On Mixed Type Duality for Multiobjective Programming Containing Support Functions

(1)

Volume 2, Number 1 (2010), pp. 35–44

On Mixed Type Duality for Multiobjective Programming Containing Support Functions

I. Husain, A. Ahmed, and Rumana, G. Mattoo

Abstract. A mixed type vector dual to a multiobjective programming problem containing support functions is formulated and various duality results are proved under generalized invexity conditions. Special cases are generated from our results.

1. Introduction

In [5], Husain et al. considered the following multiobjective programming containing support functions

(NP) Minimize( f¹(x) + S(x|C¹), . . . , f^p(x) + S(x|C^p)) subject

g^j(x) + S(x|D^j) µ 0, j = 1, 2, . . . , m.

Where

(i) fⁱ: Rⁿ→ R and g^j: Rⁿ→ R, j = 1, 2, . . . , m are differentiable functions and (ii) S(·|Cⁱ), i = 1, 2, . . . , p and S(·|D^j), j = 1, 2, . . . , m are support functions of

a compact convex set Cⁱ, i = 1, 2, . . . , p and D^j, j = 1, 2, . . . , m in Rⁿ, to be defined later.

The following Wolfe type dual to the problem (NP) is presented [5]:

(WND) Maximize

f¹(u) + u^Tz¹+ Xm

j=1

y^j(g^j(u) + u^Tw^j), . . . ,

f^p(u) + u^Tz^p+ Xm

j=1

y^j(g^j(u) + u^Tw^j)

Key words and phrases.Multiobjective programming problem; Support functions; Mixed type duality;

Generalized invexity; Related problems.

(2)

subject to Xp

i=1

λⁱ∇( fⁱ(u) + u^Tzⁱ) + Xm

j=1

y^j∇(g^j(u) + u^Tw^j) = 0, zⁱ∈ Cⁱ, i = 1, 2, . . . , p,

w^j∈ D^j, j = 1, 2, . . . , m, y ½ 0,

λ > 0, Xp

i=1

λⁱ= 1.

The problem (WND) is a dual to (NP) assuming that Xp

i=1

λⁱ( fⁱ(·) + (·)^Tzⁱ) + Xm

j=1

y^j(g^j(·) + (·)^Tw^j)

is pseudoinvex with respect to η. The authors in [5] further weakened the invexity required in Wolfe type by constructing the following Mond-Weir type vector dual.

The Mond-Weir vector type dual is the following to (NP):

(M-WNP) Maximize ( f¹(u) + u^Tz¹, . . . , f^p(u) + u^Tz^p) subject to

Xp i=1

λⁱÏ( fⁱ(u) + u^Tzⁱ) + Xm

j=1

y^j(g^j(u) + u^Tw^j) = 0, zⁱ∈ Cⁱ, i = 1, 2, . . . , p,

w^j∈ D^j, j = 1, 2, . . . , m, Xm

j=1

y^j(g^j(u) + u^Tw^j) ½ 0, λ > 0, y ½ 0.

Husain et al. [5] established usual duality theorems under the hypotheses that Pp

i=1

λⁱÏ( fⁱ(·) + (·)^Tzⁱ) is pseudoinvex and Pm j=1

y^j(g^j(·) + (·)^Tw^j) is quasi-invex with respect to the same η.

In this paper, we propose in the spirit of Husain and Jabeen [4] and Xu [7], a mixed type dual to (NP) to combine the problems (WND) and (M-WNP) and establish various duality theorems under generalized invexity conditions. Special cases are discussed to show that our results extend some earlier results in the literature.

2. Pre-requisites

Before stating our multiobjective nonlinear problem, we mention the following conventions for vectors x and y in n-dimensional Euclidian space Rⁿ to be used

(3)

throughout the analysis of this research.

x < y ⇔ x_i< y_i, i = 1, 2, . . . , n.

x µ y ⇔ x_iµ y_i, i = 1, 2, . . . , n.

x ≤ y ⇔ x_i≤ y_i, i = 1, 2, . . . , n, but x 6= y x 6≤ y, is the negation of x ≤ y

For x, y ∈ R, x ≤ y and x < y have the usual meaning.

