second order duality in multiobjective programming - Semantic Scholar

Journal of Applied Analysis Vol. 14, No. 1 (2008), pp. 131-148

SECOND ORDER DUALITY IN MULTIOBJECTIVE PROGRAMMING I. AHMAD and Z. HUSAIN Received November 10, 2006 and, in revised form, November 6, 2007

Abstract. A nonlinear multiobjective programming problem is considered. Weak, strong and strict converse duality theorems are established under generalized second order (F, α, ρ, d)-convexity for second order Mangasarian type and general Mond-Weir type vector duals.

1. Introduction In recent years, there has been an increasing interest in generalizations of convexity in connection with sufficiency and duality in optimization problems. It has been found that only a few properties of convex functions are needed for establishing sufficiency and duality theorems. Using properties needed as definitions of new classes of functions, it is possible to generalize the notion of convexity and to extend the validity of theorems to larger classes of optimization problems. Consequently, several classes of generalized convexity have been introduced. More specifically, the concept of 2000 Mathematics Subject Classification. Primary: 90C29, 90C30, 90C46. Key words and phrases. Multiobjective programming, second order duality, efficient solution, generalized (F, α, ρ, d)-convexity. The research of second author is supported by the Department of Atomic Energy, Government of India, under the NBHM Post-Doctoral Fellowship Program No. 40/9/2005-R & D II/2398.

ISSN 1425-6908

c Heldermann Verlag.

132

I. AHMAD AND Z. HUSAIN

(F, ρ)-convexity was introduced by Preda [16], an extension of F -convexity [9] and ρ-convexity [17], and he used the concept to obtain some duality results for Wolfe vector dual, Mond-Weir vector dual and general Mond-Weir vector dual to multiobjective programming problem. Gulati and Islam [8] established sufficiency and duality results for multiobjective programming problems under generalized F -convexity. Later on, Aghezzaf and Hachimi [1] and Ahmad [2] generalized these results involving generalized (F, ρ)convex functions. For a more comprehensive view of optimality conditions and duality results in multiobjective programming, we refer [6, 7, 18] and references cited therein. Mangasarian [13] first formulated the second order dual for a nonlinear programming problem and established duality results under somewhat involved assumptions. Mond [14] reproved second order duality theorems under simpler assumptions than those previously used by Mangasarian [13], and showed that the second order dual has computational advantages over the first order dual. Zhang and Mond [19] extended the class of (F, ρ)-convex functions to second order (F, ρ)-convex functions and obtained duality results for Mangasarian type, Mond-Weir type and general Mond-Weir type multiobjective dual problems. A newly introduced concept of generalized convexity, named as (F, α, ρ, d)-convexity can be viewed in [4, 10, 11], while (F, α, ρ, d)pseudoconvexity and (F, α, ρ, d)-quasiconvexity can be found in [5]. Recently, Ahmad and Husain [3] introduced the class of generalized second order (F, α, ρ, d)-convex functions and discussed duality results for MondWeir type vector dual. Consider the following nonlinear multiobjective programming problem: (MP) Minimize f (x) = [f1 (x), f2 (x), . . . , fk (x)] subject to x ∈ S = {x ∈ X : g(x) 5 0}, where f = (f1 , f2 , . . . , fk ) : X 7→ Rk , g = (g1 , g2 , . . . , gm ) : X 7→ Rm are assumed to be twice differentiable functions over X, an open subset of Rn . In this paper, we establish duality theorems under generalized second order (F, α, ρ, d)-convexity, for second order Mangasarian type and general Mond-Weir type duals associated with (MP). These results extend the results obtained by Mond and Zhang [15], Zhang and Mond [19] and Ahmad [2]. 2. Notations and preliminaries Throughout the paper, following convention for vectors x, y ∈ Rn will be followed: x = y if and only if xi = yi , i = 1, 2, . . . , n; x ≥ y if and only if x = y and x 6= y; x > y if and only if xi > yi , i = 1, 2, . . . , n.

