Hindawi Publishing Corporation
Advances in Operations Research
Volume 2012, Article ID 279215, 13 pages
doi:10.1155/2012/279215
Research Article
Generalized d-ρ-η-θ-Type I Univex Functions in
Multiobjective Optimization
Pallavi Kharbanda,1 Divya Agarwal,2 and Deepa Sinha3
1
Department of Mathematics, Dronacharya Government College (DGC), New Railway Road,
Gurgaon 122001, India
2
Department of Applied Sciences and Humanities, ITM University, Gurgaon 122017, India
3
Centre for Mathematical Sciences, Banasthali University, Rajasthan 304022, India
Correspondence should be addressed to Pallavi Kharbanda, pallavimath@yahoo.co.in
Received 13 March 2012; Revised 5 June 2012; Accepted 11 June 2012
Academic Editor: J. J. Judice
Copyright q 2012 Pallavi Kharbanda et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
A new class of generalized functions d-ρ-η-θ-type I univex is introduced for a nonsmooth multiobjective programming problem. Based upon these generalized functions, sufficient optimality
conditions are established. Weak, strong, converse, and strict converse duality theorems are also
derived for Mond-Weir-type multiobjective dual program.
1. Introduction
Generalizations of convexity related to optimality conditions and duality for nonlinear single
objective or multiobjective optimization problems have been of much interest in the recent
past and thus explored the extent of optimality conditions and duality applicability in
mathematical programming problems. Invexity theory was originated by Hanson 1. Many
authors have then contributed in this direction.
For a nondifferentiable multiobjective programming problem, there exists a generalisation of invexity to locally Lipschitz functions with gradients replaced by the Clarke
generalized gradient. Zhao 2 extended optimality conditions and duality in nonsmooth
scalar programming assuming Clarke generalized subgradients under type I functions.
However, Antczak 3 used directional derivative in association with a hypothesis of an invex
kind following Ye 4. On the other hand, Bector et al. 5 generalized the notion of convexity
to univex functions. Rueda et al. 6 obtained optimality and duality results for several
mathematical programs by combining the concepts of type I and univex functions. Mishra
7 obtained optimality results and saddle point results for multiobjective programs under
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generalized type I univex functions which were further generalized to univex type I-vectorvalued functions by Mishra et al. 8. Jayswal 9 introduced new classes of generalized
α-univex type I vector valued functions and established sufficient optimality conditions
and various duality results for Mond-Weir type dual program. Generalizing the work of
Antczak 3, recently Nahak and Mohapatra 10 obtained duality results for multiobjective
programming problem under d-ρ-η-θ invexity assumptions.
In this paper, by combining the concepts of Mishra et al. 8 and Nahak and Mohapatra
10, we introduce a new generalized class of d-ρ-η-θ-type I univex functions and establish
weak, strong, converse, and strict converse duality results for Mond-Weir type dual.
2. Preliminaries and Definitions
The following convention of vectors in Rn will be followed throughout this paper: x y ⇔
y; x > y ⇔ xi > yi , i 1, 2, . . . , n. Let D be a
xi yi , i 1, 2, . . . , n; x ≥ y ⇔ x y, x /
nonempty subset of Rn , η : D × D → Rn , xo be an arbitrary point of D and h : D → R, φ :
p
R → R, b : D × D → R . Also, we denote R≥ {y : y ∈ Rp and y ≥ 0} and Rk {y : y ∈ Rk
and y 0}.
Definition 2.1 Ben-Israel and Mond 11. Let D ⊆ Rn be an invex set. A function h is called
preinvex on D with respect to η, if for all x, xo ∈ D,
λhx 1 − λhxo h xo ληx, xo ,
∀λ ∈ 0, 1.
2.1
Definition 2.2 Clarke 12. The function h is said to be locally Lipschitz at xo ∈ D, if there
exists a neighbourhood vxo of xo and a constant k > 0 such that
h y − hx ky − x ∀x, y ∈ vxo ,
2.2
where · denotes the euclidean norm. Also, we say that h is locally Lipschitz on D if it is
locally Lipschitz at every point of D.
Definition 2.3 Bector et al. 5. A differentiable function h is said to be univex at xo if for all
x ∈ D, we have
bx, xo φhx − hxo hxo T ηx, xo .
