Introduction In this paper, we consider the following generalized nondifferentiable frac- tional optimization problem (GFP): (GFP) Minimize max ½fi(x

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Website: http://www.kcam.biz

OPTIMALITY AND DUALITY FOR GENERALIZED NONDIFFERENTIABLE FRACTIONAL PROGRAMMING

WITH GENERALIZED INVEXITY^†

MOON HEE KIM^∗AND GWI SOO KIM

Abstract. Sufficient optimality conditions for a class of generalized nondifferentiable fractional optimization programming problems are established.

Moreover, we prove the weak and strong duality theorems under (V, ρ)- invexity assumption.

AMS Mathematics Subject Classification : 90C30, 90C46.

Key words and phrases : Generalized nondifferentiable fractional opti- mization problem, (V, ρ)-invex, optimality condition, duality results.

1. Introduction

In this paper, we consider the following generalized nondifferentiable fractional optimization problem (GFP):

(GFP) Minimize max

½fi(x) + s(x|Ci)

gi(x) − s(x|Di) | i = 1, · · · , p

¾

subject to hj(x) ≤ 0, j = 1, · · · , m,

where f := (f1, · · · , fp) : Rⁿ → R^p, g := (g1, · · · , gp) : Rⁿ → R^p and h :=

(h1, · · · , hm) : Rⁿ → R^mare continuously differentiable. We assume that gi(x)−

s(x|Di) > 0, i = 1, · · · , p. For each i = 1, · · · , p, Ci and Di are compact convex set of Rⁿ and we define a support function with respect to Ci as follows:

s(x|Ci) := max{hx, yii | yi ∈ Ci}.

Further let, J(x) = {j : hj(x) = 0}, for any x ∈ Rⁿ and let ki(x) = s(x|Ci), i = 1, · · · , p.

Received April 23, 2010. Revised June 17, 2010. Accepted June 23, 2010. ^∗Corresponding author. ^†This work was supported by the Korea Science and Engineering Foundation (KOSEF) NRL program grant funded by the Korea government(MEST)(No. ROA-2008-000-20010-0).

° 2010 Korean SIGCAM and KSCAMc . 1535

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Then, ki is a convex function and we can prove that

∂ki(x) = {wi∈ Ci | hwi, xi = s(x|Ci)}, where ∂ki is the subdifferential of ki.

Many authors have introduced various concepts of generalized convexity and have obtained optimality and duality results for a fractional programming problem ([2]–[9], [11]).

Recently, Kim and Kim [4] consider the following generalized nondifferentiable fractional optimization problem.

(FP) Minimize max

½fi(x) + s(x|Ci)

gi(x) | i = 1, · · · , p

¾

subject to hj(x) ≤ 0, j = 1, · · · , m,

where f := (f1, · · · , fp) : Rⁿ → R^p, g := (g1, · · · , gp) : Rⁿ → R^p and h :=

(h1, · · · , hm) : Rⁿ → R^m are continuously differentiable. We assume that gi(x) > 0, i = 1, · · · , p. For each i = 1, · · · , p, Ci are compact convex set of Rⁿ.

In this paper, we apply the approach of Kim and Kim [4] to the generalized nondifferentiable fractional optimization problem (GFP), we establish the necessary and sufficient optimality conditions for a class of generalized nondifferentiable fractional optimization problem (GFP). Moreover, we prove the weak and strong duality theorems under (V, ρ)-invexity assumptions.

We introduce the following definition due to Kuk et al. [5].

Definition 1. A vector function f : Rⁿ → R^p is said to be (V, ρ)-invex at u ∈ Rⁿ with respect to the functions η and θi : Rⁿ× Rⁿ → Rⁿ if there exists αi: Rⁿ× Rⁿ→ R+\ {0} and ρi∈ R, i = 1, · · · , p such that for any x ∈ Rⁿ and for all i = 1, · · · , p,

α_i(x, u)£

f_i(x) − f_i(u)¤

≥ ∇f_i(u)η(x, u) + ρ_ikθ_i(x, u)k².

Definition 2. A vector function f : Rⁿ → R^p is said to be η-invex at u ∈ Rⁿ such that for any x ∈ Rⁿ and for all i = 1, · · · , p,

fi(x) − fi(u) ≥ ∇fi(u)η(x, u).

We give the following theorem due to Kim et al. [2].