SECOND ORDER DUALITY IN MULTIOBJECTIVE PROGRAMMING

133

Definition 2.1. A point x ¯ ∈ S is said to be an efficient solution of the vector minimum problem (MP), if there exists no other x ∈ S such that f (x) ≤ f (¯ x). In the sequel, we require the following definitions [3]. Definition 2.2. A functional F : X × X × Rn 7→ R is said to be sublinear in its third argument, if for all x, x ¯∈X (i) F (x, x ¯; a + b) 5 F (x, x ¯; a) + F (x, x ¯; b), for all a, b ∈ Rn , (ii) F (x, x ¯; βa) = βF (x, x ¯; a), for all β ∈ R, β = 0, and for all a ∈ Rn . Let F be sublinear and the scalar function φ : X 7→ tiable at x ¯ ∈ X and ρ ∈ R.

R be twice differen-

Definition 2.3. The function φ is said to be second order (F, α, ρ, d)convex at x ¯ on X, if for all x ∈ X, there exist vector p ∈ Rn , a real valued function α : X ×X 7→ R+ \{0}, and a real valued function d(·, ·) : X ×X 7→ R such that 1 φ(x) − φ(¯ x) + pt ∇2 φ(¯ x)p =F x, x ¯; α(x, x ¯) ∇φ(¯ x) + ∇2 φ(¯ x)p 2 + ρd2 (x, x ¯). If for all x ∈ X, x 6= x ¯, the above inequality holds as strict inequality, then φ is said to be strictly second order (F, α, ρ, d)-convex at x ¯ on X. Definition 2.4. The function φ is said to be second order (F, α, ρ, d)pseudoconvex at x ¯ on X, if for all x ∈ X, there exist vector p ∈ Rn , a real valued function α : X × X 7→ R+ \ {0}, and a real valued function d(·, ·) : X × X 7→ R such that 1 x)p ⇒ F x, x ¯; α(x, x ¯) ∇φ(¯ x) + ∇2 φ(¯ x)p φ(x) < φ(¯ x) − pt ∇2 φ(¯ 2 < −ρd2 (x, x ¯). Definition 2.5. The function φ is said to be strictly second order (F, α, ρ, d)-pseudoconvex at x ¯ on X, if for all x ∈ X, x 6= x ¯, there exist vector p ∈ Rn , a real valued function α : X × X 7→ R+ \ {0}, and a real valued function d(·, ·) : X × X 7→ R such that F x, x ¯; α(x, x ¯) ∇φ(¯ x) + ∇2 φ(¯ x)p = −ρd2 (x, x ¯) ⇒ φ(x) 1 t 2 > φ(¯ x) − p ∇ φ(¯ x)p, 2

134


or equivalently 1 φ(x) 5 φ(¯ x) − pt ∇2 φ(¯ x)p ⇒ F x, x ¯; α(x, x ¯) ∇φ(¯ x) + ∇2 φ(¯ x)p 2 < −ρd2 (x, x ¯). Definition 2.6. The function φ is said to be second order (F, α, ρ, d)quasiconvex at x ¯ on X, if for all x ∈ X, there exist vector p ∈ Rn , a real valued function α : X × X 7→ R+ \ {0}, and a real valued function d(·, ·) : X × X 7→ R such that 1 φ(x) 5 φ(¯ x) − pt ∇2 φ(¯ x)p ⇒ F x, x ¯; α(x, x ¯) ∇φ(¯ x) + ∇2 φ(¯ x)p 2 2 5 −ρd (x, x¯), or equivalently

F x, x ¯; α(x, x ¯) ∇φ(¯ x) + ∇2 φ(¯ x)p > −ρd2 (x, x ¯) ⇒ φ(x) 1 > φ(¯ x) − pt ∇2 φ(¯ x)p. 2

