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Symmetric duality for invex multiobjective fractional variational problems
J. Math. Anal. Appl. 289 (2004) 505–521 www.elsevier.com/locate/jmaa
Symmetric duality for invex multiobjective fractional variational problems Do Sang Kim,a,1 Won Jung Lee,a and Siegfried Schaible b,∗ a Department of Applied Mathematics, Pukyong National University, Pusan 608-737, South Korea b A.G. Anderson Graduate School of Management, University of California, Riverside, CA 92521, USA
Received 10 October 2002 Submitted by J.P. Dauer
Abstract We introduce a symmetric dual for multiobjective fractional variational problems. Under certain invexity assumptions we establish weak, strong and converse duality theorems as well as self-duality relations. The paper includes an extension of previous symmetric duality results for the static case obtained by Weir to the dynamic case. 2003 Elsevier Inc. All rights reserved.
1. Introduction The classical dual in linear programming is symmetric in the sense that the dual of the dual is the original linear program. Such symmetry is usually not found in duality concepts for nonlinear programming, not even in quadratic programming [12]. In [6] Dorn introduced a different dual for quadratic programming which is symmetric. Extending these results to general convex programming, Dantzig et al. [5] formulated a symmetric dual and established weak and strong duality relations. Symmetric duality results under generalized convexity were given by Mond and Weir in [15] for new types of a dual. Then in [22] Weir and Mond introduced two distinct symmetric duals for multiobjective programs. Under additional assumptions the multiobjective programs are shown to be self-dual. Very recently * Corresponding author.
E-mail address: [email protected] (S. Schaible). 1 This author was supported by the Korea Research Foundation Grant KRF-2001-013-D00007. He gratefully
acknowledges the support of his visit by the A.G. Anderson Graduate School of Management during 2002. 0022-247X/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2003.08.035
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Yang et al. [23] introduced a symmetric dual for a class of nondifferentiable multiobjective fractional programs and obtained weak and strong duality relations. In [13] Mond and Hanson first extended the symmetric duality results of [5] to variational problems by introducing continuous analogues of the earlier concepts. Since Hanson defined invexity in [7] as a new generalization of convexity, several authors have introduced concepts of invexity and generalized invexity for use in continuous programming; e.g., [1, 10,14,16,20]. Smart and Mond [20] extended symmetric duality results to variational problems by employing a continuous version of invexity. Recently, Kim and Lee [8] presented a symmetric dual in the sense of a dual proposed by Mond and Weir [15], not in the sense of Wolfe’s dual as in [20], establishing duality relations for variational problems hereby assuming pseudo-invexity. Subsequently, Kim et al. [9] extended the results in [8] to the multiobjective case. More recently Kim and Lee [10,11] formulated a symmetric and a generalized symmetric dual for multiobjective variational problems. Weak, strong and converse duality relations are obtained under suitable invexity assumptions. In this article we focus on symmetric duality for fractional programming. Since duality results for convex programming do not apply to fractional programs in general [17], duality concepts for such nonconvex programs had to be defined separately; e.g., [17,18]. Most duals in fractional programming are not symmetric [17–19]. Chandra et al. [2] first introduced a symmetric dual in nonlinear fractional programming. Chandra and Husain [3] then extended these results to continuous nonlinear fractional programming obtaining symmetric duality and self-duality relations. For the multiobjective case of a static nonlinear fractional program symmetric duality was introduced by Weir [21], and weak and strong duality relations were derived under convexity assumptions. Very recently Chen [4] formulated a symmetric dual for multiobjective fractional programs in the dynamic case and obtained duality relations under so-called partial invexity. In this article we introduce a symmetric dual for multiobjective fractional variational problems which is different from the one proposed in [4]. Under invexity assumptions we establish weak, strong and converse duality as well as self-duality relations. As in [4], we work with properly efficient solutions in the strong and converse duality theorems. The weak duality theorem involves efficient solutions. The proof of the weak duality theorem differs from the one in [4]. Furthermore the strong and converse duality theorems are derived in a more direct way in the present paper than in [4]. In the strong and converse duality theorems two assumptions are introduced as in [4]. While assumption (I) is similar to assumption (I) in [4], assumption (II) below is quite different from (II) in [4]. In the present article (II) is a linear independence assumption. Such a condition seems natural since it is in keeping with earlier studies in the static and dynamic case. In fact our strong and converse duality theorems can be viewed as an extension of the results for the static case obtained by Weir in [21] to the dynamic case. The article is organized as follows. In Section 2 we introduce the multiobjective fractional variational problem and its proposed symmetric dual as well as invexity for such problems. Section 3 contains the weak, strong and converse duality theorems as well as a self-duality theorem. Finally, in Section 4 we specialize these results to the static case. Among others, we reobtain the symmetric duality relations (i.e., weak and strong duality) derived earlier by Weir in [21].
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2. Notation and definitions The following notation will be used for vectors in R n : x
⇔
xi < yi ,
i = 1, 2, . . . , n,
x y
⇔
xi yi ,
i = 1, 2, . . . , n,
x y
⇔
xi yi ,
i = 1, 2, . . . , n,
x y
is the negation of x y.
but x = y,
Let I = [a, b] be a real interval, f : I × R n × R n × R m × R m → R k and g : I × R n × × R m × R m → R k . Consider the vector-valued function f (t, x, x , y, y ), where t ∈ I , x and y are functions of t with x(t) ∈ R n and y(t) ∈ R m , and x and y denote the derivatives of x and y, respectively, with respect to t. Assume that f has continuous fourth-order partial derivatives with respect to x, x , y and y . fx and fx denote the k × n matrices of first-order partial derivatives with respect to x and x ; i.e., ∂fi ∂fi ∂fi ∂fi fix = and fix = , i = 1, 2, . . . , k. ,..., , . . . , ∂x1 ∂xn ∂x1 ∂xn Rn
Similarly, fy and fy denote the k × m matrices of first-order partial derivatives with respect to y and y . Consider the following multiobjective fractional variational problem: b f (t, x(t), x (t)) dt (FVP) Minimize ab a g(t, x(t), x (t)) dt b b fk (t, x(t), x (t)) dt a f1 (t, x(t), x (t)) dt = b , . . . , ab a g1 (t, x(t), x (t)) dt a gk (t, x(t), x (t)) dt subject to
x(a) = α, x(b) = β, h t, x(t), x (t) 0,
where h : I × R n × R n → R l . Assume that gi (t, x, x ) > 0 and fi (t, x, x ) 0 for all i = 1, 2, . . . , k. Let X denote the set of all feasible solutions of (FVP). Definition 2.1. A point x ∗ ∈ X is said to be an efficient (Pareto optimal) solution of (FVP) if for all x ∈ X, b b ∗ ∗ a f (t, x, x ) dt a f (t, x , x ) dt . b b ) dt ∗ , x ∗ ) dt g(t, x, x g(t, x a a Definition 2.2. A point x ∗ ∈ X is said to be a properly efficient solution of (FVP) if it is efficient for (FVP) and if there exists a scalar M > 0 such that, for all i ∈ {1, 2, . . ., k},
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b ∗ ∗ fi (t, x(t), x (t)) dt a fi (t, x (t), x (t)) dt − ab b ∗ ∗ a gi (t, x (t), x (t)) dt a gi (t, x(t), x (t)) dt b b ∗ ∗ a fj (t, x(t), x (t)) dt a fj (t, x (t), x (t)) dt M b − b ∗ ∗ a gj (t, x(t), x (t)) dt a gj (t, x (t), x (t)) dt
b
for some j , such that b b ∗ ∗ a fj (t, x(t), x (t)) dt a fj (t, x (t), x (t)) dt > b b ∗ ∗ a gj (t, x(t), x (t)) dt a gj (t, x (t), x (t)) dt whenever x ∈ X, and b b fi (t, x ∗ (t), x ∗ (t)) dt a fi (t, x(t), x (t)) dt < ab . b ∗ ∗ a gi (t, x(t), x (t)) dt a gi (t, x (t), x (t)) dt Definition 2.3. A point x ∗ ∈ X is said to be a weakly efficient solution if there exists no other feasible point x for which b b ∗ ∗ f (t, x(t), x (t)) dt a f (t, x (t), x (t)) dt > ab . b ∗ ∗ a g(t, x (t), x (t)) dt a g(t, x(t), x (t)) dt Now we define invexity as follows. b b b Definition 2.4. The vector of functionals a f = ( a f1 , . . . , a fk ) is said to be invex in x and x if for each y : [a, b] → R m , with y piecewise smooth, there exists a function η : [a, b] × R n × R n × R n × R n → R n such that ∀i = 1, 2, . . . , k, b
fi (t, x, x , y, y ) − fi (t, u, u , y, y ) dt
a
b
d η(t, x, x , u, u )T fix (t, u, u , y, y ) − fix (t, u, u , y, y ) dt dt
a
for all x : [a, b] → R n , u : [a, b] → R n , where (x (t), u (t)) is piecewise smooth on [a, b]. b Definition 2.5. The vector of functionals − a f is said to be invex in y and y if for each x : [a, b] → R n , with x piecewise smooth, there exists a function ξ : [a, b] × R m × R m × R m × R m → R m such that ∀i = 1, 2, . . . , k, b −
fi (t, x, x , v, v ) − fi (t, x, x , y, y ) dt
a
b
−
d ξ(t, v, v , y, y ) fiy (t, x, x , y, y ) − fiy (t, x, x , y, y ) dt dt
T
a
for all v : [a, b] → R m , y : [a, b] → R m , where (v (t), y (t)) is piecewise smooth on [a, b].