2.3
We consider the following nonlinear multiobjective programming problem:
Minimize
fx f1 x, f2 x, . . . , fp x
subject to gx 0,
MP
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3
where x ∈ D and the functions f : D → Rp , g : D → Rk . Let X {x ∈ D : gx 0} be a set
of feasible solutions of MP . For xo ∈ D, if we denote by
Jxo j ∈ {1, 2, . . . , k} : gj xo 0 ,
o j ∈ {1, 2, . . . , k} : gj xo < 0 ,
Jx
Jxo j ∈ {1, 2, . . . , k} : gj xo > 0 ,
2.4
then
o ∪ Jxo {1, 2, . . . , k}.
Jxo ∪ Jx
2.5
Since the objectives in multiobjective programming problems generally conflict with
one another, an optimal solution is chosen from the set of efficient or weak efficient solution
in the following sense by Miettinen 13.
Definition 2.4. A point xo ∈ X is said to be an efficient solution of MP , if there exists no
x ∈ X such that
fx ≤ fxo .
2.6
Definition 2.5. A point xo ∈ X is said to be a weak efficient solution of MP , if there exists no
x ∈ X such that
2.7
fx < fxo .
Now we define a new class of d-ρ-η-θ-type I univex functions which generalize the
work of Mishra et al. 8 and Nahak and Mohapatra 10. Let functions f f1 , . . . , fp : D →
Rp and g g1 , . . . , gk : D → Rk are directionally differentiable at xo ∈ X, η : X × D → Rn ,
bo and b1 are nonnegative functions defined on X × D, φo : Rp → Rp and φ1 : Rk → Rk , while
ρ ∈ Rpk and θ·, · : X × D → Rn be vector-valued functions.
Definition 2.6. f, g is said to be d-ρ-η-θ-type I univex at xo ∈ D if for all x ∈ X
bo x, xo φo fx − fxo f xo ; ηx, xo ρ1 θx, xo 2 ,
2
−b1 x, xo φ1 gxo g xo ; ηx, xo ρ θx, xo .
2
2.8
If the inequalities in f are strict whenever x /
xo , then f, g is said to be semistrictly
d-ρ-η-θ-type I univex at xo .
Remark 2.7. i If ρ1 , ρ2 0, bo x, xo b1 x, xo 1, φo t t, φ1 t t, then above definition
becomes that of d-type I function 14.
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ii If in the above definition, the functions f and g are differentiable functions such
that f xo ; ηx, xo fxo T ηx, xo , g xo ; ηx, xo gxo T ηx, xo ; ρ1 , ρ2 0;
bo x, xo b1 x, xo 1, φo t t, φ1 t t, then we obtain the definition of type I function
15.
Definition 2.8. f, g is said to be a weak strictly pseudo-quasi d-ρ-η-θ-type I univex at xo ∈
D if for all x ∈ X
bo x, xo φo fx − fxo ≤ 0 ⇒ f xo ; ηx, xo < −ρ1 θx, xo 2 ,
b1 x, xo φ1 gxo 0 ⇒ g xo ; ηx, xo −ρ2 θx, xo 2 .
2.9
Definition 2.9. f, g is said to be strong pseudo-quasi d-ρ-η-θ-type I univex at xo ∈ D if for
all x ∈ X
bo x, xo φo fx − fxo ≤ 0 ⇒ f xo ; ηx, xo ≤ −ρ1 θx, xo 2 ,
b1 x, xo φ1 gxo 0 ⇒ g xo ; ηx, xo −ρ2 θx, xo 2 .
2.10
Definition 2.10. f, g is said to be weak strictly-pseudo d-ρ-η-θ-type I univex at xo ∈ D if
for all x ∈ X
bo x, xo φo fx − fxo ≤ 0 ⇒ f xo ; ηx, xo < −ρ1 θx, xo 2 ,
b1 x, xo φ1 gxo 0 ⇒ g xo ; ηx, xo < −ρ2 θx, xo 2 .
2.11
Definition 2.11. f, g is said to be a weak quasistrictly-pseudo d-ρ-η-θ-type I univex at xo ∈
D if for all x ∈ X
bo x, xo φo fx − fxo ≤ 0 ⇒ f xo ; ηx, xo −ρ1 θx, xo 2 ,
b1 x, xo φ1 gxo 0 ⇒ g xo ; ηx, xo ≤ −ρ2 θx, xo 2 .