Theorem 1. Assume that f and g are vector-valued differentiable functions defined on X0 and f (x) + hw, xi ≥ 0, g(x) − h ew, xi > 0 for all x ∈ X0. If

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f (·) + hw, ·i and −g(·) + h ew, ·i are (V, ρ)-invex at x0 ∈ X0, then f (·)+hw,·i g(·)−he^w,·i is (V, ρ)-invex at x0, where

¯

αi(x, x0) = gi(x) − h ewi, xi

gi(x0) − h ewi, x0iαi(x, x0), θ¯i(x, x0) =

³ 1

gi(x0) − h ewi, x0i

´¹

2θi(x, x0),

that is, for all i,

αi(x, x0)h fi(x) + hwi, xi

gi(x) − h ewi, xi−fi(x0) + hwi, x0i gi(x0) − h ewi, x0i i

≥ gi(x0) − h ewi, x0i gi(x) − h ewi, xi

h

∇³ fi(x0) + hwi, x0i gi(x0) − h ewi, x0i

´

η_i(x, x₀)

+ρik

³ 1

´¹

2θi(x, x0)k² i

.

Proof. Let ki(x) = s(x|Ci) and eki(x) = s(x|Di), i = 1, . . . , p. Choose wi ∈ ∂ki(x0) and ewi ∈ ∂eki(x0). Let x, x0 ∈ X0. By the (V, ρ)-invexity of f (·) + hw, ·i and −g + h ew, ·i,

αi(x, x0)h f_i(x) + hwi, xi

gi(x) − h ewi, xi−fi(x0) + hwi, x0i gi(x0) − h ewi, x0i

i

= αi(x, x0)h fi(x) + hwi, xi − fi(x0) − hwi, x0i gi(x) − h ewi, xi

−(fi(x0) + hwi, x0i)gi(x) − h ewi, xi − gi(x0) + h ewi, x0i (gi(x) − h ewi, xi)(gi(x0) − h ewi, x0i)

i

≥ 1

g_i(x) − h ew_i, xi h

(∇fi(x0) + wi)ηi(x, x0) + ρikθi(x, x0)k²i + fi(x0) + hwi, x0i

(gi(x) − h ewi, xi)(gi(x0) − h ewi, x0i) h

(−∇gi(x0) + ewi)ηi(x, x0) + ρikθi(x, x0)k² i

.

Since g(x) − h ew, xi > 0 for all x ∈ X0, we see that

≥g_i(x₀) − h ew_i, x₀i gi(x) − h ewi, xi

h ∇f_i(x₀) + w_i

gi(x0) − h ewi, x0iηi(x, x0)+ρik

³ 1

´¹

2θi(x, x0)k²

− fi(x0) + hwi, x0i

(gi(x0) − h ewi, x0i)²(∇gi(x0)− ewi)ηi(x, x0)+ρik³ f_i(x0) + hwi, x0i (gi(x0) − h ewi, x0i)²

´¹

2θi(x, x0)k² i

.

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Thus, we have

≥gi(x0) − h ewi, x0i gi(x) − h ewi, xi

h (∇f_i(x0) + wi)(gi(x0) − h ewi, x0i) − (fi(x0) + h ewi, x0i)(∇gi(x0) − ewi)

(gi(x0) − h ewi, x0i)² ηi(x, x0) +ρik³ 1

g_i(x₀) − h ew_i, x₀i

´¹

2θi(x, x0)k²+ ρik³ fi(x0) + hwi, x0i (g_i(x₀) − h ew_i, x₀i)²

´¹

2θi(x, x0)k²i

=gi(x0) − h ewi, x0i gi(x) − h ewi, xi

h

´

η_i(x, x₀)

+ρik

³ 1

´¹

2³

1 +³ f_i(x0) + hwi, x0i gi(x0) − h ewi, x0i

´¹

2´

θi(x, x0)k² i

. Considering that

1 +³ fi(x0) + hwi, x0i gi(x0) − h ewi, x0i

´¹

2 ≥ 1, i = 1, 2, . . . , p,

we have for all i,

≥gi(x0) − h ewi, x0i gi(x) − h ewi, xi

h

´

ηi(x, x0)+ρik³ 1

´¹

2θi(x, x0)k²i .

Therefore, the function f (x)+hw,xi

g(x)−he^w,xi is (V, ρ)-invex, where

¯

αi(x, x0) = gi(x) − h ewi, xi

gi(x0) − h ewi, x0iαi(x, x0), θ¯i(x, x0) =

³ 1

´¹

2θi(x, x0).