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In the sequel, we will write η(x, u) for η(t, x, x , u, u ) and ξ(v, y) for ξ(t, v, v , y, y ). We consider the problem of finding functions x : [a, b] → R n and y : [a, b] → R m , where (x (t), y (t)) is piecewise smooth on [a, b], to solve the following pair of symmetric dual multiobjective fractional variational problems introduced below. b (MFP)
Minimize
subject to
a f (t, x(t), x (t), y(t), y (t)) dt b a g(t, x(t), x (t), y(t), y (t)) dt b f1 (t, x(t), x (t), y(t), y (t)) dt ,..., = ab (t), y(t), y (t)) dt g (t, x(t), x 1 a b a fk (t, x(t), x (t), y(t), y (t)) dt b a gk (t, x(t), x (t), y(t), y (t)) dt
x(a) = 0 = x(b), y(a) = 0 = y(b), x (a) = 0 = x (b), y (a) = 0 = y (b), k ωi (fiy − Dfiy )Gi (x, y) − (giy − Dgiy )Fi (x, y) 0, i=1
b
y(t)T
k ωi (fiy − Dfiy )Gi (x, y)
− (giy − Dgiy )Fi (x, y) dt 0, ωT e = 1, t ∈ I,
i=1
a
ω > 0, b f (t, u(t), u (t), v(t), v (t)) dt (MFD) Maximize ab a g(t, u(t), u (t), v(t), v (t)) dt b f1 (t, u(t), u (t), v(t), v (t)) dt ,..., = ab a g1 (t, u(t), u (t), v(t), v (t)) dt b a fk (t, u(t), u (t), v(t), v (t)) dt b a gk (t, u(t), u (t), v(t), v (t)) dt subject to
u(a) = 0 = u(b), v(a) = 0 = v(b), u (a) = 0 = u (b), v (a) = 0 = v (b), k ωi (fiu − Dfiu )Gi (u, v) − (giu − Dgiu )Fi (u, v) 0, i=1
b u(t)T a
ω > 0,
k ωi (fiu − Dfiu )Gi (u, v)
− (giu − Dgiu )Fi (u, v) dt 0, ωT e = 1, t ∈ I.
i=1
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We make the following assumptions: (i) f and g are twice continuously differentiable with respect to (x, x ) and (y, y ), x : I → R n and y : I → R m are piecewise twice continuously differentiable; let fx = fx t, x(t), x (t), y(t), y (t) , fx = fx t, x(t), x (t), y(t), y (t) , etc.; b b (ii) Fi = a fi (t, x(t), x (t), y(t), y (t)) dt and − a gi (t, x(t), x (t), y(t), y (t)) dt are b b invex in x and x , and − a fi (t, x(t), x (t), y(t), y (t)) dt and Gi = a gi (t, x(t), x (t), y(t), y (t)) dt are invex in y and y ; (iii) In the above problems (MFP) and (MFD), the numerators are nonnegative and denominators are positive; the differential operator D is given by t y = Dx
⇔
x(t) = α +
y(s) ds, a
and x(a) = α, x(b) = β are given boundary values; thus D = d/dt except at discontinuities. All the above statements for Fi and Gi will be assumed to hold for subsequent results. It is to be noted here that Dfiy = fiy t + fiy y y + fiy y y + fiy x x + fiy x x and consequently ∂ ∂ Dfiy = Dfiy y , Dfiy = Dfiy y + fiy y , ∂y ∂y ∂ ∂ Dfiy = Dfiy x , Dfiy = Dfiy x + fiy x , ∂x ∂x In order to simplify the notation we introduce b f (t, x, x , y, y ) dt F (x, y) p= = ab G(x, y) g(t, x, x , y, y ) dt
∂ Dfiy = fiy y , ∂y ∂ Dfiy = fiy x . ∂x
a
and
b f (t, u, u , v, v ) dt F (u, v) = ab q= G(u, v) a g(t, u, u , v, v ) dt
and express problems (MFP) and (MFD) equivalently as (MFP ) Minimize p = (p1 , . . . , pk )T subject to x(a) = 0 = x(b),
(1) y(a) = 0 = y(b),
(2)
D.S. Kim et al. / J. Math. Anal. Appl. 289 (2004) 505–521
x (a) = 0 = x (b), y (a) = 0 = y (b), b b fi (t, x, x , y, y ) dt − pi gi (t, x, x , y, y ) dt = 0, a
511
(3) (4)
a
k
ωi (fiy − Dfiy ) − pi (giy − Dgiy ) 0,
(5)
i=1
b y(t)T
k
ωi (fiy − Dfiy ) − pi (giy − Dgiy ) dt 0,
a
i=1
ω > 0,
ωT e = 1,
t ∈ I,
(6) (7)
(MFD ) Maximize q = (q1 , . . . , qk )T subject to u(a) = 0 = u(b),
(8) v(a) = 0 = v(b),
(9)
u (a) = 0 = u (b), v (a) = 0 = v (b), b b fi (t, u, u , v, v ) dt − qi gi (t, u, u , v, v ) dt = 0, a
(10) (11)
a
k
ωi (fiu − Dfiu ) − qi (giu − Dgiu ) 0,
(12)
i=1
b u(t)T
k
ωi (fiu − Dfiu ) − qi (giu − Dgiu ) dt 0,
a
i=1
ω > 0,
ωT e = 1,
t ∈ I.
In the above problems (MFP ) and (MFD ), it is to be noted that p
(13) (14)
and q are nonnegative
as a consequence of the assumption (iii).
3. Duality theorems In this section, we state duality theorems for problems (MFP ) and (MFD ) which lead to corresponding relations between (MFP) and (MFD). We establish weak, strong and converse duality as well as self-duality relations between (MFP ) and (MFD ). The proof of the weak duality relation is different from the proof of the corresponding result in [4]. Theorem 3.1 (Weak duality). Let (x(t), y(t), p, ω) be feasible for (MFP ) and let (u(t), b b v(t), q, ω) be feasible for (MFD ). Assume that a fi and − a gi are invex in x and x , and b b − a fi and a gi are invex in y and y , with η(x, u) + u(t) 0 and ξ(v, y) + y(t) 0 for all t ∈ I (except possibly at corners of (x (t), y (t)) or (u (t), v (t))). Then one has p q.
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D.S. Kim et al. / J. Math. Anal. Appl. 289 (2004) 505–521
Proof. The invexity assumptions of 1, 2, . . . , k, are invex. Then we have b
b a
fi and −
b a
gi imply that
b a
(fi − qi gi ), i =
fi (t, x, x , v, v ) − qi gi (t, x, x , v, v ) dt
a
b −
fi (t, u, u , v, v ) − qi gi (t, u, u , v, v ) dt
a
b
η(x, u)T
fix (t, u, u , v, v ) − qi gix (t, u, u , v, v )
a
− D fix (t, u, u , v, v ) − qi gix (t, u, u , v, v ) dt b =
η(x, u)T
fix (t, u, u , v, v ) − Dfix (t, u, u , v, v )
a
− qi gix (t, u, u , v, v ) − Dgix (t, u, u , v, v ) dt. From (7), (12) and (13) with η(x, u) + u(t) 0, we obtain k
b ωi
i=1
fi (t, x, x , v, v ) − qi gi (t, x, x , v, v ) dt
a k
b ωi
i=1
fi (t, u, u , v, v ) − qi gi (t, u, u , v, v ) dt.
a
The invexity assumptions of − are invex. Then we have b
b a
fi and
b a
gi imply that −
b a
(fi − pi gi ), i = 1, 2, . . . , k,
fi (t, x, x , v, v ) − pi gi (t, x, x , v, v ) dt
a
b −
fi (t, x, x , y, y ) − pi gi (t, x, x , y, y ) dt
a
b
(15)
ξ(v, y)T
fiy (t, x, x , y, y ) − Dfiy (t, x, x , y, y )
a
− pi giy (t, x, x , y, y ) − Dgiy (t, x, x , y, y ) dt. From (5), (6) and (14) with ξ(v, y) + y(t) 0, we obtain
D.S. Kim et al. / J. Math. Anal. Appl. 289 (2004) 505–521 k
b ωi
i=1
513
fi (t, x, x , v, v ) − pi gi (t, x, x , v, v ) dt
a
k
b ωi
i=1
fi (t, x, x , y, y ) − pi gi (t, x, x , y, y ) dt.