Before presenting our mixed type dual (Mix D), we mention some definitions of invexity and generalized invexity for easy reference.

Definition 2.1 (Invexity). The function φ : Rⁿ→ R is said to be invex with respect to η at ¯x if there exists a vector function η(x, ¯x) ∈ Rⁿ, such that for all x and

¯x ∈ Rⁿ

φ(x) − φ(¯x) ½ η(x, ¯x)^TÏφ(¯x) .

Definition 2.2 (Pseudoinvex). The function φ : Rⁿ→ R is said to be pseudoinvex with respect to η at ¯x if there exists a vector function η(x, ¯x) ∈ Rⁿ, such that for all x and ¯x ∈ Rⁿ

η(x, ¯x)^TÏφ(¯x) ½ 0 implies

φ(x) ½ φ(¯x) .

Definition 2.3 (Quasi-invex). The function φ : Rⁿ → R is said to be quasi-invex with respect to η at ¯x if there exists a vector function η(x, ¯x) ∈ Rⁿ, such that for all x and ¯x ∈ Rⁿ

φ(x) µ φ(¯x) implies

η(x, ¯x)^TÏφ(¯x) µ 0 .

Definition 2.4 (Support function). Let K be a compact set in Rⁿ, then the support function of K is defined by

S(x|K) = max{x^Tv ∈ K} .

A support function, being convex everywhere finite, has a subdifferential in the sense of convex analysis. The subdifferential of s(x|K) is given by

∂ S(x|K) = {z ∈ K|z^Tx = S(x|K)}.

For a set K, the normal cone to K at a point x ∈ K is defines by N_k(x) = { y| y^T(z − x) ≤ 0, for all z ∈ K}.

When K is a compact convex set, y is in N_k(x) if and only if S( y|K) = x^Ty i.e., x is a subdifferential of s at y.

(4)

Definition 2.5 (Efficient solution). A feasible solution ¯x is efficient for (NP) if there exist no other feasible x for (VPE) such that for some i ∈ P = {1, 2, . . . , p},

fⁱ(x) + S(x|Cⁱ) < fⁱ(¯x) + S(¯x|Cⁱ) and

f^j(x) + S(x|C^j) µ f^j(¯x) + S(¯x|C^j) for all j ∈ P, j 6= i.

In order to prove the strong duality theorem we will invoke the following lemma due to Chankong and Haimes [1]. In the subsequent analysis we shall denote the set of feasible solutions of the problem (NP) by X .

Lemma 2.6. A point ¯x ∈ X is an efficient for (NP), if and only if ¯x ∈ X solves the following problem:

(P_k(¯x)) Minimize f^k(x) + s(S|C^k) subject to

fⁱ(x) + S(x|Cⁱ) µ fⁱ(¯x) + S(¯x|Cⁱ) ∀ i ∈ P g^j(x) + S(x|D^j) µ 0, j = 1, 2, . . . , m.

3. Mixed type duality

We formulate the following type dual (Mix D) to (NP):

(Mix D) Maximize

f¹(u) + u^Tz¹+ X

j∈J_◦

y^j(g^j(u) + u^Tw^j), . . . ,

f^p(u) + u^Tz^p+ X

j∈J_◦

y^j(g^j(u) + u^Tw^j)) subject to

Xp i=1

λⁱ∇( fⁱ(u) + u^Tzⁱ) + Xm

j=1

y^j∇(g^j(u) + u^Tw^j) = 0, (1)

X

j∈Jα

y^j(g^j(u) + u^Tw^j) ½ 0, α = 1, 2, . . . , r, (2)

zⁱ∈ Cⁱ, i = 1, 2, . . . , p, (3)

w^j∈ D^j, j = 1, 2, . . . , m, (4)

y ½ 0, (5)

λ ∈ Λ, (6)

where Λ =

λ ∈ R^p

λ > 0,

Pp i=1

λⁱ= 1

.

where J_α⊆ M = {1, 2, . . . , m}, α = 0, 1, 2, . . . , r with Sr α=0

J_α= M and J_αT J_β = φ, if α 6= β. If J_◦= M, then (Mix D) becomes Wolfe type dual considered in [5], if J_◦= φ and J_α= M for some α ∈ {1, 2, . . . , r}, then (Mix D) becomes Mond-Weir type dual considered in [5].