2

2. Optimality Conditions

Now, we establish the Kuhn-Tucker necessary and sufficient conditions for a solution of (GFP).

Theorem 2. (Kuhn-Tucker Necessary Optimality Theorem) If x0 is a solution of (GFP), and assume that 0 6∈ co{∇hj(x0) | j ∈ J(x0)}, then there exist

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λi ≥ 0, i ∈ I(x0) := {i | maxn

fi(x0)+s(x0|Ci)

gi(x0)−s(x0|Di) | i = 1, · · · , po }, P

i∈I(x0)λi = 1, µj≥ 0, j = 1, · · · , m and wi∈ Ci, ewi∈ Di, i ∈ I(x0) such that

X

i∈I(x0)

λi∇³ fi(x0) + hwi, x0i g_i(x₀) − h ew_i, x₀i

´ +

Xm j=1

µj∇hj(x0) = 0, hwi, x0i = s(x0|Ci), h ewi, x0i = s(x0|Di),

Xm j=1

µ_jh_j(x₀) = 0.

Proof. Let ϕ_i(x) = _g^fⁱ^(x)+s(x|Cⁱ⁾

i(x)−s(x|Di), i = 1, · · · , p. Let x₀ be a solution of (GFP) and let I(x0) = {i | max{ϕi(x0) | i = 1, · · · , p}}. Then by Proposition 2.3.12 in [1] and Corollary 5.1.8 in [10], there exists µj≥ 0, j = 1, · · · , m,

0 ∈ co{∂^cϕi(x0) | i ∈ I(x0)} + Xm j=1

µj∂^chj(x0) and µjhj(x0) = 0.

Thus there exists λi≥ 0, i ∈ I(x0), P

i∈I(x0)λi = 1 such that

0 ∈ X

i∈I(x0)

λi∂^cϕi(x0) + Xm j=1

µj∇hj(x0) (2.1) and µjhj(x0) = 0.

By Proposition 2.3.14 in [1],

∂^cϕi(x0) = 1

(gi(x0) − s(x0|Di))²

³

(gi(x0) − s(x0|Di))(∇fi(x0) + ∂s(x0|Ci))

−(fi(x0) + s(x0|Ci))(∇gi(x0) − ∂s(x0|Di))´ . Since

∂^cϕi(x0) =

½ 1

(gi(x0) − h ewi, x0i)²

³

(gi(x0) − h ewi, x0i)(∇fi(x0) + wi)

−(fi(x0) + hwi, x0i)(∇gi(x0) − ewi)´

| wi∈ Ci, hwi, x0i = s(x0|Ci), h ewi, x0i = s(x0|Di), i ∈ I(x0)}

=

½

´

|w_i∈ C_i, ew_i ∈ D_i, hw_i, x₀i = s(x₀|C_i), h ewi, x0i = s(x0|Di), i ∈ I(x0)}

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and hence from (2.1), there exist λi ≥ 0, i ∈ I(x0), P

i∈I(x0)λi = 1, µj ≥ 0, j = 1, · · · , m and wi∈ Ci, i ∈ I(x0) such that

X

i∈I(x0)

λi∇³ fi(x0) + hwi, x0i gi(x0) − h ewi, x0i

´ +

Xm j=1

µj∇hj(x0) = 0, hwi, x0i = s(x0|Ci), h ewi, x0i = s(x0|Di),

Xm j=1

µjhj(x0) = 0.

2

Theorem 3. (Kuhn-Tucker Sufficient Optimality Theorem) Let x0 be a feasible solution of (GFP). Suppose that there exist λi ≥ 0, i ∈ I(x0), P

i∈I(x0)λi = 1, µj ≥ 0, j = 1, · · · , m and wi ∈ Ci, ewi ∈ Di, i ∈ I(x0) such that

X

i∈I(x0)

λi∇³ f_i(x0) + hwi, x0i gi(x0) − h ewi, x0i

´ +

Xm j=1

µj∇hj(x0) = 0, (2.2) hwi, x0i = s(x0|Ci), h ewi, x0i = s(x0|Di),

Xm j=1

µjhj(x0) = 0.