A k-dimensional vector function ψ = (ψ1 , ψ2 , . . . , ψk ) is said to be second order (F, α, ρ, d)-convex, if each ψi , i = 1, 2, . . . , k, is second order (F, α, ρi , d)-convex for the same sublinear functional F . Other definitions follow similarly. Remark 2.1. Let α(x, x ¯) = 1. Then second order (F, α, ρ, d)-convexity becomes the second order (F, ρ)-convexity introduced by Zhang and Mond [19]. In addition, if we set second order term equal to zero i.e., p = 0, it reduces to (F, ρ)-convexity in [2, 16]. In [12], Maeda derived the following necessary conditions for a feasible solution x∗ to be an efficient solution of (MP) under generalized Guignard constraint qualification (GGCQ). We need these conditions in the proof of strong duality theorems. Theorem 2.1 (Kuhn-Tucker Type Necessary Conditions). Assume that x∗ is an efficient solution for (M P ) at which the generalized Guignard constraint qualification (GGCQ) is satisfied. Then there exist λ∗ ∈ Rk and y ∗ ∈ Rm , such that λ∗ t ∇f (x∗ ) + y ∗ t ∇g(x∗ ) = 0, y ∗ t g(x∗ ) = 0, y∗

= 0,

λ∗ > 0, λ∗ t e = 1.


135

3. Mangasarian type second order duality In this section, we consider the following Mangasarian type second order dual associated with multiobjective problem (MP) and establish weak, strong and strict converse duality theorems under generalized second order (F, α, ρ, d)-convexity: 1 (WD) Maximize f1 (u) + y t g(u) − pt ∇2 f1 (u) + y t g(u) p, . . . , 2 1 t fk (u) + y g(u) − pt ∇2 fk (u) + y t g(u) p 2 subject to ∇λt f (u) + ∇2 λt f (u)p + ∇y t g(u) + ∇2 y t g(u)p = 0,

= 0,

(3.2)

λ > 0,

(3.3)

λt e = 1,

(3.4)

y

where e = (1, 1, . . . , 1) ∈ dimensional vector.

Rk ,

(3.1)

λ is a k-dimensional vector, and is an m-

Theorem 3.1 (Weak Duality). Suppose that for all feasible x in (M P ) and all feasible (u, y, λ, p) in (W D) (i) fi , i = 1, 2, . . . , k, is second order (F, α, ρi , d)-convex at u, and gj , j = 1, 2, . . . , m, is second order (F, α, σj , d)-convex at u, and   k m X X 1  λi ρi + σj yj  = 0; α(x, u) i=1

j=1

or (ii) λt f + y t g is second order (F, α, ρ, d)-pseudoconvex at u, and ρ α(x, u)

= 0.

Then, the following cannot hold fi (x) and

5 fi(u) + ytg(u) − 12 pt∇2

fi (u) + y t g(u) p, for all i ∈ K,

(3.5)

1 fj (x) < fj (u)+y t g(u)− pt ∇2 fj (u) + y t g(u) p, for some j ∈ K. (3.6) 2

136


Proof. Let x be feasible for (MP) and (u, y, λ, p) feasible for (WD). Suppose contrary to the result that (3.5) and (3.6) hold. By y = 0 and g(x) 5 0, we have 1 fi (x) + y t g(x) 5 fi (u) + y t g(u) − pt ∇2 fi (u) + y t g(u) p, 2 for all i ∈ K, (3.7)

and

1 fj (x) + y t g(x) < fj (u) + y t g(u) − pt ∇2 fj (u) + y t g(u) p, 2 for some j ∈ K. (3.8) (i) In view of the hypothesis λ > 0 and λt e = 1, we get 1 λt f (x) + y t g(x) < λt f (u) + y t g(u) − pt ∇2 λt f (u) + y t g(u) p. (3.9) 2 The second order (F, α, ρi , d)-convexity of fi , i = 1, 2, . . . , k, and the second order (F, α, σj , d)-convexity of gj , j = 1, 2, . . . , m, at u imply 1 fi (x) − fi (u) + pt ∇2 fi (u)p 2 = F x, u; α(x, u) ∇fi(u) + ∇2fi(u)p + ρid2(x, u), i = 1, 2, . . . , k, and