(5)

Theorem 3.1 (Weak duality). Let ¯x be feasible for (NP) and (u, y, z¹, . . . , z^p, w¹, . . ., w^m, λ) be feasible for (Mix D). If for all feasible (x, u, y, z¹, . . . , z^p, w¹, . . . , w^m, λ), Pp

i=1

λⁱ∇( fⁱ(·) + (·)^Tzⁱ)+P

j∈J◦

y^j(g^j(·) + (·)^Tw^j) is pseudoinvex and P

j∈Jα

y^j(g^j(·) + (·)^Tw^j), α = 1, 2, . . . , r is quasi-invex with respect to η, then the following cannot hold.

fⁱ(x) + s(x|Cⁱ) ≤ fⁱ(u) + u^Tzⁱ+X

j∈J◦

y^j(g^j(u) + u^Tw^j) (7)

for all i ∈ {1, . . . , p}, and

f^k(x) + s(x|C^k) < f^k(u) + u^Tz^k+ X

j∈J_◦

y^j(g^j(u) + u^Tw^j) (8)

for some k.

Proof. Suppose that (7) and (8) hold. Then in view of λ > 0 and Pp i=1

λⁱ = 1, (7) and (8) together with x^Tzⁱ µ s(x|Cⁱ), i = 1, 2, . . . , p and x^Tw^j µ s(x|D^j), j = 1, 2, . . . , m and the feasibility for (NP) and (Mix D) imply

Xp i=1

λⁱ( fⁱ(x) + x^Tzⁱ) + X

j∈J_◦

y^j(g^j(x) + x^Tw^j)

<

Xp i=1

λⁱ( fⁱ(u) + u^Tzⁱ) + X

j∈J_◦

y^j(g^j(u) + u^Tw^j)

This in view of the pseudoinvexity of Xp

i=1

λⁱ( fⁱ(·) + (·)^Tzⁱ) + X

j∈J_◦

y^j(g^j(·) + (·)^Tw^j)

with respect to η, implies, η^T(x, u)

Xp

i=1

λⁱÏ( fⁱ(u) + u^Tzⁱ) +X

j∈J◦

y^jÏ(g^j(u) + u^Tw^j)

< 0 (9)

Since ¯x is feasible for (VP), (u, y, z¹, . . . , z^p, w¹, . . . , w^m, λ) is feasible for (Mix D), and x^Tw^jµ s(x|D^j), j = 1, 2, . . . , m, we have

X

j∈Jα

y^j(g^j(x) + x^Tw^j) µX

j∈Jα

y^j(g^j(u) + u^Tw^j), α = 1, 2, . . . , r .

This in view of quasi-invexity of P

j∈J_α

y^j(g^j(·) + (·)^Tw^j), α = 1, 2, . . . , r with respect to η, gives

η^T(x, u) X

j∈J_α

y^jÏ(g^j(x) + x^Tw^j)

µ 0, α = 1, 2, . . . , r

Hence

η^T(x, u)Ï

X

j∈M−J_◦

y^j(g^j(x) + x^Tw^j)

µ 0, α = 1, 2, . . . , r (10)

(6)

Combining (9) and (10), we have η^T(x, u)

X_p

i=1

j=1

< 0 (11)

From the equality constraint of (Mix D), we have η^T(x, u)

Xp

i=1

j=1

= 0 (12)

The relation (12) contradicts (11). Hence the conclusion of the theorem is true.

