Page 133 - IJOCTA-15-1

Duality for robust multi-dimensional vector variational control problem under invexity
h i
T T
−D κ λ θ ζ κ (Π, ζ κ , b)+τ β ζ κ (Π, ζ κ , µ) =0,
Z
¯ T
¯
¯
+ D κ η(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω)λ θ σ κ (Λ, ¯σ κ , b)dν+ (10)
T
T
T
Ω χ Ψ ϱ (Π, ¯ a)+λ θ ϱ (Π, ζ κ , b)+τ β ϱ (Π, ζ κ , µ)=0,
Z
¯ T
¯
¯
ξ(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω)λ θ ω (Λ, ¯σ κ , b)dν (8) (11)
ζ(t 0 )=δ 0 , ζ(t 1 )=δ 1 , (12)
Ω
p
Z
T
Further, invexity of ¯ τ β(., ., ¯µ)dν and robust χ≧0, X χ i =1, λ≧0, (13)
Ω
feasibility of (ˆσ, ˆω)∈T gives i=1
for ¯a ∈A, b∈B, µ∈M .
Z
T
¯
0≧ η(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω)¯τ β σ (Λ, ¯σ κ , ¯µ)dν Denote T w ={(ζ,ϱ,χ,λ,τ,¯ a,b,µ) satisfying (10)−
Ω (13)} as the set of feasible solutions of (WD).
Z
¯
T
+ D κ η(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω)¯τ β σ κ (Λ, ¯σ κ , ¯µ)dν+ Theorem 3. [Weak Duality Theorem] Let
Ω
(σ, ω) and (ζ, ϱ, χ, λ, τ, ¯ a, b, µ) be the ro-
Z
T
¯
ξ(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω)¯τ β ω (Λ, ¯σ κ , ¯µ)dν. (9) bust feasible solutions of (RUVP) and
Z
Ω
(WD), respectively. Further, if Ψ(., ¯ a)dν,
Adding inequalities (7), (8) and (9), we have
Ω
Z Z
Z
T T
¯
T
0> η(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω) ¯χ Ψ σ Λ, ¯ a λ θ(., ., b)dν and τ β(., ., µ)dν are in-
Ω Ω Ω q
vex w.r.t. η(t, σ, ζ, σ κ , ζ κ , ω, ϱ)∈R and

¯
¯
¯
T
¯ T
+λ θ σ (Λ, ¯σ κ , b)+ ¯τ β σ (Λ, ¯σ κ , ¯µ) dν ξ(t, σ, ζ, σ κ , ζ κ , ω, ϱ)∈R at (ζ, ϱ), then
r
Z

¯
¯
¯ T
+ D κ η(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω) λ θ σ κ (Λ, ¯σ κ , b)dν Z Z
T
Ω Ψ(Λ, ¯ a)dν < Ψ(Π, ¯ a)+λ θ(Π, ζ κ , b)+

T ¯ Ω Ω
+¯τ β σ κ (Λ, ¯σ κ , ¯µ) dν

T
Z τ β(Π, ζ κ , µ) dν

¯
T
+ ξ(t, ¯σ, ˆσ, ¯σ κ , ˆσ κ , ¯ω, ˆω) ¯χ Ψ ω Λ, ¯ a +
Ω cannot hold.

¯
¯
¯
¯ T
T
λ θ ω (Λ, ¯σ κ , b)+ ¯τ β ω (Λ, ¯σ κ , ¯µ) dν Proof. Suppose, on the contrary
Z Z
which is a contradiction to (6). Hence, (¯σ, ¯ω) is T
Ψ(Λ, ¯ a)dν < Ψ(Π, ¯ a)+λ θ(Π, ζ κ , b)
a robust weak efficient solution for the problem Ω Ω
(RUVP). T
+τ β(Π, ζ κ , µ) dν.
p
3. Wolfe type dual and duality X
As (σ, ω)∈T , χ≧0, χ i =1 and λ≧0,
theorems
i=1
Taking maxΨ(Π, a)=Ψ(Π, ¯ a), where Π= Z
T
T
a∈A Ψ(Λ, ¯ a)+λ θ(Λ, σ κ , b)+τ β(Λ, σ κ , µ) dν
(t, ζ(t), ϱ(t)), the Wolfe-type dual for the problem Ω
(RUVP) is formulated as follows : Z
T
T
< Ψ(Π, ¯ a)+λ θ(Π, ζ κ , b) +τ β(Π, ζ κ , µ) dν,
Ω
Z
(14)
T
(WD) max Ψ(Π, ¯ a)+λ θ(Π, ζ κ , b)+ Z Z
T
(ζ(.),ϱ(.)) Ω Since Ψ(., ¯ a)dν, λ θ(., ., b)dν and
Ω Ω
T
τ β(Π, ζ κ , µ) dν = Z T
τ β(., ., µ)dν are invex at (ζ, ϱ), there-

Z Ω

T
T
Ψ 1 (Π, ¯ a)+λ θ(Π, ζ κ , b)+τ β(Π, ζ κ , µ) dν, fore there exist η(t, σ, ζ, σ κ , ζ κ , ω, ϱ) and
Ω ξ(t, σ, ζ, σ κ , ζ κ , ω, ϱ) such that
Z

T
... , Ψ p (Π, ¯ a)+λ θ(Π, ζ κ , b)+ Z T Z T
Ω χ Ψ(Λ, ¯ a)dν − χ Ψ(Π, ¯ a) dν ≧
!
Ω Ω
Z
T
τ β(Π, ζ κ , µ) dν T
η(t, σ, ζ, σ κ , ζ κ , ω, ϱ)χ Ψ ζ (Π, ¯ a)dν
Ω
subject to Z
T
T
T
T
χ Ψ ζ (Π, ¯ a)+λ θ ζ (Π, ζ κ , b)+τ β ζ (Π, ζ κ , µ) + ξ(t, σ, ζ, σ κ , ζ κ , ω, ϱ)χ Ψ ϱ (Π, ¯ a)dν, (15)
Ω
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