If f (·)+hw, ·i and −g(·)+h ew, ·i are (V, ρ)-invex at x0, and h is η-invex at x0with respect to the same η, andP

i∈I(x0)λiρik¯θi(x, x0)k² ≥ 0, then x0 is a solution of (GFP).

Proof. Suppose that x0is not a solution of (GFP). Then there exist a feasible solution x of (GFP) such that

1≤i≤pmax

fi(x) + s(x|Ci)

gi(x) − s(x|Di)< max

1≤i≤p

fi(x0) + s(x0|Ci) gi(x0) − s(x0|Di). Then

fi(x) + s(x|Ci)

gi(x) − s(x|Di)< fi(x0) + s(x0|Ci)

gi(x0) − s(x0|Di), for all i ∈ I(x0).

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Since hwi, x0i = s(x0|Ci), wi ∈ Ci, and h ewi, x0i = s(x0|Di), ewi ∈ Di, we have for all i ∈ I(x0),

fi(x) + hwi, xi

gi(x) − h ewi, xi ≤ fi(x) + s(x|Ci) gi(x) − s(x|Di)

< fi(x0) + s(x0|Ci) g_i(x₀) − s(x₀|D_i)

= fi(x0) + hwi, x0i gi(x0) − h ewi, x0i and hence ¯αi(x, x0) > 0,

¯

< 0.

By the (V, ρ)-invexity of f (·) + hw, ·i and −g(·) + h ew, ·i at x0, and by Theorem 1, we have

∇³ f_i(x0) + hwi, x0i gi(x0) − h ewi, x0i

´

η(x, x0) + ρik¯θi(x, x0)k²< 0.

Hence, we have X

i∈I(x0)

λi∇³ f_i(x0) + hwi, x0i gi(x0) − h ewi, x0i

´

η(x, x0) + X

i∈I(x0)

λiρik¯θi(x, x0)k²< 0.

SinceP

i∈I(x0)λiρik¯θi(x, x0)k²≥ 0, X

i∈I(x0)

λi∇³ fi(x0) + hwi, x0i gi(x0) − h ewi, x0i

´

η(x, x0) < 0

and so, it follows from (2.2) that Xm j=1

µj∇hj(x0)η(x, x0) > 0.

Then, by the η-invexity of h, we have Xm

j=1

µjhj(x) − Xm j=1

µjhj(x0) > 0.

SinceP_m

j=1µjhj(x0) = 0, we have P_m

j=1µjhj(x) > 0, which is a contradiction since µ_j ≥ 0, j = 1, · · · , m and x is a feasible solution of (GFP). Consequently,

x0 is a solution of (GFP). 2

3. Duality Theorems

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Now, we propose the following Mond-Weir type dual problem (DGFP):

(DGFP)

Maximize max

½fi(u) + s(u|Ci)

gi(u) − s(u|Di) | i = 1, · · · , p

¾

subject to X

i∈I(u)

λi∇³ f_i(u) + hwi, ui gi(u) − h ewi, ui

´ +

Xm j=1

µj∇hj(u) = 0, (3.1) wi ∈ Ci, ewi∈ Di, hwi, ui = s(u|Ci), h ewi, ui = s(u|Di), i ∈ I(u)

Xm j=1

µjhj(u) = 0,

λi ≥ 0, i ∈ I(u), X

i∈I(u)

λi= 1, µj ≥ 0, j = 1, · · · , m.

Now we show that the following weak duality theorem holds between (GFP) and (DGFP).

Theorem 4. (Weak Duality) Let x be a feasible for (GFP) and let (u, λ, µ, w) be feasible for (DGFP). Assume that f (·) + hw, ·i and −g(·) + h ew, ·i are (V, ρ)- invex at u, and let h is η-invex at u with respect to the same η, andP

i∈I(u)λiρik¯θi(x, u)k²≥ 0. Then the following holds:

max

½fi(x) + s(x|Ci)

gi(x) − s(x|Di) | i = 1, · · · , p

¾

≥ max

½fi(u) + s(u|Ci)

gi(u) − s(u|Di) | i = 1, · · · , p

¾ .

Proof. Let x be any feasible for (GFP) and let (u, λ, µ, w) be any feasible for (DGFP). Then we have

Xm j=1

µjhj(x) ≤ 0 ≤ Xm j=1

µjhj(u).

By the η-invexity of hj(u), j = 1, · · · , m, we have Xm

j=1

µj∇hj(u)η(x, u) ≤ 0.

