Page 137 - IJOCTA-15-1

Duality for robust multi-dimensional vector variational control problem under invexity

(ζ, ϱ, χ, λ, τ, ¯ a, b, µ)∈T w such that Z
¯
¯
¯
T
+ ξ(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ) ¯χ Ψ ϱ Π, ¯ a +
Z

T
T
Ψ(Π, ¯ a)+λ θ(Π, ζ κ , b) +τ β(Π, ζ κ , µ) dν ≧ Ω
!
Ω
¯ T
Z ¯ ¯ ¯ T ¯ ¯
λ θ ϱ (Π, ζ κ , b)+ ¯τ β ϱ (Π, ζ κ , ¯µ) dν =0
¯
¯
¯
¯
¯ T
T

Ψ Λ, ¯ a dν +λ θ(Λ, ¯σ κ , b) + ¯τ β(Λ, ¯σ κ , ¯µ) dν
Ω
which is equivalent to the following, using the fact
¯ m
¯
From (3), λ(t)∈R + and feasibility of (¯σ, ¯ω) to that η =0 when ¯σ(t)=ζ(t) ,
(RUVP), we obtain Z

¯
¯
¯
T
η(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ) ¯χ Ψ ζ Π, ¯ a
Ω
Z

T
T
Ψ(Π, ¯ a)+λ θ(Π, ζ κ , b) +τ β(Π, ζ κ , µ) dν ≧ ¯ T ¯ ¯ ¯ T ¯ ¯
+λ θ ζ (Π, ζ κ , b)+ ¯τ β ζ (Π, ζ κ , ¯µ) dν
Ω
Z Z

¯
¯
¯
¯ T
¯ ¯ ¯

Ψ Λ, ¯ a dν + D κ η(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ) λ θ ζ κ (Π, ζ κ , b)
Ω Ω

¯ ¯
contradicting Theorem 3. T (Π, ζ κ , ¯µ) dν
+¯τ β ζ κ
Z
¯
¯
¯
¯
¯
T
Hence, (¯σ, ¯ω, ¯χ,λ,¯τ,¯ a,b, ¯µ) is a robust weak effi- + ξ(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ) ¯χ Ψ ϱ Π, ¯ a +

cient solution to (WD). Ω

Further, from (3) it follows that the objective λ θ ϱ (Π, ζ κ , b)+ ¯τ β ϱ (Π, ζ κ , ¯µ) dν =0 (19)
¯ ¯ ¯
¯ ¯
T
¯ T
function values of (RUVP) and (WD) are equal Z
¯
¯
at (¯σ, ¯ω, ¯χ, λ, ¯τ, ¯ a, b, ¯µ). □
As Ψ(., ¯ a)dν is strictly invex w.r.t. η and ξ at
Ω
¯
Theorem 5. [Strict Converse Duality Theorem] (ζ, ¯ϱ) and ¯χ>0,
¯
¯
¯
Assume (¯σ, ¯ω) and (ζ, ¯ϱ, ¯χ, λ, ¯τ, ¯ a, b, ¯µ) to be ro-
bust weak efficient solutions of (RUVP) and (WD) Z T ¯ Z T ¯
respectively, such that ¯ χ Ψ Λ, ¯ a dν − ¯ χ Ψ Π, ¯ a dν >
Ω Ω
Z
¯
¯
¯
T

η(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ)¯χ Ψ ζ Π, ¯ a dν

Z Z Ω
¯
¯ T
¯ ¯
¯ ¯ ¯
Ψ Λ, ¯ a dν ≦ Ψ Π, ¯ a +λ θ(Π, ζ κ , b)+ Z
¯
¯
¯
T

Ω Ω + ξ(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ)¯χ Ψ ϱ Π, ¯ a dν, (20)
Ω
!
¯ ¯
T
¯ τ β(Π, ζ κ , ¯µ) dν (18) Similarly, as Z ¯ T ¯ Z ¯ τ β(., ., ¯µ)dν
T
λ θ(., ., b)dν and
Ω Ω
¯
Z are invex w.r.t. η and ξ at (ζ, ¯ϱ),
¯
Further, if Ψ(., ¯ a)dν is strictly in-
Ω
Z Z Z Z
¯
¯ ¯ ¯
¯
¯ T
¯ T
¯
¯ T
T
vex, λ θ(., ., b)dν and ¯ τ β(., ., ¯µ)dν λ θ(Λ, ¯σ κ , b)dν − λ θ(Π, ζ κ , b)dν ≧
Ω Ω Ω Ω
¯
¯
are invex w.r.t. η(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ) and Z
¯ T
¯
¯
¯ ¯ ¯
¯
¯
¯
ξ(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ) at (ζ, ¯ϱ), then (¯σ, ¯ω)= η(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ)λ θ ζ (Π, ζ κ , b)dν
¯
(ζ, ¯ϱ). Z Ω
¯ ¯ ¯
¯
¯ T
¯
+ D κ η(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ)λ θ ζ κ (Π, ζ κ , b)dν+
Ω
Z
Proof. Suppose to the contrary that (¯σ, ¯ω)̸= ξ(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ)λ θ ϱ (Π, ζ κ , b)dν (21)
¯
¯
¯ T
¯ ¯ ¯
¯
¯
¯
¯
(ζ, ¯ϱ). Since (ζ, ¯ϱ, ¯χ, λ, ¯τ, ¯ a, b, ¯µ) is a robust Ω
weak efficient solution of (WD), therefore by us- and
ing equation (10), (11) and multiplying them by
¯
¯
¯
¯
η(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ) and ξ(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ), Z Z
¯ ¯
T
T
¯
respectively, adding them and thereafter integrat- ¯ τ β(Λ, ¯σ κ , ¯µ)dν − ¯ τ β(Π, ζ κ , ¯µ)dν ≧
ing, we obtain Ω Ω
Z
¯ ¯
T
¯
¯
η(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ)¯τ β ζ (Π, ζ κ , ¯µ)dν

Z
Ω
¯
¯
T
¯

η(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ) ¯χ Ψ ζ Π, ¯ a Z
¯
¯ ¯
T
¯
Ω + D κ η(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ)¯τ β ζ κ (Π, ζ κ , ¯µ)dν+
¯ ¯
¯ ¯ ¯
T
¯ T
+λ θ ζ (Π, ζ κ , b)+ ¯τ β ζ (Π, ζ κ , ¯µ) Z Ω
ξ(t, ¯σ, ζ, ¯σ κ , ζ κ , ¯ω, ¯ϱ)¯τ β ϱ (Π, ζ κ , ¯µ)dν. (22)
! ¯ ¯ T ¯ ¯
h i
¯ ¯
¯ T ¯ ¯ ¯ T (Π, ζ κ , ¯µ) dν Ω
−D κ λ θ ζ κ (Π, ζ κ , b)+ ¯τ β ζ κ
Adding the inequalities (20)−(22),
131