1 gj (x) − gj (u) + pt ∇2 gj (u)p 2 = F x, u; α(x, u) ∇gj (u) + ∇2gj (u)p + σj d2(x, u),

j = 1, 2, . . . , m. On multiplying the first inequality by λi > 0 and second by yj = 0, and then summing up to get 1 λt f (x) + y t g(x) − λt f (u) − y t g(u) + pt ∇2 λt f (u) + y t g(u) p 2 t 2 t = F x, u; α(x, u) ∇λ f (u) + ∇ λ f (u)p + F x, u; α(x, u) ∇y t g(u) + ∇2 y t g(u)p   k m X X + λi ρi + σj yj  d2 (x, u), i=1

j=1

which in view of (3.9) and the sublinearity of F with α(x, u) > 0 gives F x, u; ∇λt f (u) + ∇2 λt f (u)p + ∇y t g(u) + ∇2 y t g(u)p   k m X X 1  λi ρi + σj yj  d2 (x, u). 0 gives F x, u; ∇λt f (u) + ∇2 λt f (u)p + ∇y t g(u) + ∇2 y t g(u)p ρ 0, λ ¯ t e = 1. λ ¯ p¯ = 0) is feasible for (WD) and the corresponding obTherefore, (¯ x, y¯, λ, jective values of (MP) and (WD) are equal. The efficiency of this feasible solution for (WD) thus follows from weak duality (Theorem 3.1). ¯ p¯) be the effiTheorem 3.3 (Strict Converse Duality). Let x ¯ and (¯ u, y¯, λ, cient solutions of (M P ) and (W D) respectively, such that t 1 ¯ t f (¯ ¯ t f (¯ ¯ f (¯ λ x) = λ u) + y¯t g(¯ u) − p¯t ∇2 λ u) + y¯t g(¯ u) p¯. (3.10) 2 Suppose that fi , i = 1, 2, . . . , k, is strictly second order (F, α, ρi , d)-convex at u ¯, and gj , j = 1, 2, . . . , m, is second order (F, α, σj , d)-convex at u ¯, and   k m X X 1 ¯  λi ρi + σj y¯j  = 0. α(¯ x, u ¯) i=1

j=1

Then x ¯=u ¯; that is, u ¯ is an efficient solution of (M P ).

Proof. We assume that x ¯ 6= u ¯ and exhibit a contradiction. Since fi , i = 1, 2, . . . , k, is strictly second order (F, α, ρi , d)-convex at u ¯, and gj , j = 1, 2, . . . , m, is second order (F, α, σj , d)-convex at u ¯, we have 1 fi (¯ x) − fi (¯ u) + p¯t ∇2 fi (¯ u)¯ p 2 >F x ¯, u ¯; α(¯ x, u ¯) ∇fi (¯ u) + ∇2 fi (¯ u)¯ p + ρi d2 (¯ x, u ¯),

i = 1, 2, . . . , k, and

1 gj (¯ x) − gj (¯ u) + p¯t ∇2 gj (¯ u)¯ p 2 = F x¯, u¯; α(¯x, u¯) ∇gj (¯u) + ∇2gj (¯u)¯p + σj d2(¯x, u¯),