Using (3.1), we obtain X

i∈I(u)

λi∇³ f_i(u) + hwi, ui gi(u) − h ewi, ui

´

η(x, u) ≥ 0. (3.2)

Now suppose that max

½fi(x) + s(x|Ci)

gi(x) − s(x|Di) | i = 1, · · · , p

¾

< max

½fi(u) + s(u|Ci)

gi(u) − s(u|Di) | i = 1, · · · , p

¾ .

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Then fi(x) + s(x|Ci)

gi(x) − s(x|Di) < fi(u) + s(u|Ci)

gi(u) − s(u|Di), for all i ∈ I(u).

Since hwi, ui = s(u|Ci) and h ewi, ui = s(u|Di), we have for all i ∈ I(u), fi(x) + hwi, xi

gi(x) − h ewi, xi < fi(u) + hwi, ui gi(u) − h ewi, ui.

By the (V, ρ)-invexity of f (·) + hw, ·i and −g(·) + h ew, ·i at x0, and by Theorem 1, we have,

0 > α¯i(x, u)h f_i(x) + hwi, xi

gi(x) − h ewi, xi −fi(u) + hwi, ui gi(u) − h ewi, ui i

≥ ∇³ f_i(u) + hwi, ui gi(u) − h ewi, ui

´

η(x, u) + ρik¯θi(x, u)k². By using λi≥ 0, i ∈ I(u), we have,

X

i∈I(u)

λi∇³ fi(u) + hwi, ui gi(u) − h ewi, ui

´

η(x, u) + X

i∈I(u)

λiρik¯θi(x, u)k²< 0.

SinceP

i∈I(u)λiρik¯θi(x, u)k²≥ 0, we have X

i∈I(u)

λi∇³ f_i(u) + hw_i, ui gi(u) − h ewi, ui

´

η(x, u) < 0,

which contradicts (3.2). Hence the result holds. 2 Now we give a strong duality theorem which holds between (GFP) and (DGFP).

Theorem 5. (Strong Duality) If ¯x be a solution of (GFP) and suppose that 0 6∈ co{∇hj(¯x) | j ∈ J(¯x)}. Then there exist ¯λ ∈ R^p, ¯µ ∈ R^m and ¯w ∈ C such that (¯x, ¯λ, ¯µ, ¯w, ew) is feasible for (DGFP). Moreover if the weak duality holds, then (¯x, ¯λ, ¯µ, ¯w, ew) is a solution of (DGFP).

Proof. By Theorem 2, there exist ¯λ ∈ R^p, ¯µ ∈ R^m and ¯wi∈ Ci, ew ∈ Di, i ∈ I(¯x), such that

X

i∈I(¯x)

λ¯i∇³ f_i(¯x) + h ¯w_i, ¯xi gi(¯x) −D

e wi, ¯xE´

+ Xm j=1

¯

µj∇hj(¯x) = 0,

h ¯wi, ¯xi = s(¯x|Ci), D

e wi, u

E

= s(¯x|Di), Xm

j=1

µjhj(¯x) = 0,

λi≥ 0, i ∈ I(¯x), X

i∈I(¯x)

λi = 1.

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Thus (¯x, ¯λ, ¯µ, ¯w, ew) is a feasible for (DGFP). On the other hand, by weak duality (Theorem 4),

max

½fi(¯x) + s(¯x|Ci)

gi(¯x) − s(¯x|Di) | i = 1, · · · , p

¾

≥ max

½fi(u) + s(u|Ci)

gi(u) − s(u|Di) | i = 1, · · · , p

¾

for any (DGFP) feasible solution (u, λ, µ, w, ew). Hence (¯x, ¯λ, ¯µ, ¯w, ew) is a solution

of (DGFP). 2

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Moon Hee Kim received her Ph. D at Pukyong National University under the direction of Professor Gue Myung Lee. Since 2006 she has been at the Tongmyong University. Her research interests focus on vector optimization problem and variational inequality.

Department of Multimedia Engineering, Tongmyong University, Pusan 608-711, Korea.

e-mail: mooni@tu.ac.kr

Gwi Soo Kim received her Ph. D at Pukyong National University under the direction of Professor Gue Myung Lee. Since 2005 she has been at the Pukyong National University.

Her research interests focus on vector optimization problem.

Department of Applied Mathematics, Pukyong National University, Pusan 608-737, Korea.

e-mail: gwisoo1103@hanmail.net