )T
l
i =1
λ¯ i (fx˙ix˙ − q¯ i gx˙i x˙ )}]
is linearly independent.
Then there exist z¯ i ∈ Rm , r¯ i ∈ Rm , i = 1, 2, . . . , l, such that (¯u, v¯ , q¯ , z¯ , r¯ ) is feasible for (FVP)λ¯ , and the objective values of (FVP)λ¯ and (FVD) are equal. Furthermore, if the hypotheses of a weak duality theorem are satisfied for all feasible solutions of (FVP)λ¯ and (FVD)λ¯ , then (¯u, v¯ , q¯ , z¯ , r¯ ) is an efficient solution of (FVP)λ¯ . Proof. Follows on the lines of Theorem 3.3.
Acknowledgments The authors wish to thank the referees for their valuable and constructive suggestions which have considerably improved the presentation of the paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
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