j = 1, 2, . . . , m. ¯ i > 0 and second by y¯j = 0 and On multiplying the first inequality by λ then summing up to get t 1 ¯ t f (¯ ¯ t f (¯ ¯ f (¯ λ x) + y¯t g(¯ x) − λ u) − y¯t g(¯ u) + p¯t ∇2 λ u) + y¯t g(¯ u) p¯ 2 t 2 ¯t ¯ >F x ¯, u ¯; α(¯ x, u ¯) ∇λ f (¯ u) + ∇ λ f (¯ u)¯ p t 2 t +F x ¯, u ¯; α(¯ x, u ¯) ∇¯ y g(¯ u) + ∇ y¯ g(¯ u)¯ p   k m X X ¯ i ρi + + λ σj y¯j  d2 (¯ x, u ¯), i=1

j=1


139

which in view of (3.10) and the feasibility of x ¯ for (MP) implies t ¯ f (¯ ¯ t f (¯ F x ¯, u ¯; α(¯ x, u ¯) ∇λ u) + ∇2 λ u)¯ p t +F x ¯, u ¯; α(¯ x, u ¯) ∇¯ y g(¯ u) + ∇2 y¯t g(¯ u)¯ p   k m X X ¯ i ρi + < − λ σj y¯j  d2 (¯ x, u ¯). j=1

i=1

Since F is sublinear and α(¯ x, u ¯) > 0, then t ¯ ¯ t f (¯ F x ¯, u ¯; ∇λ f (¯ u) + ∇2 λ u)¯ p + ∇¯ y t g(¯ u) + ∇2 y¯t g(¯ u)¯ p   k m X X 1 ¯ i ρi +  0,

(4.4)

t

(4.5)

i∈Iβ

i∈Iβ

y

λ e = 1,

(4.1)

140


where S Iβ ⊆ M = {1, 2, . . . , m}, β = 0, 1, 2, . . . , r, with Iβ ∩ Iγ = ∅ if β 6= γ and rβ=0 Iβ = M . Theorem 4.1 (Weak Duality). Suppose that for all feasible x in (M P ) and all feasible (u, y, λ, p) in (GM D) P (i) i∈Iβ yi gi , β = 1, 2, . . . , r, is second order (F, α, σβ , d)-quasiconvex at u, and assume that any one ofPthe following conditions holds: (ii) I0 6= M , for all i ∈ K, fi + i∈I0 yi gi is P second order (F, α1 , ρi , d)quasiconvex and for some j ∈ K, fj + i∈I0 yi gi is second order (F, α1 , ρj , d)-pseudoconvex at u, and   r k X X 1  1 λi ρi  = 0; σβ + α(x, u) α1 (x, u) i=1

β=1

(iii) I0 = 6 M , λt f + u, and

P  

i∈I0

yi gi is second order (F, α2 , ρ, d)-pseudoconvex at

1 α(x, u)

r X

β=1

Then, the following cannot hold fi (x) 5 fi (u) +

X

i∈I0

σβ +



ρ  = 0. α2 (x, u)



1 yi gi (u) − pt ∇2 fi (u) + 2

X

i∈I0



yi gi (u) p,

for all i ∈ K,

(4.6)

and fj (x) < fj (u) +

X

i∈I0

  X 1 t 2 yi gi (u) − p ∇ fj (u) + yi gi (u) p, 2 i∈I0

for some j ∈ K. (4.7)

Proof. (i) Let x be any feasible solution in (MP) and (u, y, λ, p) be any feasible solution in (GMD). Then y = 0, g(x) 5 0 and (4.2) yields X

i∈Iβ

yi gi (x) 5 0 5

X

i∈Iβ

X 1 yi gi (u) − pt ∇2 yi gi (u)p, 2 i∈Iβ

β = 1, 2, . . . , r. (4.8)


141

P Since i∈Iβ yi gi , β = 1, 2, . . . , r, is second order (F, α, σβ , d)-quasiconvex at u, then (4.8) gives     X  X F x, u; α(x, u) ∇ yi gi (u) + ∇2 yi gi (u)p  5 −σβ d2 (x, u),   i∈Iβ

i∈Iβ

β = 1, 2, . . . , r.