(16)
a
Combining (15) and (16) along with (4) and (11) gives k
b ωi (pi − qi )
i=1
gi (t, x, x , v, v ) dt 0.
If, for some i, pi < qi and for all j = i, pj qj , then 1, 2, . . . , k, implies that k
(17)
a
b ωi (pi − qi )
i=1
b a
gi (t, x, x , v, v ) dt > 0, i =
gi (t, x, x , v, v ) dt < 0,
a
which contradicts (17). Hence p q.
✷
We now present proofs of a strong and a converse duality theorem which are more direct than the corresponding ones in [4]. As to the assumptions of the strong and converse duality theorem, our condition (I) is similar to (I) in [4], while our condition (II) is quite different from (II) in [4]. Moreover we use as condition (II) a linear independence assumption in keeping with earlier studies in the static and dynamic case. With that the duality relations become a natural extension of earlier results in [21] for the static case. We mention that symmetric duality results for the static case which would lead to the results in the dynamic case in [4] are not known, at least to the authors. Theorem 3.2 (Strong duality). Let (x0 (t), y0 (t), p0 , ω0 ) be a properly efficient solution for (MFP ). Fix ω = ω0 in (MFD ) and define b fi (t, x0 , x0 , y0 , y0 ) dt p0i = ab , i = 1, 2, . . . , k. a gi (t, x0 , x0 , y0 , y0 ) dt Assume that (I)
k i=1
b Ψ (t)T
ω0i
(fiyy − p0i giyy ) − D(fiy y − p0i giy y )
a
− D (fiyy − Dfiy y − fiy y ) − p0i (giyy − Dgiy y − giy y ) + D 2 −(fiy y − p0i giy y ) Ψ (t) dt = 0
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implies Ψ (t) = 0, for all t ∈ I , and b (f1y − p01 g1y ) dt, . . . ,
(II) a
b (fky − p0k gky ) dt a
is linearly independent. If the invexity conditions of Theorem 3.1 are satisfied, then (x0 (t), y0 (t), p0 , ω0 ) is properly efficient for (MFD ). Proof. Since (x0 (t), y0 (t), p0 , ω0 ) is a properly efficient solution of (MFP ), it is also a weakly efficient solution. Hence there exist a ∈ R k , b ∈ R k , s ∈ R, t ∈ R k , piecewise smooth γ : I → R m and z : I → R n such that ai − bi gi − ω0i (giy − Dgiy )T γ (t) − sy0 (t) = 0, i = 1, 2, . . . , k, (18) k bi (fix − p0i gix ) − D(fix − p0i gix ) i=1
+
k
ω0i (fiyx − p0i giyx ) − D(fiy x − p0i giy x ) γ (t) − sy0 (t) − z(t)
i=1
−D
k
ω0i (fiyx − Dfiy x − fiy x )
i=1
− p0i (giy x − Dgiy x − giy x ) γ (t) − sy0 (t) +D
−
2
k
ω0i (fiy x − p0i giy x ) γ (t) − sy0 (t) = 0,
(19)
i=1 k (bi − sω0i ) (fiy − p0i giy ) − D(fiy − p0i giy ) i=1
+
k
ω0i (fiyy − p0i giyy ) − D(fiy y − p0i giy y ) γ (t) − sy0 (t)
i=1
−D
k
ω0i (fiyy − Dfiy y − fiy y )
i=1
− p0i (giyy − Dgiy y − giy y ) γ (t) − sy0 (t) +D
2
−
k
ω0i (fiy y − p0i giy y ) γ (t) − sy0 (t) = 0,
(20)
i=1 T
γ (t) − sy0 (t)
i = 1, 2, . . . , k,
(fiy − Dfiy ) − p0i (giy − Dgiy ) − ti = 0, (21)
D.S. Kim et al. / J. Math. Anal. Appl. 289 (2004) 505–521 k
515
b (fi − p0i gi ) dt = 0,
bi
i=1
(22)
a
γ (t)T
k
ω0i (fiy − Dfiy ) − p0i (giy − Dgiy ) = 0,
(23)
i=1
sy(t)
T
k
b ω0i
i=1
(fiy − Dfiy ) − p0i (giy − Dgiy ) dt = 0,
(24)
a
t T ω0 = 0,
(25)
z(t) x0 (t) = 0, a, γ (t), s, t, z(t) 0, a, γ (t), s, t, z(t) = 0.
(26)
T
(27) (28)
Since ω0 > 0 and t 0, we have t = 0. Multiplying (20) by (γ (t) − sy0 (t))T and applying (21) gives k
b ω0i
i=1
γ (t) − sy0 (t)
T
(fiyy − p0i giyy ) − D(fiy y − p0i giy y )
a
− D (fiyy − Dfiy y − fiy y ) − p0i (giyy − Dgiy y − giy y ) + D 2 −(fiy y − p0i giy y ) γ (t) − sy0 (t) dt = 0,
which by virtue of the hypothesis (I) yields γ (t) − sy0 (t) = 0 for all t ∈ I.
(29)
From (20) along with (29), we obtain k
b (bi − sω0i )
i=1
(fiy − p0i giy ) − D(fiy − p0i giy ) dt = 0.
a
By the hypothesis (II), b = sω0 .
(30)
If s = 0, then b = 0. From (18) we get a = 0, from (29) γ (t) = 0 and from (19) z(t) = 0. This combined with t = 0 contradicts (28). Hence s > 0 and b > 0. From (27) and (29), we get y0 (t) 0
for all t ∈ I, as s > 0.
(31)
From (19), (27) and (30), we obtain k i=1
ω0i (fix − p0i gix ) − D(fix − p0i gix ) 0.
(32)
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D.S. Kim et al. / J. Math. Anal. Appl. 289 (2004) 505–521
From (19), (26), (27) and (30), it also follows that k
b ω0i
i=1
x0 (t)T (fix − p0i gix ) − D(fix − p0i gix ) dt = 0.
(33)
a
Equation (22) with b > 0 implies b
fi (t, x0 , x0 y0 , y0 ) − p0i gi (t, x0 , x0 y0 , y0 ) dt = 0.
(34)
a
Thus the relations (31)–(34) imply that (x0 (t), y0 (t), p0 , ω0 ) is feasible for (MFD ) and the objective values of (MFP ) and (MFD ) are equal there. Clearly, (x0 , y0 , q0 , ω0 ) is efficient for (MFD ). If (x0 , y0 , q0 , ω0 ) is not properly efficient, then for some feasible (ui , vi , q0 , ω0 ) with b fi (ui , ui , vi , vi ) dt q0i = ab , i = 1, 2, . . . , k, a gi (ui , ui , vi , vi ) dt and for some i, q0i − p0i > M for any M > 0. b Since a gi (t, x, x , v, v ) dt, i = 1, 2, . . . , k, is bounded, it follows that k
b ω0i (q0i − p0i )
i=1
gi (t, x, x , v, v ) dt < 0,
a
which contradicts weak duality; see (17). Thus (x0 , y0 , q0 , ω0 ) is properly efficient for (MFD ).