Theorem 3.2 (Strong duality). Let ¯x be an efficient solution of (NP) and for at least one i, i ∈ {1, 2, . . . , p}, ¯x satisfies the regularity condition [3] for the problem (P_k(¯x)). Then there exist λ ∈ R^p with λ^T = (¯λ¹, . . . , ¯λⁱ, . . . , ¯λ^p), ¯y ∈ R^m with

¯y^T = ( ¯y¹, . . . , ¯yⁱ, . . . , ¯y^m), zⁱ ∈ Rⁿ, i = {1, 2, . . . , p} and w^j ∈ Rⁿ, j = 1, 2, . . . , m such that (x, u, y, z¹, . . . , z^p, w¹, . . . , w^m, λ) is feasible for (Mix D) and the objectives of (NP) and (Mix D) are equal.

Further, if the hypotheses of Theorem 1 are satisfied, then (x, u, y, z¹, . . . , z^p, w¹, . . . , w^m, λ) is an efficient solution of (Mix D).

Proof. Since ¯x is an efficient solution for (P_k(¯x)), this implies that there exists ξ ∈ R^p, υ ∈ R^m with ¯υ^T = ( ¯υ¹, . . . , ¯υⁱ, . . . , ¯υ^m) and zⁱ∈ Rⁿ, i = {1, 2, . . . , p} such that

ξ¯^kÏ( f^k(x) + ¯x^T¯z^k) + Xp i=1i6=k

ξ¯ⁱÏ( fⁱ(x) + ¯x^T¯zⁱ)

+ Xm

j=1

y^jÏ(g^j(x) + x^Tw^j) = 0, (13)

Xm j=1

¯

υ^jÏ(g^j(x) + x^Tw^j) = 0, (14)

¯x^T¯zⁱ= S(¯x|Cⁱ), i = 1, 2, . . . , p, (15)

¯x^Tw¯^j= S(¯x|D^j), j = 1, 2, . . . , m, (16)

zⁱ∈ Cⁱ, i = 1, 2, . . . , p, (17)

w^j∈ D^j, j = 1, 2, . . . , m, (18)

ξ > 0, ¯υ ½ 0 (19)

Dividing (13), (14) and (19) by Pp i=1

ξⁱ6= 0, and putting ¯λⁱ= Pp^ξ^¯ⁱ i=1

ξⁱ

and ¯yⁱ= Pp^υ^¯ⁱ i=1

ξⁱ

, we have

(7)

Xp i=1

λ¯ⁱÏ( fⁱ(¯x) + ¯x^T¯zⁱ) + Xm

j=1

¯y^jÏ(g^j(x) + ¯x^Tw¯^j) = 0 (20)

Xm j=1

¯y^jÏ(g^j(x) + ¯x^Tw¯^j) = 0 (21)

λ > 0, Xp

i=1

λⁱ= 1, ¯y ½ 0 (22)

The equation (21) implies X

j∈J_◦

¯y^j(g^j(x) + ¯x^Tw¯^j) = 0 (23)

and X

j∈J_α

¯y^j(g^j(x) + ¯x^Tw¯^j) = 0, α = 1, 2, . . . , r (24)

The relation (20), (22) and (24) imply that (x, u, y, z¹, . . . , z^p, w¹, . . . , w^m, λ) is feasible for (Mix D).

fⁱ(¯x) + ¯x^T¯zⁱ+ X

j∈J_◦

¯y^j(g^j(x) + ¯x^Tw¯^j) = fⁱ(¯x) + S(¯x|Cⁱ), i = 1, 2, . . . , p.

This implies the objective of the primal and dual problems are equal.

Further, in view of the assumptions Theorem 1, the efficiency of ¯x for (NP) is immediate.