The sublinearity of F with α(x, u) > 0 implies   X X yi gi (u) + ∇2 F x, u; ∇ yi gi (u)p

5

r X

β=1



i∈M \I0

F x, u; ∇

5 − α(x,1 u) (

r X

i∈M \I0

X

i∈Iβ

yi gi (u) + ∇2

σβ )d2 (x, u).

X

i∈Iβ



yi gi (u)p (4.9)

β=1

Now suppose contrary to the result that (4.6) and (4.7) hold. By y = 0 and g(x) 5 0, it follows that   X X X 1 t 2 yi gi (u) p, fi (x) + yi gi (x) 5 fi (u) + yi gi (u) − p ∇ fi (u) + 2 i∈I0

i∈I0

i∈I0

for all i ∈ K,

(4.10)

and fj (x) +

X

yi gi (x) < fj (u) +

i∈I0

X

i∈I0

  X 1 t 2 yi gi (u) − p ∇ fj (u) + yi gi (u) p, 2 i∈I0

for some j ∈ K. (4.11)

P (ii) Using the second order (F, α1 , ρi , d)-quasiconvexity of fi + i∈I0 yi gi , for all P i ∈ K, and the second order (F, α1 , ρj , d)-pseudoconvexity of fj + i∈I0 yi gi , for some j ∈ K, we have from (4.10) and (4.11)      X X F x, u; α1 (x, u) ∇fi (u) + ∇2 fi (u)p + ∇ yi gi (u) + ∇2 yi gi (u)p   

5 −ρid (x, u), 2

i∈I0

i∈I0

142


for all i ∈ K, and      X X F x, u; α1 (x, u) ∇fj (u) + ∇2 fj (u)p + ∇ yi gi (u) + ∇2 yi gi (u)p    i∈I0

i∈I0

< −ρj d2 (x, u),

for some j ∈ K. The sublinearity of F , α1 (x, u) > 0, λ > 0 and λt e = 1 imply   X X yi gi (u) + ∇2 F x, u; ∇λt f (u) + ∇2 λt f (u)p + ∇ yi gi (u)p 0 and λt e = 1, (4.10) and (4.11) imply   X X X 1 λt f (x)+ yi gi (x) < λt f (u)+ yi gi (u)− pt ∇2 λt f (u) + yi gi (u)p, 2 i∈I0

i∈I0

i∈I0

which by the second order (F, α2 , ρ, d)-pseudoconvexity of λt f + at u gives

P

i∈I0

yi gi


143

    X X F x, u; α2 (x, u) ∇λt f (u)+∇2 λt f (u)p+∇ yi gi (u)+∇2 yi gi (u)p    

i∈I0

i∈I0

2

< −ρd (x, u).

(4.13)

Using (4.9), (4.13) and the sublinearity of F with α2 (x, u) > 0, we get F x, u; ∇λt f (u) + ∇2 λt f (u)p + ∇y t g(u) + ∇2 y t g(u)p   X X 5 F x, u; ∇λtf (u) + ∇2λtf (u)p + ∇ yigi(u) + ∇2 yigi(u)p 

i∈I0

+ F x, u; ∇ 

< −

Since

we have

X

i∈M \I0

yi gi (u) + ∇2

i∈M \I0



r

X

 

1 α(x, u)

r X

β=1

σβ +

i∈I0

yi gi (u)p

X 1 ρ  d2 (x, u). σβ + α(x, u) α2 (x, u) β=1





ρ  = 0, α2 (x, u)

F x, u; ∇λt f (u) + ∇2 λt f (u)p + ∇y t g(u) + ∇2 y t g(u)p < 0,

again a contradiction to (4.1), since F (x, u; 0) = 0.