✷
A converse duality theorem is stated now. The proof is analogous to that of Theorem 3.2. Theorem 3.3 (Converse duality). Let (x0 (t), y0 (t), p0 , ω0 ) be a properly efficient solution for (MFD ). Fix ω = ω0 in (MFP ) and define b fi (t, x0 , x0 , y0 , y0 ) dt p0i = ab , i = 1, 2, . . . , k. a gi (t, x0 , x0 , y0 , y0 ) dt Assume that (I)
k i=1
b Ψ (t)T
ω0i
(fixx − p0i gixx ) − D(fix x − p0i gix x )
a
− D (fiyy − Dfiy y − fiy y ) − p0i (giyy − Dgiy y − giy y ) + D 2 −(fix x − p0i gix x ) Ψ (t) dt = 0 implies Ψ (t) = 0 for all t ∈ I , and b (f1x − p01 g1x ) dt, . . . ,
(II) a
b (fkx − p0k gkx ) dt a
D.S. Kim et al. / J. Math. Anal. Appl. 289 (2004) 505–521
517
is linearly independent. If the invexity conditions of Theorem 3.1 are satisfied, then (x0 (t), y0 (t), p0 , ω0 ) is properly efficient for (MFP ). Remark 3.1. Our weak, strong and converse duality theorems can also be obtained under b b pseudo-invexity assumptions on ki=1 ωi a (fi − qi gi ) in x and x and − ki=1 ωi a (fi − pi gi ) in y and y . In order to present a better view of the concept of self-duality, we will consider here problems (MFP) and (MFD) instead of their equivalents (MFP ) and (MFD ). Assume that x(t) and y(t) have the same dimension. The function f (t, x(t), x (t), y(t), y (t)) is said to be skew-symmetric if f (t, x, x , y, y ) = −f (t, y, y , x, x ) for all x(t) and y(t) in the domain of f and the function g(t, x, x , y, y ) will be called symmetric if g(t, x, x , y, y ) = g(t, y, y , x, x ) for all x(t) and y(t) in the domain of g. Theorem 3.4 (Self-duality). If f (t, x, x , y, y ) is skew-symmetric and g(t, x, x , y, y ) is symmetric, then (MFP) and (MFD) are self-dual. If (MFP) and (MFD) are dual problems, then with (x0 (t), y0 (t), p0 , ω0 ) also (y0 (t), x0 (t), p0 , ω0 ) is a joint optimal solution and the common optimal value is 0. Proof. With f skew-symmetric, we have fx (t, x, x , y, y ) = −fy (t, y, y , x, x ), fy (t, x, x , y, y ) = −fx (t, y, y , x, x ), fx (t, x, x , y, y ) = −fy (t, y, y , x, x ), fy (t, x, x , y, y ) = −fx (t, y, y , x, x ), and with g symmetric, we have gx (t, x, x , y, y ) = gy (t, y, y , x, x ), gy (t, x, x , y, y ) = gx (t, y, y , x, x ), gx (t, x, x , y, y ) = gy (t, y, y , x, x ), gy (t, x, x , y, y ) = gx (t, y, y , x, x ). Expressing the dual problem (MFD) as a minimization problem and making use of the above relations, we have b f (t, v, v , u, u ) dt Minimize ab a g(t, v, v , u, u ) dt subject to u(a) = 0 = u(b),
u (a) = 0 = u (b), k i=1
v(a) = 0 = v(b), v (a) = 0 = v (b),
ωi ((fiv − Dfiv ))Gi (v, u) − ((giv − Dgiv ))Fi (v, u) 0,
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b u(t)T
k
ωi ((fiv − Dfiv ))Gi (v, u)
a
i=1
ω > 0,
ωT e = 1,
− ((giv − Dgiv ))Fi (v, u) dt 0, t ∈ I,
which is just the primal problem (MFP). Thus if (x0 (t), y0 (t), p0 , ω0 ) is an optimal solution of (MFD), then (y0 (t), x0 (t), p0 , ω0 ) is an optimal solution of (MFD). Since f is skew-symmetric and g is symmetric, respectively, we have b b f (t, x0 (t), x0 (t), y0 (t), y0 (t)) dt a f (t, y0 (t), y0 (t), x0 (t), x0 (t)) dt = − ab . b a g(t, y0 (t), y0 (t), x0 (t), x0 (t)) dt a g(t, x0 (t), x0 (t), y0 (t), y0 (t)) dt Hence b a
b a
and so
g(t, x0 (t), x0 (t), y0 (t), y0 (t)) dt b b f (t, x0 (t), x0 (t), y0 (t), y0 (t)) dt a f (t, y0 (t), y0 (t), x0 (t), x0 (t)) dt = − ab , = b a g(t, y0 (t), y0 (t), x0 (t), x0 (t)) dt a g(t, x0 (t), x0 (t), y0 (t), y0 (t)) dt
b a
b a
f (t, x0 (t), x0 (t), y0 (t), y0 (t)) dt
f (t, x0 (t), x0 (t), y0 (t), y0 (t)) dt
g(t, x0 (t), x0 (t), y0 (t), y0 (t)) dt b f (t, y0 (t), y0 (t), x0 (t), x0 (t)) dt = 0. = ab a g(t, y0 (t), y0 (t), x0 (t), x0 (t)) dt
✷
4. The static case If the time dependence of programs (MFP) and (MFD) is removed and f and g are considered to have domain R n × R m , we obtain the symmetric dual fractional pair given by f (x, y) f1 (x, y) fk (x, y) (SMFP) Minimize = ,..., g(x, y) g1 (x, y) gk (x, y) subject to
k
ωi {gi fiy − fi giy } 0,
i=1
yT
k
ωi {gi fiy − fi giy } 0,
i=1
ω > 0,
ωT e = 1,
D.S. Kim et al. / J. Math. Anal. Appl. 289 (2004) 505–521
(SMFD) Maximize subject to
f (u, v) = g(u, v) k
f1 (u, v) fk (u, v) ,..., g1 (u, v) gk (u, v)
519
ωi {gi fiu − fi giu } 0,
i=1
uT
k
ωi {gi fiu − fi giu } 0,
i=1
ω > 0,
ωT e = 1.
The following duality theorems are specializations of Theorems 3.1–3.3. Theorem 4.1 (Weak duality). Let (x, y, p, ω) be feasible for (SMFP) and let (u, v, q, ω) be feasible for (SMFD). Assume that fi and −gi are invex in x, and −fi and gi are invex in y, with η(x, u) + u 0 and ξ(v, y) + y 0. Then f (x, y) f (u, v) . g(x, y) g(u, v) Theorem 4.2 (Strong duality). Let (x0 , y0 , p0 , ω0 ) be a properly efficient solution for (SMFP). Fix ω = ω0 in (SMFD) and define fi (x0 , y0 ) , gi (x0 , y0 )
p0i =
i = 1, 2, . . . , k.
Assume that k
(I)
ω0i (fiyy − p0i giyy ) is positive or negative definite
i=1
and (II)
(f1y − p01 g1y ), . . . , (fky − p0k gky ) is linearly independent.
If the invexity conditions of Theorem 4.1 are satisfied, then (x0 , y0 , p0 , ω0 ) is properly efficient for (SMFD). Hence the weak and strong duality results in Section 3 reduce to the symmetric duality relations in the static case obtained by Weir in [21]. Theorem 4.3 (Converse duality). Let (x0 , y0 , p0 , ω0 ) be a properly efficient solution for (SMFD). Fix ω = ω0 in (SMFP) and define p0i =
fi (x0 , y0 ) , gi (x0 , y0 )
i = 1, 2, . . . , k.
Assume that (I)
k i=1
ω0i (fixx − p0i gixx ) is positive or negative definite
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and (II)
(f1u − p01 g1u ), . . . , (fku − p0k gku ) is linearly independent.
If the invexity conditions of Theorem 4.1 are satisfied, then (x0 , y0 , p0 , ω0 ) is properly efficient for (SMFP). The pair (SMFP) and (SMFD) will be self-dual if m = n and f is skew-symmetric (i.e., f (x, y) = −f (y, x)), and g is symmetric (i.e., g(x, y) = g(y, x)). Finally we obtain through specialization of Theorem 3.4. Theorem 4.4 (Self-duality). If f (x, y) is skew-symmetric and g(x, y) is symmetric, then (SMFP) and (SMFD) are self-dual. If (SMFP) and (SMFD) are dual problems, then with (x0 , y0 , p0 , ω0 ) also (y0 , x0 , p0 , ω0 ) is a joint optimal solution and the common optimal value is 0.
References [1] A. Ben-Israel, B. Mond, What is invexity? J. Austral. Math. Soc. Ser. B 28 (1986) 1–9. [2] S. Chandra, B.D. Craven, B. Mond, Symmetric dual fractional programming, Z. Oper. Res. 29 (1985) 59–64. [3] S. Chandra, I. Husain, Symmetric dual continuous fractional programming, J. Inform. Optim. Sci. 10 (1989) 241–255. [4] X.H. Chen, Symmetric duality for the multiobjective fractional variational problems with partial invexity, J. Math. Anal. Appl. 245 (2000) 105–123. [5] G.B. Dantzig, E. Eisenberg, R.W. Cottle, Symmetric dual nonlinear programs, Pacific J. Math 15 (1965) 809–812. [6] W.S. Dorn, A symmetric dual theorem for quadratic programs, J. Oper. Res. Soc. Japan 2 (1960) 93–97. [7] M.A. Hanson, On sufficiency of the Kuhn–Tucker conditions, J. Math. Anal. Appl. 80 (1981) 545–550. [8] D.S. Kim, G.M. Lee, Symmetric duality with pseudo-invexity in variational problems, Optimization 28 (1993) 9–16. [9] D.S. Kim, G.M. Lee, W.J. Lee, Symmetric duality for multiobjective variational problems with pseudoinvexity, in: Nonlinear Analysis and Convex Analysis, in: RIMS Kokyuroku, Vol. 985, RIMS of Kyoto University, Kyoto, 1997, pp. 106–117. [10] D.S. Kim, W.J. Lee, Symmetric duality for multiobjective variational problems with invexity, J. Math. Anal. Appl. 218 (1998) 34–48. [11] D.S. Kim, W.J. Lee, Generalized symmetric duality for multiobjective variational problems with invexity, J. Math. Anal. Appl. 234 (1999) 147–164. [12] O.L. Mangasarian, Nonlinear Programming, McGraw–Hill, New York, 1969. [13] B. Mond, M.A. Hanson, Symmetric duality for variational problems, J. Math. Anal. Appl. 23 (1968) 161– 172. [14] B. Mond, I. Husain, Sufficient optimality criteria and duality for variational problem with generalized invexity, J. Austral. Math. Soc. Ser. B 31 (1989) 108–121. [15] 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. [16] R.N. Mukherjee, S.K. Mishra, Sufficient optimality criteria and duality for multiobjective variational problems with V -invexity, Indian J. Pure Appl. Math 25 (1994) 801–813. [17] S. Schaible, Duality in fractional programming: a unified approach, Oper. Res. 24 (1976) 452–461. [18] S. Schaible, Fractional programming: I, Duality, Management Sci. 22 (1976) 858–867. [19] S. Schaible, Fractional Programming, in: R. Horst, P.M. Pardalos (Eds.), Handbook of Global Optimization, Kluwer Academic, Dordrecht, 1995, pp. 495–608.