2.12
Remark 2.12. In the above definitions, if f and g are differentiable functions such that
f xo ; ηx, xo fxo T ηx, xo ; g xo ; ηx, xo gxo T ηx, xo ; ρ1 ρ2 0, then
we obtain the functions given in Mishra et al. 8.
3. Sufficient Optimality Conditions
In this section, we discuss sufficient optimality conditions for a point to be an efficient
solution of MP under generalized d-ρ-η-θ-type I univex assumptions. In the following
theorems, μ μ1 , . . . , μp ∈ Rp and λ λ1 , . . . , λJxo ∈ Jxo .
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Theorem 3.1. Suppose there exists a feasible solution xo for MP, vector functions η : X × D → Rn
and vectors μ > 0 and λ 0, such that
i
p
i1
μi fi xo ; ηx, xo j∈Jxo λj gj xo ; ηx, xo 0,
ii f, gJxo is a strong pseudo-quasi d-ρ-η-θ-type I univex at xo ,
iii for any u ∈ Rp , u ≤ 0 ⇒ φo u ≤ 0 and v ∈ RJxo , v 0 ⇒ φ1 v 0; bo x, xo > 0,
b1 x, xo 0,
p
iv i1 μi ρi1 j∈Jxo λj ρj2 0,
then xo is an efficient solution of MP.
Proof. Suppose xo is not an efficient solution of MP, then there exists x ∈ X such that
fx ≤ fxo .
Since gj xo 0, j ∈ Jxo , therefore by hypothesis iii, we get
bo x, xo φo fx − fxo ≤ 0,
3.1
b1 x, xo φ1 gJxo xo 0.
which using hypothesis ii yields
f xo ; ηx, xo ≤ −ρ1 θx, xo 2 ,
2
gJx
xo ; ηx, xo −ρJx
θx, xo 2 .
o
o
3.2
Also μ > 0 and λ 0, so, we get
p
p
μi fi xo ; ηx, xo < − μi ρi1 θx, xo 2 ,
i1
j∈Jxo i1
λj gj xo ; ηx, xo −
3.3
j∈Jxo λj ρj2 θx, xo 2 .
Adding the above inequalities, we obtain
p
i1
μi fi xo ; ηx, xo j∈Jxo ⎛
⎞
p
λj gj xo ; ηx, xo < −⎝ μi ρi1 λj ρj2 ⎠θx, xo 2
i1
0
which contradicts hypothesis i. Hence the proof.
j∈Jxo By hypothesis iv ,
3.4
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Theorem 3.2. Suppose there exists a feasible solution xo for MP, vector functions η : X × D → Rn
and vectors μ ≥ 0 and λ 0, such that
i
p
i1
μi fi xo ; ηx, xo j∈Jxo λj gj xo ; ηx, xo 0,
ii f, gJxo is a weak strictly-pseudo-quasi d-ρ-η-θ-type I univex at xo ,
iii for any u ∈ Rp , u ≤ 0 ⇒ φo u ≤ 0 and v ∈ RJxo , v 0 ⇒ φ1 v 0; bo x, xo > 0,
b1 x, xo 0,
p
iv i1 μi ρi1 j∈Jxo λj ρj2 0,
then xo is an efficient solution of MP.
Proof. Suppose xo is not an efficient solution of MP, then there exists x ∈ X such that fx ≤
fxo .
As gj xo 0, j ∈ Jxo , so, hypothesis iii yields
bo x, xo φo fx − fxo ≤ 0,
3.5
b1 x, xo φ1 gJxo xo 0.
By hypothesis ii, the above inequalities imply
f xo ; ηx, xo < −ρ1 θx, xo 2 ,
2
gJx
θx, xo 2 .
xo ; ηx, xo −ρJx
o
o
3.6
Since μ ≥ 0 and λ 0, we get
p
μi fi
p
xo ; ηx, xo < − μi ρi1 θx, xo 2 ,
i1
j∈Jxo i1
λj gj
3.7
xo ; ηx, xo −
j∈Jxo λj ρj2 θx, xo 2 .