We now merely state the following strong duality theorem as its proof would run analogously to that of Theorem 3.2. Theorem 4.2 (Strong Duality). Let x ¯ be an efficient solution of (M P ) at which the generalized Guignard constraint qualification (GGCQ) is satisfied. ¯ ∈ Rk , such that (¯ ¯ p¯ = 0) is feasible for Then there exist y¯ ∈ Rm and λ x, y¯, λ, (GM D) and the corresponding objective values of (M P ) and (GM D) are equal. If, in addition, the assumptions of weak duality (Theorem 4.1) hold for ¯ p¯ = 0) is an efficient all feasible solutions of (MP) and (GMD), then (¯ x, y¯, λ, solution of (GMD). ¯ p¯) be the effiTheorem 4.3 (Strict Converse Duality). Let x ¯ and (¯ u, y¯, λ, cient solutions of (M P ) and (GM D) respectively, such that

144


¯ t f (¯ ¯ t f (¯ λ x) =λ u) +

X

y¯i gi (¯ u)

i∈I0

  X 1 t 2 ¯t − p¯ ∇ λ f (¯ u) + y¯i gi (¯ u) p¯. 2

(4.14)

i∈I0

Suppose that any one of the following conditions is satisfied: P (i) I0 6= M , ¯i gi , β = 1, 2, . . . , r is second order (F, α, σβ , d)i∈Iβ y ¯tf + P quasiconvex at u ¯ and λ ¯i gi is strictly second order i∈I0 y (F, α1 , ρ, d)-pseudoconvex at u ¯, and   r X ρ  1  = 0; σβ + α(¯ x, u ¯) α1 (¯ x, u ¯) β=1

(ii) I0 6= M ,

P

y¯i gi , β = 1, 2, . . . , r, is strictly second order (F, α, σβ , d)¯tf + P pseudoconvex at u ¯ and λ ¯i gi is second order (F, α1 , ρ, d)i∈I0 y quasiconvex at u ¯, and   r X ρ 1   = 0. σβ + α(¯ x, u ¯) α1 (¯ x, u ¯) i∈Iβ

β=1

Then x ¯=u ¯; that is, u ¯ is an efficient solution of (M P ).

Proof. We assume that x ¯ 6= u ¯ and exhibit a contradiction. Let x ¯ be feasible ¯ for (MP) and (¯ u, y¯, λ, p¯) be feasible for (GMD). Then y¯ = 0, g(¯ x) 5 0 and (4.2) yields X X X 1 y¯i gi (¯ u)¯ p, y¯i gi (¯ x) 5 0 5 y¯i gi (¯ u) − p¯t ∇2 2 i∈Iβ

i∈Iβ

i∈Iβ

β = 1, 2, . . . , r, (4.15) P which by the second order (F, α, σβ , d)-quasiconvexity of ¯i gi at u ¯ i∈Iβ y gives     X  X F x ¯, u ¯; α(¯ x, u ¯) ∇ y¯i gi (¯ u) + ∇2 y¯i gi (¯ u)¯ p  5 −σβ d2 (¯ x, u ¯),   i∈Iβ

i∈Iβ

β = 1, 2, . . . , r. (4.16)

The sublinearity of F and (4.16) with α(¯ x, u ¯) > 0 imply


F (¯ x, u ¯; ∇ r X

5

β=1

X

y¯i gi (¯ u) + ∇2

i∈M \I0

F (¯ x, u ¯; ∇

5 − α(¯x1, u¯) (

X

y¯i gi (¯ u)¯ p)

i∈M \I0

y¯i gi (¯ u) + ∇2

i∈Iβ

r X

X

145

X

y¯i gi (¯ u)¯ p)

i∈Iβ

σβ )d2 (¯ x, u ¯).