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[20] I. Smart, B. Mond, Symmetric duality with invexity in variational problems, J. Math. Anal. Appl. 152 (1990) 536–545. [21] T. Weir, Symmetric dual multiobjective fractional programming, J. Austral. Math. Soc. Ser. A 50 (1991) 67–74. [22] T. Weir, B. Mond, Symmetric and self duality in multiple objective programming, Asia-Pacific J. Oper. Res. 5 (1988) 124–133. [23] X.M. Yang, S.Y. Wang, X.T. Deng, Symmetric duality for a class of multiobjective fractional programming problems, J. Math. Anal. Appl. 274 (2002) 279–295.
Symmetric duality for invex multiobjective fractional variational problems Do Sang Kim,a,1 Won Jung Lee,a and Siegfried Schaible b,∗ a Department of Applied Mathematics, Pukyong National University, Pusan 608-737, South Korea b A.G. Anderson Graduate School of Management, University of California, Riverside, CA 92521, USA
Received 10 October 2002 Submitted by J.P. Dauer
Abstract We introduce a symmetric dual for multiobjective fractional variational problems. Under certain invexity assumptions we establish weak, strong and converse duality theorems as well as self-duality relations. The paper includes an extension of previous symmetric duality results for the static case obtained by Weir to the dynamic case. 2003 Elsevier Inc. All rights reserved.
1. Introduction The classical dual in linear programming is symmetric in the sense that the dual of the dual is the original linear program. Such symmetry is usually not found in duality concepts for nonlinear programming, not even in quadratic programming [12]. In [6] Dorn introduced a different dual for quadratic programming which is symmetric. Extending these results to general convex programming, Dantzig et al. [5] formulated a symmetric dual and established weak and strong duality relations. Symmetric duality results under generalized convexity were given by Mond and Weir in [15] for new types of a dual. Then in [22] Weir and Mond introduced two distinct symmetric duals for multiobjective programs. Under additional assumptions the multiobjective programs are shown to be self-dual. Very recently * Corresponding author.
E-mail address: [email protected] (S. Schaible). 1 This author was supported by the Korea Research Foundation Grant KRF-2001-013-D00007. He gratefully
acknowledges the support of his visit by the A.G. Anderson Graduate School of Management during 2002. 0022-247X/$ – see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2003.08.035
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Yang et al. [23] introduced a symmetric dual for a class of nondifferentiable multiobjective fractional programs and obtained weak and strong duality relations. In [13] Mond and Hanson first extended the symmetric duality results of [5] to variational problems by introducing continuous analogues of the earlier concepts. Since Hanson defined invexity in [7] as a new generalization of convexity, several authors have introduced concepts of invexity and generalized invexity for use in continuous programming; e.g., [1, 10,14,16,20]. Smart and Mond [20] extended symmetric duality results to variational problems by employing a continuous version of invexity. Recently, Kim and Lee [8] presented a symmetric dual in the sense of a dual proposed by Mond and Weir [15], not in the sense of Wolfe’s dual as in [20], establishing duality relations for variational problems hereby assuming pseudo-invexity. Subsequently, Kim et al. [9] extended the results in [8] to the multiobjective case. More recently Kim and Lee [10,11] formulated a symmetric and a generalized symmetric dual for multiobjective variational problems. Weak, strong and converse duality relations are obtained under suitable invexity assumptions. In this article we focus on symmetric duality for fractional programming. Since duality results for convex programming do not apply to fractional programs in general [17], duality concepts for such nonconvex programs had to be defined separately; e.g., [17,18]. Most duals in fractional programming are not symmetric [17–19]. Chandra et al. [2] first introduced a symmetric dual in nonlinear fractional programming. Chandra and Husain [3] then extended these results to continuous nonlinear fractional programming obtaining symmetric duality and self-duality relations. For the multiobjective case of a static nonlinear fractional program symmetric duality was introduced by Weir [21], and weak and strong duality relations were derived under convexity assumptions. Very recently Chen [4] formulated a symmetric dual for multiobjective fractional programs in the dynamic case and obtained duality relations under so-called partial invexity. In this article we introduce a symmetric dual for multiobjective fractional variational problems which is different from the one proposed in [4]. Under invexity assumptions we establish weak, strong and converse duality as well as self-duality relations. As in [4], we work with properly efficient solutions in the strong and converse duality theorems. The weak duality theorem involves efficient solutions. The proof of the weak duality theorem differs from the one in [4]. Furthermore the strong and converse duality theorems are derived in a more direct way in the present paper than in [4]. In the strong and converse duality theorems two assumptions are introduced as in [4]. While assumption (I) is similar to assumption (I) in [4], assumption (II) below is quite different from (II) in [4]. In the present article (II) is a linear independence assumption. Such a condition seems natural since it is in keeping with earlier studies in the static and dynamic case. In fact our strong and converse duality theorems can be viewed as an extension of the results for the static case obtained by Weir in [21] to the dynamic case. The article is organized as follows. In Section 2 we introduce the multiobjective fractional variational problem and its proposed symmetric dual as well as invexity for such problems. Section 3 contains the weak, strong and converse duality theorems as well as a self-duality theorem. Finally, in Section 4 we specialize these results to the static case. Among others, we reobtain the symmetric duality relations (i.e., weak and strong duality) derived earlier by Weir in [21].
D.S. Kim et al. / J. Math. Anal. Appl. 289 (2004) 505–521
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2. Notation and definitions The following notation will be used for vectors in R n : x
⇔
xi < yi ,
i = 1, 2, . . . , n,
x y
⇔
xi yi ,
i = 1, 2, . . . , n,
x y
⇔
xi yi ,
i = 1, 2, . . . , n,
x y
is the negation of x y.
