WITH GENERALIZED (F , b, φ, ρ, θ)-UNIVEX n-SET FUNCTIONS

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EFFICIENCY CONDITIONS IN MULTIOBJECTIVE FRACTIONAL PROGRAMMING

WITH GENERALIZED (F , b, φ, ρ, θ)-UNIVEX n-SET FUNCTIONS

I.M. STANCU-MINASIAN and ALINA PARASCHIV

We consider some types of generalized convexity and discuss semiparametric sufficient efficiency conditions for a multiobjective fractional programming problem involvingn-set functions.

AMS 2000 Subject Classification: 90C29, 90C30, 90C32, 90C46.

Key words: multiobjective programming, n-set function, optimality condition, generalized convexity, fractional programming.

1. INTRODUCTION

In this paper we present a few global semiparametric sufficient efficiency conditions under various generalized (F, b, φ, ρ, θ)-univexity hypotheses for the multiobjective fractional subset programming problem

(P) min

F₁(S)

G1(S), F₂(S)

G2(S), . . . ,F_p(S) Gp(S)

subject to Hj(S)50, j ∈q, S∈Aⁿ,

whereAⁿis then-fold product of theσ-algebraA of subsets of a given setX, Fi, Gi, i ∈ p = {1,2, . . . , p}, and Hj, j ∈ q = {1,2, . . . , q}, are real-valued functions defined on Aⁿ , and Gi(S) > 0, for all i ∈ p and S ∈ Aⁿ such that H_j(S) 5 0, j ∈ q. Let F={S ∈ Aⁿ : H_j(S) 5 0, j ∈ q} be the set of all feasible solutions to (P). We further assume that F is nonempty. Until now, F was assumed to be a sublinear function in the third argument. In our approach, we suppose that F is a convex function in the third argument.

Let us introduce some notation. Forx= (x1, x2, . . . , xm),y= (y1, y2, . . . , ym) ∈R^m, we denote x5y iff xi 5yi for eachi∈M ={1,2, . . . , m}; x≤y iff x_i 5y_i for each i∈M and x 6=y; x < y iff x_i < y_i for each i∈M. Here x≮y is the negation ofx < y. We note that x∈R^m₊ iffx=0.

REV. ROUMAINE MATH. PURES APPL.,54(2009),4, 331–345

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In Problem (P) minimality is taken in terms of efficient solution as defined below.

A feasible solution S⁰ ∈ F is said to be an efficient solution to (P) if there is no other S ∈F such that

F1(S)

G₁(S), F2(S)

G₂(S), . . . ,Fp(S) G_p(S)

≤

F1(S⁰)

G₁(S⁰),F2(S⁰)

G₂(S⁰), . . . , Fp(S⁰) G_p(S⁰)

. Optimization problems on a measure space are optimization problems involving functions defined on aσ-algebra of subsets of a finite atomless measure space. Such problems appear in several applications including electrical insulator design, optimal distribution of crops subject to rainfall in a given region, shape optimization, fluid flow, optimal plasma confinement, and statistics. The investigations in the field of optimization problems on a measure space followed closely the lines of development in nonlinear programming with point-functions. A good account of optimality conditions and duality for programming problems involving set andn-set functions can be found in the paper of Stancu-Minasian and Preda [21] and the references therein.

Kellerer [7] discussed the linear programming in a measure theoretic model. Jacobs and Seiffert [5] also investigated a measure theoretic model for flows in networks.

The definition of a convex set function was given by Dolecki and Kur- cyusz [4]. Morris [14] defined a convex set function using a special class of sequences (now known as Morris-sequences). Morris [14] defined the notions of local and global convexity and differentiability for set functions and established optimality conditions and Lagrangian duality relations for a nonlinear programming problem involving set functions. He also discussed some algo- rithms for the numeric solution of nonlinear programs with set functions.

Corley [3] started to give the concepts of partial derivatives and derivatives of real-valued n-set functions.

Lee [9] introduced the concepts of quasiconvexity and pseudoconvexity of set functions and established optimality conditions for optimization problems involving these functions.

Liu [10] extended ton-set functions the concepts and properties of quasiconvex and pseudoconvex functions and developed necessary and sufficient conditions for the existence of an optimal solution of a nonconvex program.

In order to relax convexity assumptions imposed in theorems on sufficient optimality conditions and duality, various generalized convexity notions have been proposed.

Liu [11] considers ρ-convex, ρ-quasiconvex and ρ-pseudoconvex subdifferentiable set functions and for a multiobjective programming problem proves sufficient optimality conditions and Wolfe-type and Mond-Weir type dual problems.