Theorem 3.3 (Converse duality). Let (¯x, ¯y, ¯z¹, . . . , ¯z^p, ¯w¹, . . . , ¯w^m, ¯λ) be an efficient solution for (Mix D). Assume that

(A₁) f and g are twice continuously differentiable, (A₂) Ï fⁱ(¯x) + ¯zⁱ+ P

j∈J_◦

¯y^j(Ïg^j(¯x) + ¯w^j) are linearly independent, (A₃) Ï²(λ^Tfⁱ(¯x) + ¯y^Tg(¯x)) is positive or negative definite.

Further, if the assumptions of Theorem 1 are met, then ¯x is an efficient solution.

Proof. Since (¯x, ¯y, ¯z¹, . . . , ¯z^p, ¯w¹, . . . , ¯w^m, ¯λ) be an efficient solution of (Mix D), then there exist τ ∈ R^p, β ∈ Rⁿ, γ ∈ R for each γ constraints, η ∈ R^p with η^T= (η¹, . . . , ηⁱ, . . . , η^p) and µ ∈ R^msuch that the following Fritz-John optimality conditions [2] are satisfied,

− Xp

i=1

τⁱ

Ï( fⁱ(¯x) + ¯x^T¯zⁱ) + X

j∈J_◦

¯y^jÏ(g(¯x) + ¯x^Tw¯^j)

+β^TÏ²(λ^Tf (¯x) + ¯y^Tg(¯x)) − γ Xr α=1

X

j∈J_◦

¯y^jÏ(g^j(x) + ¯x^Tw¯^j) = 0 (25)

−(τ^Te)(g^j(¯x) + ¯x^Tw¯^j+ β^TÏ(g^j(¯x) + ¯x^Tw¯^j)) − µ^j= 0, j ∈ J_◦ (26)

(8)

−γ(g^j(¯x) + ¯x^Tw¯^j+ β^TÏ(g^j(¯x) + ¯x^Tw¯^j)) − µ^j= 0, j ∈ J_α, α = 1, . . . , r (27)

β^TÏ( f (¯x) + ¯x^T¯zⁱ) +X

j∈J◦

¯y^j(Ïg^j(¯x) + ¯w^j) − ηⁱ= 0 (28)

(λⁱβ − τⁱ¯x) ∈ N_Ci(¯zⁱ), i = 1, . . . , p (29)

(β − (τ^Te)¯x) ¯y^j∈ N_Dj( ¯w^j), j ∈ J_◦ (30)

(β − γ¯x) ¯y^j∈ N_Dj( ¯w^j), j ∈ J_α, α = i, . . . , r (31)

µ^T¯y = 0 (32)

η^Tλ = 0 (33)

γX

j∈J_α

¯y^jÏ(g^j(¯x) + ¯x^Tw¯^j) = 0, α = 1, . . . , r (34)

(τ, µ, η, γ) ½ 0 (35)

(τ, β, µ, η, γ) 6= 0 (36)

Since λ > 0, (33) implies η = 0. Consequently (28) implies

Ï( fⁱ(¯x) + ¯x^T¯zⁱ) +X

j∈J◦

¯y^j(Ïg^j(¯x) + ¯w^j)

β = 0 (37)

Using the equality constraint of (Mix D) in (25), we have

− Xp

i=1

(τⁱ− γλⁱ)

Ï fⁱ(¯x) + ¯zⁱ+ X

j∈J_◦

¯y^T(Ïg^j(¯x) + ¯w^j)

+β^TÏ²(λ^Tf (¯x) + ¯y g(¯x)) = 0 (38)

Postmultiplying (38) by β and then using (37), we have β^TÏ²(λ^Tf (¯x) + ¯y^Tg(x))β = 0

This because of (A₃), yields β = 0

(39)

Using (39) along with (A₂), we have τⁱ− γλⁱ= 0, i = 1, 2, . . . , p (40)

Suppose γ = 0, then from (40) we have τ = 0. Consequently we have from (26) and (27), µ = 0.

Thus (τ, β, µ, η, γ) = 0, contradicting (36).

Hence γ > 0 and τ > 0.