but x = y,
Let I = [a, b] be a real interval, f : I × R n × R n × R m × R m → R k and g : I × R n × × R m × R m → R k . Consider the vector-valued function f (t, x, x , y, y ), where t ∈ I , x and y are functions of t with x(t) ∈ R n and y(t) ∈ R m , and x and y denote the derivatives of x and y, respectively, with respect to t. Assume that f has continuous fourth-order partial derivatives with respect to x, x , y and y . fx and fx denote the k × n matrices of first-order partial derivatives with respect to x and x ; i.e., ∂fi ∂fi ∂fi ∂fi fix = and fix = , i = 1, 2, . . . , k. ,..., , . . . , ∂x1 ∂xn ∂x1 ∂xn Rn
Similarly, fy and fy denote the k × m matrices of first-order partial derivatives with respect to y and y . Consider the following multiobjective fractional variational problem: b f (t, x(t), x (t)) dt (FVP) Minimize ab a g(t, x(t), x (t)) dt b b fk (t, x(t), x (t)) dt a f1 (t, x(t), x (t)) dt = b , . . . , ab a g1 (t, x(t), x (t)) dt a gk (t, x(t), x (t)) dt subject to
x(a) = α, x(b) = β, h t, x(t), x (t) 0,
where h : I × R n × R n → R l . Assume that gi (t, x, x ) > 0 and fi (t, x, x ) 0 for all i = 1, 2, . . . , k. Let X denote the set of all feasible solutions of (FVP). Definition 2.1. A point x ∗ ∈ X is said to be an efficient (Pareto optimal) solution of (FVP) if for all x ∈ X, b b ∗ ∗ a f (t, x, x ) dt a f (t, x , x ) dt . b b ) dt ∗ , x ∗ ) dt g(t, x, x g(t, x a a Definition 2.2. A point x ∗ ∈ X is said to be a properly efficient solution of (FVP) if it is efficient for (FVP) and if there exists a scalar M > 0 such that, for all i ∈ {1, 2, . . ., k},
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D.S. Kim et al. / J. Math. Anal. Appl. 289 (2004) 505–521
b ∗ ∗ fi (t, x(t), x (t)) dt a fi (t, x (t), x (t)) dt − ab b ∗ ∗ a gi (t, x (t), x (t)) dt a gi (t, x(t), x (t)) dt b b ∗ ∗ a fj (t, x(t), x (t)) dt a fj (t, x (t), x (t)) dt M b − b ∗ ∗ a gj (t, x(t), x (t)) dt a gj (t, x (t), x (t)) dt
b
for some j , such that b b ∗ ∗ a fj (t, x(t), x (t)) dt a fj (t, x (t), x (t)) dt > b b ∗ ∗ a gj (t, x(t), x (t)) dt a gj (t, x (t), x (t)) dt whenever x ∈ X, and b b fi (t, x ∗ (t), x ∗ (t)) dt a fi (t, x(t), x (t)) dt < ab . b ∗ ∗ a gi (t, x(t), x (t)) dt a gi (t, x (t), x (t)) dt Definition 2.3. A point x ∗ ∈ X is said to be a weakly efficient solution if there exists no other feasible point x for which b b ∗ ∗ f (t, x(t), x (t)) dt a f (t, x (t), x (t)) dt > ab . b ∗ ∗ a g(t, x (t), x (t)) dt a g(t, x(t), x (t)) dt Now we define invexity as follows. b b b Definition 2.4. The vector of functionals a f = ( a f1 , . . . , a fk ) is said to be invex in x and x if for each y : [a, b] → R m , with y piecewise smooth, there exists a function η : [a, b] × R n × R n × R n × R n → R n such that ∀i = 1, 2, . . . , k, b
fi (t, x, x , y, y ) − fi (t, u, u , y, y ) dt
a
b
d η(t, x, x , u, u )T fix (t, u, u , y, y ) − fix (t, u, u , y, y ) dt dt
a
for all x : [a, b] → R n , u : [a, b] → R n , where (x (t), u (t)) is piecewise smooth on [a, b]. b Definition 2.5. The vector of functionals − a f is said to be invex in y and y if for each x : [a, b] → R n , with x piecewise smooth, there exists a function ξ : [a, b] × R m × R m × R m × R m → R m such that ∀i = 1, 2, . . . , k, b −
fi (t, x, x , v, v ) − fi (t, x, x , y, y ) dt
a
b
−
d ξ(t, v, v , y, y ) fiy (t, x, x , y, y ) − fiy (t, x, x , y, y ) dt dt
T
a
for all v : [a, b] → R m , y : [a, b] → R m , where (v (t), y (t)) is piecewise smooth on [a, b].
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509
In the sequel, we will write η(x, u) for η(t, x, x , u, u ) and ξ(v, y) for ξ(t, v, v , y, y ). We consider the problem of finding functions x : [a, b] → R n and y : [a, b] → R m , where (x (t), y (t)) is piecewise smooth on [a, b], to solve the following pair of symmetric dual multiobjective fractional variational problems introduced below. b (MFP)
Minimize
subject to
a f (t, x(t), x (t), y(t), y (t)) dt b a g(t, x(t), x (t), y(t), y (t)) dt b f1 (t, x(t), x (t), y(t), y (t)) dt ,..., = ab (t), y(t), y (t)) dt g (t, x(t), x 1 a b a fk (t, x(t), x (t), y(t), y (t)) dt b a gk (t, x(t), x (t), y(t), y (t)) dt
x(a) = 0 = x(b), y(a) = 0 = y(b), x (a) = 0 = x (b), y (a) = 0 = y (b), k ωi (fiy − Dfiy )Gi (x, y) − (giy − Dgiy )Fi (x, y) 0, i=1
b
y(t)T
k ωi (fiy − Dfiy )Gi (x, y)
− (giy − Dgiy )Fi (x, y) dt 0, ωT e = 1, t ∈ I,
i=1
a
ω > 0, b f (t, u(t), u (t), v(t), v (t)) dt (MFD) Maximize ab a g(t, u(t), u (t), v(t), v (t)) dt b f1 (t, u(t), u (t), v(t), v (t)) dt ,..., = ab a g1 (t, u(t), u (t), v(t), v (t)) dt b a fk (t, u(t), u (t), v(t), v (t)) dt b a gk (t, u(t), u (t), v(t), v (t)) dt subject to
u(a) = 0 = u(b), v(a) = 0 = v(b), u (a) = 0 = u (b), v (a) = 0 = v (b), k ωi (fiu − Dfiu )Gi (u, v) − (giu − Dgiu )Fi (u, v) 0, i=1
b u(t)T a
ω > 0,
k ωi (fiu − Dfiu )Gi (u, v)
− (giu − Dgiu )Fi (u, v) dt 0, ωT e = 1, t ∈ I.
i=1
510
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We make the following assumptions: (i) f and g are twice continuously differentiable with respect to (x, x ) and (y, y ), x : I → R n and y : I → R m are piecewise twice continuously differentiable; let fx = fx t, x(t), x (t), y(t), y (t) , fx = fx t, x(t), x (t), y(t), y (t) , etc.; b b (ii) Fi = a fi (t, x(t), x (t), y(t), y (t)) dt and − a gi (t, x(t), x (t), y(t), y (t)) dt are b b invex in x and x , and − a fi (t, x(t), x (t), y(t), y (t)) dt and Gi = a gi (t, x(t), x (t), y(t), y (t)) dt are invex in y and y ; (iii) In the above problems (MFP) and (MFD), the numerators are nonnegative and denominators are positive; the differential operator D is given by t y = Dx
⇔
x(t) = α +
y(s) ds, a
and x(a) = α, x(b) = β are given boundary values; thus D = d/dt except at discontinuities. All the above statements for Fi and Gi will be assumed to hold for subsequent results. It is to be noted here that Dfiy = fiy t + fiy y y + fiy y y + fiy x x + fiy x x and consequently ∂ ∂ Dfiy = Dfiy y , Dfiy = Dfiy y + fiy y , ∂y ∂y ∂ ∂ Dfiy = Dfiy x , Dfiy = Dfiy x + fiy x , ∂x ∂x In order to simplify the notation we introduce b f (t, x, x , y, y ) dt F (x, y) p= = ab G(x, y) g(t, x, x , y, y ) dt
∂ Dfiy = fiy y , ∂y ∂ Dfiy = fiy x . ∂x
a
and
b f (t, u, u , v, v ) dt F (u, v) = ab q= G(u, v) a g(t, u, u , v, v ) dt
and express problems (MFP) and (MFD) equivalently as (MFP ) Minimize p = (p1 , . . . , pk )T subject to x(a) = 0 = x(b),
(1) y(a) = 0 = y(b),
(2)
D.S. Kim et al. / J. Math. Anal. Appl. 289 (2004) 505–521
x (a) = 0 = x (b), y (a) = 0 = y (b), b b fi (t, x, x , y, y ) dt − pi gi (t, x, x , y, y ) dt = 0, a
511
(3) (4)
a
k
ωi (fiy − Dfiy ) − pi (giy − Dgiy ) 0,
(5)
i=1
b y(t)T
k
ωi (fiy − Dfiy ) − pi (giy − Dgiy ) dt 0,
a
i=1
ω > 0,
ωT e = 1,
t ∈ I,
(6) (7)
(MFD ) Maximize q = (q1 , . . . , qk )T subject to u(a) = 0 = u(b),
(8) v(a) = 0 = v(b),
(9)
u (a) = 0 = u (b), v (a) = 0 = v (b), b b fi (t, u, u , v, v ) dt − qi gi (t, u, u , v, v ) dt = 0, a
(10) (11)
a
k
ωi (fiu − Dfiu ) − qi (giu − Dgiu ) 0,
(12)
i=1
b u(t)T
k
ωi (fiu − Dfiu ) − qi (giu − Dgiu ) dt 0,
a
i=1
ω > 0,
ωT e = 1,
t ∈ I.
In the above problems (MFP ) and (MFD ), it is to be noted that p
(13) (14)
and q are nonnegative
as a consequence of the assumption (iii).
3. Duality theorems In this section, we state duality theorems for problems (MFP ) and (MFD ) which lead to corresponding relations between (MFP) and (MFD). We establish weak, strong and converse duality as well as self-duality relations between (MFP ) and (MFD ). The proof of the weak duality relation is different from the proof of the corresponding result in [4]. Theorem 3.1 (Weak duality). Let (x(t), y(t), p, ω) be feasible for (MFP ) and let (u(t), b b v(t), q, ω) be feasible for (MFD ). Assume that a fi and − a gi are invex in x and x , and b b − a fi and a gi are invex in y and y , with η(x, u) + u(t) 0 and ξ(v, y) + y(t) 0 for all t ∈ I (except possibly at corners of (x (t), y (t)) or (u (t), v (t))). Then one has p q.