Adding the above inequalities, we obtain
p
i1
μi fi xo ; ηx, xo j∈Jxo ⎛
⎞
p
λj gj xo ; ηx, xo < −⎝ μi ρi1 λj ρj2 ⎠θx, xo 2
i1
0
which contradicts hypothesis i. Hence the proof.
j∈Jxo using hypothesis iv ,
3.8
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Theorem 3.3. Suppose there exists a feasible solution xo for MP, vector functions η : X × D → Rn
and vectors μ ≥ 0 and λ 0, such that
i
p
i1
μi fi xo ; ηx, xo j∈Jxo λj gj xo ; ηx, xo 0,
ii f, gJxo is a weak strictly-pseudo d-ρ-η-θ-type I univex at xo ,
iii for any u ∈ Rp , u ≤ 0 ⇒ φo u ≤ 0 and v ∈ RJxo , v 0 ⇒ φ1 v 0; bo x, xo > 0,
b1 x, xo 0,
p
iv i1 μi ρi1 j∈Jxo λj ρj2 0,
then xo is an efficient solution of MP.
Proof. Suppose xo is not an efficient solution of MP, then there exists x ∈ X such that
fx ≤ fxo .
As gj xo 0, j ∈ Jxo , so hypothesis iii implies
bo x, xo φo fx − fxo ≤ 0,
b1 x, xo φ1 gJxo xo 0.
3.9
Since hypothesis ii holds, above inequalities imply
f xo ; ηx, xo < −ρ1 θx, xo 2 ,
gJx
o
xo ; ηx, xo <
2
−ρJx
θx, xo 2 .
o
3.10
Also μ ≥ 0 and λ 0, so we obtain
p
p
μi fi xo ; ηx, xo < − μi ρi1 θx, xo 2 ,
i1
j∈Jxo i1
λj gj xo ; ηx, xo −
3.11
j∈Jxo λj ρj2 θx, xo 2 .
On adding and using hypothesis iv, above inequalities yield
⎞
⎛
p
p
μi fi xo ; ηx, xo λj gj xo ; ηx, xo < −⎝ μi ρi1 λj ρj2 ⎠θx, xo 2 0
i1
j∈Jxo i1
j∈Jxo 3.12
which contradicts hypothesis i. Hence the proof.
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Theorem 3.4. Suppose there exists a feasible solution xo for MP, vector functions η : X × D → Rn
and vectors μ 0 and λ > 0, such that
i
p
i1
μi fi xo ; ηx, xo j∈Jxo λj gj xo ; ηx, xo 0,
ii f, gJxo is weak quasi-strictly-pseudo d-ρ-η-θ-type I univex at xo ,
iii for any u ∈ Rp , u ≤ 0 ⇒ φo u ≤ 0 and v ∈ RJxo , v 0 ⇒ φ1 v 0; bo x, xo >
0, b1 x, xo 0,
p
iv i1 μi ρi1 j∈Jxo λj ρj2 0,
then xo is an efficient solution of MP.
Proof. Suppose xo is not an efficient solution of MP, then there exists x ∈ X such that fx ≤
fxo .
Since gj xo 0, j ∈ Jxo , therefore hypothesis iii yields
bo x, xo φo fx − fxo ≤ 0,
b1 x, xo φ1 gJxo xo 0.
3.13
f xo ; ηx, xo −ρ1 θx, xo 2 ,
2
xo ; ηx, xo ≤ −ρJx
gJx
θx, xo 2 .
o
o
3.14
By hypothesis ii, we get
Also μ 0 and λ > 0, so, we obtain
p
i1
j∈Jxo p
μi fi xo ; ηx, xo − μi ρi1 θx, xo 2 ,
3.15
i1
λj gj xo ; ηx, xo < −
λj ρj2 θx, xo 2 .
j∈Jxo 3.16
On adding and using hypothesis iv, above inequalities yield
⎞
⎛
p
p
μi fi xo ; ηx, xo λj gj xo ; ηx, xo < −⎝ μi ρi1 λj ρj2 ⎠θx, xo 2
i1
j∈Jxo i1
j∈Jxo 3.17
0,
which contradicts hypothesis i. Hence the proof.
Now, following Antczak 3, we state following necessary optimality conditions.