β=1

The first dual constraint and the above inequality along with the sublinearity of F imply   X X ¯ t f (¯ ¯ t f (¯ y¯i gi (¯ u)¯ p y¯i gi (¯ u) + ∇2 ¯, u ¯; ∇λ u) + ∇2 λ u)¯ p+∇ F x 

= α(¯x1, u¯) 

r X

β=1

Since

σβ )d2 (¯ x, u ¯ .  

we have 

i∈I0

i∈I0



1 α(¯ x, u ¯)

r X

σβ +

β=1

ρ  = 0, α1 (¯ x, u ¯)

¯ t f (¯ ¯ t f (¯ F x ¯, u ¯; ∇λ u) + ∇2 λ u)¯ p+∇

= − α (¯ρx, u¯) d2(¯x, u¯). 1



X

i∈I0

y¯i gi (¯ u) + ∇2

X

i∈I0



y¯i gi (¯ u)¯ p

Since α1 (¯ x, u ¯) > 0, we obtain      X X ¯ t f (¯ ¯ t f (¯ F x ¯, u ¯; α1 (¯ x, u ¯) ∇λ u)+∇2 λ u)¯ p +∇ y¯i gi (¯ u)+! + ∇2 y¯i gi (¯ u)¯ p   i∈I0

= −ρd (¯x, u¯),

i∈I0

2

which Pby the strict second ¯tf + λ ¯i gi at u ¯ yields i∈I0 y ¯ t f (¯ λ x)+

X

i∈I0

¯ t f (¯ y¯i gi (¯ x) > λ u)+

X

i∈I0

The above inequality in view of

order

(F, α1 , ρ, d)-pseudoconvexity 

1 ¯ t f (¯ y¯i gi (¯ u)− p¯t ∇2 λ u) + 2

P

i∈I0

X

i∈I0

of 

y¯i gi (¯ u) p¯.

y¯i gi (¯ x) 5 0 contradicts (4.14).

146


On the other hand, when hypothesis (ii) holds, the strict second order P (F, α, σβ , d)-pseudoconvexity of i∈Iβ y¯i gi and (4.15) yield     X  X y¯i gi (¯ u) + ∇2 y¯i gi (¯ u)¯ p  < −σβ d2 (¯ x, u ¯), F x ¯, u ¯; α(¯ x, u ¯) ∇   i∈Iβ

i∈Iβ

β = 1, 2, . . . , r.

Using the sublinearity of F and α(¯ x, u ¯) > 0, it follows from the above inequality that   r X X X 1 2   ( σβ )d2 (¯ x, u ¯) F x ¯, u ¯; ∇ y¯i gi (¯ u) + ∇ y¯i gi (¯ u)¯ p 0, we have      X X ¯ t f (¯ ¯ t f (¯ u)+∇2 λ u)+∇ F x ¯, u ¯; α1 (¯ x, u ¯ ) ∇λ y¯i gi (¯ u)+∇2 y¯i gi (¯ u)¯ p    i∈I0

i∈I0

> −ρd2 (¯ x, u ¯),

which Pby the virtue of second order (F, α1 , ρ, d)-quasiconvexity of ¯tf + λ ¯i gi at u ¯ becomes i∈I0 y   X X X 1 ¯ t f (¯ ¯ t f (¯ ¯ t f (¯ λ x)+ y¯i gi (¯ x) > λ u)+ y¯i gi (¯ u)− p¯t ∇2 λ u) + y¯i gi (¯ u) p¯. 2 i∈I0

i∈I0

i∈I0

Since x ¯ is feasible for (MP) and y¯ = 0, we have   X X 1 ¯ t f (¯ ¯ t f (¯ ¯ t f (¯ u) + y¯i gi (¯ u) p¯, λ x) > λ u) + y¯i gi (¯ u) − p¯t ∇2 λ 2 i∈I0

i∈I0

again contradicting (4.14). Hence, in both cases x ¯=u ¯.

Acknowledgment. The authors express their deep gratitude to the anonymous referee for his/her valuable suggestions which have improved the presentation of the paper.


147

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Izhar Ahmad Department of Mathematics Aligarh Muslim University Aligarh–202 002, India e-mail: [email protected]

Zakir Husain Department of Mathematics Aligarh Muslim University Aligarh–202 002, India e-mail: zhusain [email protected]