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Proof. The invexity assumptions of 1, 2, . . . , k, are invex. Then we have b
b a
fi and −
b a
gi imply that
b a
(fi − qi gi ), i =
fi (t, x, x , v, v ) − qi gi (t, x, x , v, v ) dt
a
b −
fi (t, u, u , v, v ) − qi gi (t, u, u , v, v ) dt
a
b
η(x, u)T
fix (t, u, u , v, v ) − qi gix (t, u, u , v, v )
a
− D fix (t, u, u , v, v ) − qi gix (t, u, u , v, v ) dt b =
η(x, u)T
fix (t, u, u , v, v ) − Dfix (t, u, u , v, v )
a
− qi gix (t, u, u , v, v ) − Dgix (t, u, u , v, v ) dt. From (7), (12) and (13) with η(x, u) + u(t) 0, we obtain k
b ωi
i=1
fi (t, x, x , v, v ) − qi gi (t, x, x , v, v ) dt
a k
b ωi
i=1
fi (t, u, u , v, v ) − qi gi (t, u, u , v, v ) dt.
a
The invexity assumptions of − are invex. Then we have b
b a
fi and
b a
gi imply that −
b a
(fi − pi gi ), i = 1, 2, . . . , k,
fi (t, x, x , v, v ) − pi gi (t, x, x , v, v ) dt
a
b −
fi (t, x, x , y, y ) − pi gi (t, x, x , y, y ) dt
a
b
(15)
ξ(v, y)T
fiy (t, x, x , y, y ) − Dfiy (t, x, x , y, y )
a
− pi giy (t, x, x , y, y ) − Dgiy (t, x, x , y, y ) dt. From (5), (6) and (14) with ξ(v, y) + y(t) 0, we obtain
D.S. Kim et al. / J. Math. Anal. Appl. 289 (2004) 505–521 k
b ωi
i=1
513
fi (t, x, x , v, v ) − pi gi (t, x, x , v, v ) dt
a
k
b ωi
i=1
fi (t, x, x , y, y ) − pi gi (t, x, x , y, y ) dt.
(16)
a
Combining (15) and (16) along with (4) and (11) gives k
b ωi (pi − qi )
i=1
gi (t, x, x , v, v ) dt 0.
If, for some i, pi < qi and for all j = i, pj qj , then 1, 2, . . . , k, implies that k
(17)
a
b ωi (pi − qi )
i=1
b a
gi (t, x, x , v, v ) dt > 0, i =
gi (t, x, x , v, v ) dt < 0,
a
which contradicts (17). Hence p q.
✷
We now present proofs of a strong and a converse duality theorem which are more direct than the corresponding ones in [4]. As to the assumptions of the strong and converse duality theorem, our condition (I) is similar to (I) in [4], while our condition (II) is quite different from (II) in [4]. Moreover we use as condition (II) a linear independence assumption in keeping with earlier studies in the static and dynamic case. With that the duality relations become a natural extension of earlier results in [21] for the static case. We mention that symmetric duality results for the static case which would lead to the results in the dynamic case in [4] are not known, at least to the authors. Theorem 3.2 (Strong duality). Let (x0 (t), y0 (t), p0 , ω0 ) be a properly efficient solution for (MFP ). Fix ω = ω0 in (MFD ) and define b fi (t, x0 , x0 , y0 , y0 ) dt p0i = ab , i = 1, 2, . . . , k. a gi (t, x0 , x0 , y0 , y0 ) dt Assume that (I)
k i=1
b Ψ (t)T
ω0i
(fiyy − p0i giyy ) − D(fiy y − p0i giy y )
a
− D (fiyy − Dfiy y − fiy y ) − p0i (giyy − Dgiy y − giy y ) + D 2 −(fiy y − p0i giy y ) Ψ (t) dt = 0
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D.S. Kim et al. / J. Math. Anal. Appl. 289 (2004) 505–521
implies Ψ (t) = 0, for all t ∈ I , and b (f1y − p01 g1y ) dt, . . . ,
(II) a
b (fky − p0k gky ) dt a
is linearly independent. If the invexity conditions of Theorem 3.1 are satisfied, then (x0 (t), y0 (t), p0 , ω0 ) is properly efficient for (MFD ). Proof. Since (x0 (t), y0 (t), p0 , ω0 ) is a properly efficient solution of (MFP ), it is also a weakly efficient solution. Hence there exist a ∈ R k , b ∈ R k , s ∈ R, t ∈ R k , piecewise smooth γ : I → R m and z : I → R n such that ai − bi gi − ω0i (giy − Dgiy )T γ (t) − sy0 (t) = 0, i = 1, 2, . . . , k, (18) k bi (fix − p0i gix ) − D(fix − p0i gix ) i=1
+
k
ω0i (fiyx − p0i giyx ) − D(fiy x − p0i giy x ) γ (t) − sy0 (t) − z(t)
i=1
−D
k
ω0i (fiyx − Dfiy x − fiy x )
i=1
− p0i (giy x − Dgiy x − giy x ) γ (t) − sy0 (t) +D
−
2
k
ω0i (fiy x − p0i giy x ) γ (t) − sy0 (t) = 0,
(19)
i=1 k (bi − sω0i ) (fiy − p0i giy ) − D(fiy − p0i giy ) i=1
+
k
ω0i (fiyy − p0i giyy ) − D(fiy y − p0i giy y ) γ (t) − sy0 (t)
i=1
−D
k
ω0i (fiyy − Dfiy y − fiy y )
i=1
− p0i (giyy − Dgiy y − giy y ) γ (t) − sy0 (t) +D
2
−
k
ω0i (fiy y − p0i giy y ) γ (t) − sy0 (t) = 0,
(20)
i=1 T
γ (t) − sy0 (t)
i = 1, 2, . . . , k,
(fiy − Dfiy ) − p0i (giy − Dgiy ) − ti = 0, (21)
D.S. Kim et al. / J. Math. Anal. Appl. 289 (2004) 505–521 k
515
b (fi − p0i gi ) dt = 0,
bi
i=1
(22)
a
γ (t)T
k
ω0i (fiy − Dfiy ) − p0i (giy − Dgiy ) = 0,
(23)
i=1
sy(t)
T
k
b ω0i
i=1
(fiy − Dfiy ) − p0i (giy − Dgiy ) dt = 0,
(24)
a
t T ω0 = 0,
(25)
z(t) x0 (t) = 0, a, γ (t), s, t, z(t) 0, a, γ (t), s, t, z(t) = 0.
(26)
T
(27) (28)
Since ω0 > 0 and t 0, we have t = 0. Multiplying (20) by (γ (t) − sy0 (t))T and applying (21) gives k
b ω0i
i=1
γ (t) − sy0 (t)
T
(fiyy − p0i giyy ) − D(fiy y − p0i giy y )
a
− D (fiyy − Dfiy y − fiy y ) − p0i (giyy − Dgiy y − giy y ) + D 2 −(fiy y − p0i giy y ) γ (t) − sy0 (t) dt = 0,
which by virtue of the hypothesis (I) yields γ (t) − sy0 (t) = 0 for all t ∈ I.
(29)
From (20) along with (29), we obtain k
b (bi − sω0i )
i=1
(fiy − p0i giy ) − D(fiy − p0i giy ) dt = 0.
a
By the hypothesis (II), b = sω0 .
(30)
If s = 0, then b = 0. From (18) we get a = 0, from (29) γ (t) = 0 and from (19) z(t) = 0. This combined with t = 0 contradicts (28). Hence s > 0 and b > 0. From (27) and (29), we get y0 (t) 0
for all t ∈ I, as s > 0.
(31)
From (19), (27) and (30), we obtain k i=1
ω0i (fix − p0i gix ) − D(fix − p0i gix ) 0.
(32)
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From (19), (26), (27) and (30), it also follows that k
b ω0i
i=1
x0 (t)T (fix − p0i gix ) − D(fix − p0i gix ) dt = 0.
(33)
a
Equation (22) with b > 0 implies b
fi (t, x0 , x0 y0 , y0 ) − p0i gi (t, x0 , x0 y0 , y0 ) dt = 0.
(34)
a
Thus the relations (31)–(34) imply that (x0 (t), y0 (t), p0 , ω0 ) is feasible for (MFD ) and the objective values of (MFP ) and (MFD ) are equal there. Clearly, (x0 , y0 , q0 , ω0 ) is efficient for (MFD ). If (x0 , y0 , q0 , ω0 ) is not properly efficient, then for some feasible (ui , vi , q0 , ω0 ) with b fi (ui , ui , vi , vi ) dt q0i = ab , i = 1, 2, . . . , k, a gi (ui , ui , vi , vi ) dt and for some i, q0i − p0i > M for any M > 0. b Since a gi (t, x, x , v, v ) dt, i = 1, 2, . . . , k, is bounded, it follows that k
b ω0i (q0i − p0i )
i=1
gi (t, x, x , v, v ) dt < 0,
a
which contradicts weak duality; see (17). Thus (x0 , y0 , q0 , ω0 ) is properly efficient for (MFD ).