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Theorem 3.5 Karush-Kuhn-Tucker type necessary optimality conditions. If
i xo is a weakly efficient solution of MP,
o ,
ii gj is continuous at xo for j ∈ Jx
iii there exists a vector functions η : X × D → Rn ,
iv for all i 1, p and j ∈ Jxo , fi and gj are directionally differentiable at xo and the
functions fi xo ; ηx, xo , i 1, p and gj xo ; ηx, xo , j ∈ Jxo are preinvex functions
of x on X,
v the function g satisfies the generalized Slater’s constraint qualification at xo ,
p
then there exists μ ∈ R and λ ∈ Rk such that
p
k
μi fi xo ; ηx, xo λj gj xo ; ηx, xo 0
i1
∀x ∈ X,
j1
λj gj xo 0,
3.18
j 1, k.
4. Mond-Weir Type Duality
In this section, we consider Mond-Weir type dual of MP and establish weak, strong, converse, and strict converse duality theorems. In this section, we denote gλ λ1 g1 , . . . , λk gk .
f y
Max
subject to
p
k
μi fi y, η x, y λj gj y, η x, y 0 ∀x ∈ X,
i1
λj gj y 0,
MWD
j1
j 1, k,
p
where y ∈ D, μ ∈ R , λ ∈ Rk , η : X × D → Rn . Let W be the set of feasible points of MWD.
Theorem 4.1 Weak Duality. Let x and y, μ, λ, η be the feasible solutions for MP and MWD
respectively. If
i f, gλ is a weak strictly-pseudo-quasi d-ρ-η-θ-type I univex at y,
ii for any u ∈ Rp , u ≤ 0 ⇒ φo u ≤ 0 and v ∈ Rk , v 0 ⇒ φ1 v 0; bo x, y >
0, b1 x, y 0,
p
iii i1 μi ρi1 kj1 ρj2 0,
then
fx f y .
4.1
fx ≤ f y .
4.2
Proof. Suppose to the contrary that
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Since λj gj y 0, j 1, k, hypothesis ii yields
bo x, y φo fx − f y ≤ 0,
b1 x, y φ1 gλ y 0.
4.3
As hypothesis i holds, therefore the above inequalities imply
2
f y; η x, y < −ρ1 θx, y ,
2
gλ y; η x, y −ρ2 θx, y .
4.4
p
Also μ ∈ R , so, we obtain
p
p
2
μi fi y; η x, y < − μi ρi1 θx, y ,
i1
k
j1
i1
λj gj
y; η x, y
−
k
2
x, y .
ρj2 θ
4.5
j1
On adding above inequalities and using hypothesis iii, we get
⎞
⎛
p
p
k
k
2
μi fi y; η x, y λj gj y; η x, y < −⎝ μi ρi1 ρj2 ⎠θx, y 0,
i1
j1
i1
4.6
j1
which is a contradiction to the dual constraint. Hence the proof.
The proofs of the following weak duality theorems are similar to Theorem 4.1 and
hence are omitted.
Theorem 4.2 Weak Duality. Let x and y, μ, λ, η be the feasible solutions for MP and MWD,
respectively, with μi > 0, i 1, p. If
i f, gλ is a strong pseudo-quasi d-ρ-η-θ-type I univex at y,
ii for any u ∈ Rp , u ≤ 0 ⇒ φo u ≤ 0 and v ∈ Rk , v 0 ⇒ φ1 v 0; bo x, y >
0, b1 x, y 0,
p
iii i1 μi ρi1 kj1 ρj2 0,
then fx fy.
Theorem 4.3 Weak Duality. Let x and y, μ, λ, η be the feasible solutions for MP and MWD,
respectively. If
i f, gλ is weak strictly-pseudo d-ρ-η-θ-type I univex at y,
ii for any u ∈ Rp , u ≤ 0 ⇒ φo u ≤ 0 and v ∈ Rk , v 0 ⇒ φ1 v 0; bo x, y >
0, b1 x, y 0,
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iii
p
i1
μi ρi1 k
j1
11
ρj2 0,
then fx fy.
Corollary 4.4. Let xo and yo , μ, λ, η be the feasible solutions for MP and MWD, respectively,
such that fxo fyo . If the weak duality holds between MP and MWD for all feasible solutions
of two problems, then xo is efficient for MP and yo , μ, λ, η is efficient for MWD.