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Zalmai [22] introduced the concept of a differetiableρ-convexn-set function and proved the Kuhn-Tucker sufficient optimality theorem and the duality theorem either for single objective programming or for multiobjective programming.

Preda [16] defined generalized (F, ρ)-convexity and Joet al. [6] extended these notions to generalized (F, ρ, θ)-convexity and established several sufficient optimality conditions for multiobjective programming containing differentiable n-set functions.

Lai and Liu [8] considered sufficient optimality and duality for multiobjective programming involving subdifferentiable (F, ρ, θ)-convex functions.

Preda and Stancu-Minasian [17] defined (ρ, b)-vexity for n-set functions and considered some Mond-Weir duality results.

Bhatia and Kumar [2] defined pseudo-linear n-set functions, presented a characterization of these functions and for a multiobjective programming problem sufficient efficiency and duality results are given.

Preda and Stancu-Minasian [18] defined classes ofn-set functions called d-type-I, d-quasi type-I, d-pseudo type-I, d-quasi-pseudo type-I, d-pseudo- quasi type-I and established some optimality and Wolfe duality results for a multiobjective programming problem involving such functions.

Preda et al. [20] introduced the classes ofn-set functions called (ρ, ρ⁰)- V-univex type-I, (ρ, ρ⁰)-quasi V-univex type I, (ρ, ρ⁰)-pseudo V-univex type- I, (ρ, ρ⁰)-quasipseudo V-univex type-I and (ρ, ρ⁰)-pseudoquasi V-univex type- I and studied optimality conditions and generalized Mond-Weir duality for multiobjective programming problem.

Mishra [12] defined some types of generalized convexity ((F, ρ, σ, θ)-V type-I, (F, ρ, σ, θ)-V-pseudo-quasi type-I, (F, ρ, σ, θ)-V-quasi-pseudo type-I) and established optimality and duality results for a multiobjective programming problem involving such functions.

Mishraet al. [13] defined new classes of n-set functions, called (ρ, ρ⁰, d)- strong pseudo-quasi-type-I, (ρ, ρ⁰, d)-weak strictly pseudo-quasi-type-I, (ρ, ρ⁰, d)-weak strictly pseudo-type-I, (ρ, ρ⁰, d)-weak quasi-strictly-pseudo-type-I functions and for a multiobjective programming problem established optimality and Mond-Weir duality results.

Preda, Stancu-Minasian and Koller [19] presented some optimality and duality results for a multiobjective programming problem involving generalized d-type-I vector-valuedn-set functions.

Zalmai [25] introduced a new class of generalized convexn-set functions, called (F, α, ρ, θ)-V-convex functions and presented numerous sets of para- matric and semiparametric sufficient efficiency conditions for a multiobjective fractional subset programming problem.

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For the purpose of formulating and proving various collections of sufficiency criteria for Problem (P), in this paper we shall use a new class of generalized convexn-set functions, called (F, b, φ, ρ, θ)-univexn-set functions, which was first defined in Zalmai [23]. This class may be viewed as a combi- nation of several previously defined types of generalized convex functions.

2. DEFINITIONS AND PRELIMINARIES

Let (X, A, µ) be a finite atomless measure space with L1(X, A, µ) sepa- rable, and let dbe the pseudometric on Aⁿ defined by

d(R, S) =

" _n X

k=1

µ²(R_k∆S_k)

#1/2

,

where R = (R₁, . . . , R_n) and S = (S₁, . . . , S_n) ∈Aⁿ and ∆ denotes the sym- metric difference.

Thus, (Aⁿ, d) is a pseudometric space. For h∈ L1(X, A, µ) andT ∈A, the integral R

T hdµ is denoted by hh, χ_Ti, where χ_T ∈ L∞(X, A, µ) is the indicator (characteristic) function of T.

Definition 2.1 ([14]). A function F :A →R is said to be differentiable at S^∗ ∈A if there exist DF(S^∗) ∈ L1(X, A, µ), called the derivative of F at S^∗ , and V_F :A×A→Rsuch that

F(S) =F(S^∗) +hDF(S^∗), χ_S−χ_S^∗i+V_F(S, S^∗), for each S ∈A, whereV_F(S, S^∗) iso(d(S, S^∗)), that is,

lim

d(S,S^∗)→0

V_F(S, S^∗) d(S, S^∗) = 0.

Definition 2.2 ([3]). A function G : Aⁿ → R is said to have a partial derivative at S^∗ = (S₁^∗, . . . , S_n^∗) ∈Aⁿ with respect to its ith argument if the functionF(Si) =G(S₁^∗, . . . , S_i−1^∗ , Si, S_i+1^∗ , . . . , S_n^∗) has derivativeDF(S_i^∗),i∈n.