In view of (39), (29), (30) and (31) we have,

¯x^T¯zⁱ= S(¯x|Cⁱ), i = 1, 2, . . . , p (41)

¯x^Tw¯^j= S(¯x|D^j), j = 1, 2, . . . , m (42)

(9)

From (26) and (27) along with (42) and (35), we have g^j(x) + s(¯x|D^j) µ 0, j = 1, 2, . . . , m

This implies the feasibility of ¯x for (VP).

From (26) and (32), we have X

j∈J◦

¯y^jÏ(g^j(¯x) + ¯x^Tw¯^j) = 0

In view of this together with (41), we have fⁱ(¯x) + ¯x^T¯zⁱ+X

j∈J◦

¯y^j(g^j(¯x) + ¯x^Tw¯^j)ⁱ= fⁱ(¯x) + S(¯x|Cⁱ), i = 1, 2, . . . , p

This establishes the equality of objective values of (NP).

This in view of the hypothesis of Theorem 1 gives the efficiency of ¯x for (NP).

4. Special cases

In this section, we specialize our problem (NP) and its mixed dual problems (Mix D). As discussed in [6] we may write S(x|Cⁱ) = (x^TBⁱx)¹², i = 1, . . . , p and S(x|D^j) = (x^TE^jx)¹², j = 1, . . . , m and the matrices Bⁱ, i = 1, . . . , p and E^j, j = 1, . . . , m are positive semidefinite. Putting these in our problems, we have (NP)₁ Minimize( f¹(x) + (x^TB¹x)¹², . . . , f^p(x) + (x^TB^px)¹²)

subject to

g^j(x) + (x^TE^jx)¹² µ 0, j = 1, 2, . . . , m For the dual (Mix D) problem, we get

(Mix D)₁ Maximize

f¹(u) + u^TBⁱz¹+X

j∈J◦

y^j(g^j(u) + u^TE^jw^j)

f^p(u) + u^TB^pz^p+ X

j∈J_◦

y^j(g^j(u) + u^TE^jw^j)

subject to Xp i=1

λⁱÏ( fⁱ(u) + u^TBⁱzⁱ) + Xm

j=1

y^j(g^j(u) + u^TD^jw^j) = 0, X

j∈J_α

y^j(g^j(u) + u^TD^jw^j) ½ 0, α = 1, 2, . . . , r, z^TBⁱz ≤ 1, i = 1, 2, . . . , p,

(w^j)^TE^jw^j≤ 1, j = 1, 2, . . . , m, λ > 0, y ½ 0.

(10)

References

[1] V. Chankong and Y.Y. Haimes, Multiobjective Decision Making: Theory and Methodology, North-Holland, New York, 1983.

[2] B.D. Craven, Lagrangian conditions and quasiduality, Bulletin of Australian Mathematical Society 16(1977), 325–339.

[3] I. Husain, Abha and Z. Jabeen, On nonlinear programming with support function, J.

Appl. Math. & Computing 10 (1-2) (2002), 83–99.

[4] I. Husain and Z. Jabeen, Mixed type duality for a programming problem containing support function, Journal of Applied Mathematics and Computing 15 (1-2) (2004), 211–225.

[5] I. Husain, A. Ahmed and Rumana, G. Mattoo, On multiobjective nonlinear programming with support functions, to appear in Journal of Applied Analysis.

[6] B. Mond and M. Schechter, Nondifferentiable symmetric duality, Bull. Autral. Math.

Soc. 53 (1996), 177–188.

[7] Z.Xu, Mixed type duality in multiobjective programming problems, Journal of Mathematical Analysis and Applications 198 (1996), 621–635.

I. Husain, Department of Mathematics, Jaypee University of Engineering and Technology, Guna, Madhya Pradesh, India.

E-mail: [email protected]

A. Ahmed, Department of Statistics, University of Kashmmir, Srinagar, Kashmir, India.

Rumana, G. Mattoo, Department of Statistics, University of Kashmmir, Srinagar, Kashmir, India.

Received August 08, 2009 Accepted October 23, 2009