Proof. Suppose that xo is not efficient for MP, then for some x ∈ X
fx ≤ fxo f yo .
4.7
which contradicts weak duality theorems as yo , μ, λ, η is feasible for MWD and x is
feasible for MP. So, xo is efficient for MP. Similarly yo , μ, λ, η is efficient for MWD.
Theorem 4.5 Strong Duality. Let xo be a weakly efficient solution of MP, gj is continuous at xo
o , f, g are directionally differentiable at xo with f xo , ηx, xo , and g xo , ηx, xo as
for j ∈ Jx
i
j
preinvex functions on X. Also if g satisfies the generalized Slater’s constraint qualification at xo , then
p
∃ μ ∈ R , λ ∈ Rk such that xo , μ, λ, η is feasible for MWD and the objective function values of
MP and MWD are equal. Moreover, if any of weak duality theorem holds, then xo , μ, λ, η is an
efficient solution of MWD.
Proof. Since xo is a weakly efficient solution of MP, therefore by Theorem 3.5, there exists
p
μ ∈ R , λ ∈ Rk such that
p
k
μi fi xo , ηx, xo λj gj xo , ηx, xo 0
i1
∀x ∈ X,
j1
λj gj xo 0,
4.8
j 1, k.
It follows that xo , μ, λ, η ∈ W and therefore feasible for MWD. Clearly objective
function values of MP and MWD are equal at optimal points.
η
μ
, λ,
∈ W
Suppose xo , μ, λ, η is not an efficient solution for MWD. Then ∃ y,
which contradicts weak duality theorems. Therefore xo , μ, λ, η is an
such that fxo ≤ fy,
efficient solution of MWD. Hence the proof.
Theorem 4.6 Converse Duality. Let yo , μ, λ, η be a feasible solution of MWD. If
i f, gλ is a weak strictly-pseudo-quasi d-ρ-η-θ-type I univex at yo ,
ii for any u ∈ Rp , u ≤ 0 ⇒ φo u ≤ 0 and v ∈ Rk , v 0 ⇒ φ1 v 0; bo xo , yo >
0, b1 xo , yo 0,
p
iii i1 μi ρi1 kj1 ρj2 0,
then yo is an efficient solution of MP.
Proof. Suppose that yo is not an efficient solution of MP. Then ∃ xo ∈ X such that
fxo ≤ f y0 .
4.9
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Now proceeding as in Theorem 4.1 Weak Duality, we obtain a contradiction. Hence yo is an
efficient solution of MP.
Theorem 4.7 Strict Converse Duality. Let xo and yo , μ, λ, η be the feasible solutions of MP
and MWD, respectively. If
i fxo fyo ,
ii f, gλ is a weak quasi-strictly-pseudo d-ρ-η-θ-type I univex at yo ,
iii for any u ∈ Rp , u 0 ⇒ φo u ≤ 0 and v ∈ Rk , v 0 ⇒ φ1 v 0; bo xo , yo >
0, b1 xo , yo 0,
p
iv i1 μi ρi1 kj1 ρj2 0,
then xo yo .
Proof. Suppose xo /
yo .
Since yo is a feasible solution of MWD, therefore by hypothesis i and hypothesis
iii, we get
bo xo , yo φo fxo − f yo ≤ 0,
b1 xo , yo φ1 gλ yo 0.
4.10
2
f yo ; η xo , yo −ρ1 θxo , yo ,
2
gλ yo ; η xo , yo ≤ −ρ2 θ xo , yo .
4.11
By hypothesis ii, we obtain
p
Since μ ∈ R , therefore the above inequalities yield
p
p
2
μi fi yo ; η xo , yo − μi ρi1 θxo , yo ,
i1
4.12
i1
k
k
2
λj gj yo ; η xo , yo < − ρj2 θ xo , yo ,
j1
4.13
j1
which on adding gives
p
i1
μi fi yo ; η xo , yo
k
⎞
⎛
p
k
2
λj gj yo ; η xo , yo < −⎝ μi ρi1 ρj2 ⎠θxo , yo j1
i1
0
j1
using hypothesis iv ,
which is a contradiction to feasibility of yo . Hence xo yo .
4.14
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