✷
A converse duality theorem is stated now. The proof is analogous to that of Theorem 3.2. Theorem 3.3 (Converse duality). Let (x0 (t), y0 (t), p0 , ω0 ) be a properly efficient solution for (MFD ). Fix ω = ω0 in (MFP ) and define b fi (t, x0 , x0 , y0 , y0 ) dt p0i = ab , i = 1, 2, . . . , k. a gi (t, x0 , x0 , y0 , y0 ) dt Assume that (I)
k i=1
b Ψ (t)T
ω0i
(fixx − p0i gixx ) − D(fix x − p0i gix x )
a
− D (fiyy − Dfiy y − fiy y ) − p0i (giyy − Dgiy y − giy y ) + D 2 −(fix x − p0i gix x ) Ψ (t) dt = 0 implies Ψ (t) = 0 for all t ∈ I , and b (f1x − p01 g1x ) dt, . . . ,
(II) a
b (fkx − p0k gkx ) dt a
D.S. Kim et al. / J. Math. Anal. Appl. 289 (2004) 505–521
517
is linearly independent. If the invexity conditions of Theorem 3.1 are satisfied, then (x0 (t), y0 (t), p0 , ω0 ) is properly efficient for (MFP ). Remark 3.1. Our weak, strong and converse duality theorems can also be obtained under b b pseudo-invexity assumptions on ki=1 ωi a (fi − qi gi ) in x and x and − ki=1 ωi a (fi − pi gi ) in y and y . In order to present a better view of the concept of self-duality, we will consider here problems (MFP) and (MFD) instead of their equivalents (MFP ) and (MFD ). Assume that x(t) and y(t) have the same dimension. The function f (t, x(t), x (t), y(t), y (t)) is said to be skew-symmetric if f (t, x, x , y, y ) = −f (t, y, y , x, x ) for all x(t) and y(t) in the domain of f and the function g(t, x, x , y, y ) will be called symmetric if g(t, x, x , y, y ) = g(t, y, y , x, x ) for all x(t) and y(t) in the domain of g. Theorem 3.4 (Self-duality). If f (t, x, x , y, y ) is skew-symmetric and g(t, x, x , y, y ) is symmetric, then (MFP) and (MFD) are self-dual. If (MFP) and (MFD) are dual problems, then with (x0 (t), y0 (t), p0 , ω0 ) also (y0 (t), x0 (t), p0 , ω0 ) is a joint optimal solution and the common optimal value is 0. Proof. With f skew-symmetric, we have fx (t, x, x , y, y ) = −fy (t, y, y , x, x ), fy (t, x, x , y, y ) = −fx (t, y, y , x, x ), fx (t, x, x , y, y ) = −fy (t, y, y , x, x ), fy (t, x, x , y, y ) = −fx (t, y, y , x, x ), and with g symmetric, we have gx (t, x, x , y, y ) = gy (t, y, y , x, x ), gy (t, x, x , y, y ) = gx (t, y, y , x, x ), gx (t, x, x , y, y ) = gy (t, y, y , x, x ), gy (t, x, x , y, y ) = gx (t, y, y , x, x ). Expressing the dual problem (MFD) as a minimization problem and making use of the above relations, we have b f (t, v, v , u, u ) dt Minimize ab a g(t, v, v , u, u ) dt subject to u(a) = 0 = u(b),
u (a) = 0 = u (b), k i=1
v(a) = 0 = v(b), v (a) = 0 = v (b),
ωi ((fiv − Dfiv ))Gi (v, u) − ((giv − Dgiv ))Fi (v, u) 0,
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b u(t)T
k
ωi ((fiv − Dfiv ))Gi (v, u)
a
i=1
ω > 0,
ωT e = 1,
− ((giv − Dgiv ))Fi (v, u) dt 0, t ∈ I,
which is just the primal problem (MFP). Thus if (x0 (t), y0 (t), p0 , ω0 ) is an optimal solution of (MFD), then (y0 (t), x0 (t), p0 , ω0 ) is an optimal solution of (MFD). Since f is skew-symmetric and g is symmetric, respectively, we have b b f (t, x0 (t), x0 (t), y0 (t), y0 (t)) dt a f (t, y0 (t), y0 (t), x0 (t), x0 (t)) dt = − ab . b a g(t, y0 (t), y0 (t), x0 (t), x0 (t)) dt a g(t, x0 (t), x0 (t), y0 (t), y0 (t)) dt Hence b a
b a
and so
g(t, x0 (t), x0 (t), y0 (t), y0 (t)) dt b b f (t, x0 (t), x0 (t), y0 (t), y0 (t)) dt a f (t, y0 (t), y0 (t), x0 (t), x0 (t)) dt = − ab , = b a g(t, y0 (t), y0 (t), x0 (t), x0 (t)) dt a g(t, x0 (t), x0 (t), y0 (t), y0 (t)) dt
b a
b a
f (t, x0 (t), x0 (t), y0 (t), y0 (t)) dt
f (t, x0 (t), x0 (t), y0 (t), y0 (t)) dt
g(t, x0 (t), x0 (t), y0 (t), y0 (t)) dt b f (t, y0 (t), y0 (t), x0 (t), x0 (t)) dt = 0. = ab a g(t, y0 (t), y0 (t), x0 (t), x0 (t)) dt
✷
4. The static case If the time dependence of programs (MFP) and (MFD) is removed and f and g are considered to have domain R n × R m , we obtain the symmetric dual fractional pair given by f (x, y) f1 (x, y) fk (x, y) (SMFP) Minimize = ,..., g(x, y) g1 (x, y) gk (x, y) subject to
k
ωi {gi fiy − fi giy } 0,
i=1
yT
k
ωi {gi fiy − fi giy } 0,
i=1
ω > 0,
ωT e = 1,
D.S. Kim et al. / J. Math. Anal. Appl. 289 (2004) 505–521
(SMFD) Maximize subject to
f (u, v) = g(u, v) k
f1 (u, v) fk (u, v) ,..., g1 (u, v) gk (u, v)
519
ωi {gi fiu − fi giu } 0,
i=1
uT
k
ωi {gi fiu − fi giu } 0,
i=1
ω > 0,
ωT e = 1.
The following duality theorems are specializations of Theorems 3.1–3.3. Theorem 4.1 (Weak duality). Let (x, y, p, ω) be feasible for (SMFP) and let (u, v, q, ω) be feasible for (SMFD). Assume that fi and −gi are invex in x, and −fi and gi are invex in y, with η(x, u) + u 0 and ξ(v, y) + y 0. Then f (x, y) f (u, v) . g(x, y) g(u, v) Theorem 4.2 (Strong duality). Let (x0 , y0 , p0 , ω0 ) be a properly efficient solution for (SMFP). Fix ω = ω0 in (SMFD) and define fi (x0 , y0 ) , gi (x0 , y0 )
p0i =
i = 1, 2, . . . , k.
Assume that k
(I)
ω0i (fiyy − p0i giyy ) is positive or negative definite
i=1
and (II)
(f1y − p01 g1y ), . . . , (fky − p0k gky ) is linearly independent.
If the invexity conditions of Theorem 4.1 are satisfied, then (x0 , y0 , p0 , ω0 ) is properly efficient for (SMFD). Hence the weak and strong duality results in Section 3 reduce to the symmetric duality relations in the static case obtained by Weir in [21]. Theorem 4.3 (Converse duality). Let (x0 , y0 , p0 , ω0 ) be a properly efficient solution for (SMFD). Fix ω = ω0 in (SMFP) and define p0i =
fi (x0 , y0 ) , gi (x0 , y0 )
i = 1, 2, . . . , k.
Assume that (I)
k i=1
ω0i (fixx − p0i gixx ) is positive or negative definite
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and (II)
(f1u − p01 g1u ), . . . , (fku − p0k gku ) is linearly independent.
If the invexity conditions of Theorem 4.1 are satisfied, then (x0 , y0 , p0 , ω0 ) is properly efficient for (SMFP). The pair (SMFP) and (SMFD) will be self-dual if m = n and f is skew-symmetric (i.e., f (x, y) = −f (y, x)), and g is symmetric (i.e., g(x, y) = g(y, x)). Finally we obtain through specialization of Theorem 3.4. Theorem 4.4 (Self-duality). If f (x, y) is skew-symmetric and g(x, y) is symmetric, then (SMFP) and (SMFD) are self-dual. If (SMFP) and (SMFD) are dual problems, then with (x0 , y0 , p0 , ω0 ) also (y0 , x0 , p0 , ω0 ) is a joint optimal solution and the common optimal value is 0.
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