We define D_iG(S^∗) := DF(S_i^∗) and write DF(S^∗) = (D₁F(S^∗), . . . , DnF(S^∗)).

Definition 2.3 ([3]). A function G:Aⁿ →R is said to be differentiable atS^∗ if there exist DF(S^∗) andWG :Aⁿ×Aⁿ→R such that

G(S) =G(S^∗) +

n

X

i=1

D_iG(S^∗), χ_S_i−χ_S^∗

i

+W_G(S, S^∗), where WG(S, S^∗) iso(d(S, S^∗)) for allS ∈Aⁿ.

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In the following we considerF :Aⁿ×Aⁿ×Rⁿ→R and a differentiable function F :Aⁿ→ R.Let b:Aⁿ×Aⁿ → R₊, θ :Aⁿ×Aⁿ→ Aⁿ×Aⁿ such that S6=S^∗⇒θ(S, S^∗)6= (0,0), φ:R→Rand a real number ρ.

Definition 2.4 ([23]). A function F is said to be (strictly) (F, b, φ, ρ, θ)- univex at S^∗ if

φ(F(S)−F(S^∗)) (>) =F(S, S^∗;b(S, S^∗)DF(S^∗)) +ρd²(θ(S, S^∗)) for each S ∈Aⁿ.

Definition 2.5 ([23]). A function F is said to be (strictly) (F, b, φ, ρ, θ)- pseudounivex at S^∗ if

F(S, S^∗;b(S, S^∗)DF(S^∗))=−ρd²(θ(S, S^∗))⇒φ(F(S)−F(S^∗)) (>) =0 for each S ∈Aⁿ,S 6=S^∗.

Definition 2.6 ([23]). A functionFis said to be (prestrictly) (F, b, φ, ρ, θ)- quasiunivex at S^∗ if

φ(F(S)−F(S^∗)) (<) 50⇒ F(S, S^∗;b(S, S^∗)DF(S^∗))5−ρd²(θ(S, S^∗)) for each S ∈Aⁿ.

The following result is from Zalmai ([25], Theorem 2.1).

Theorem 2.1. Assume that Fi, Gi, i∈p, and Hj, j∈q, are differentiable at S^∗ ∈Aⁿ,and for each i∈p there exist Sbi∈Aⁿ such that

Hj(S^∗) +

n

X

k=1

D

D_kHj(S^∗), χ

Sc_k−χ_S^∗

k

E

<0, j∈q, and for each l∈p\{i} we have

n

X

k=1

D

G_i(S^∗)D_kF_l(S^∗)−F_i(S^∗)D_kG_l(S^∗), χ

Sc_k−χ_S^∗

k

E

<0.

If S^∗ is an efficient solution of (P) , then there exist u^∗ ∈U =n u ∈ R^p:u >0,

p

P

i=1

ui= 1 o

and v^∗∈R^p₊ such that

n

X

k=1

^p X

i=1

u^∗_i [Gi(S^∗)DkFi(S^∗)−Fi(S^∗)DkGi(S^∗)] + +

q

X

j=1

v^∗_jD_kH_j(S^∗), χ_S_k−χ_S^∗

k

=0, for all S∈Aⁿ, v^∗_jHj(S^∗) = 0, j∈q.

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3. GENERALIZED SUFFICIENT EFFICIENCY CRITERIA In this section we define and discuss several families of generalized sufficiency results for (P) by means of a partitioning scheme that was originally proposed for constructing generalized dual problems for nonlinear programs with point-functions.

Let {J₀, J₁, . . . , J_m} be a partition of the index set q; thus J_r ⊂ q for each r∈ {0,1, . . . , m}, J_r∩J_s=∅for eachr 6=s,and

m

S

r=0

J_r =q.In addition, we shall make use of the functions Ψ_i(·, S^∗, v^∗), Ψ(·, S^∗, u^∗, v^∗) and Λ_t(·, v^∗) : Aⁿ→Rdefined for fixedu^∗, v^∗ and S^∗ by

Ψ_i(T, S^∗, v^∗) = [G_i(S^∗)F_i(T)−F_i(S^∗)G_i(T)] + X

j∈J0

v^∗_jH_j(T), i∈p,

Ψ(T, S^∗, u^∗, v^∗) =

p

X

i=1

u^∗_i [Gi(S^∗)Fi(T)−Fi(S^∗)Gi(T)] + X

j∈J0

v^∗_jHj(T), Λ_t(T, v^∗) =X

j∈J_t

v_j^∗H_j(T), t∈m∪ {0}.

Letm=|m|, m₁ = m₁

, m₂= m₂

,where{m₁, m₂} is a partition of m={1, . . . , m}.

LetS^∗ ∈F withF_i(S^∗) =0, i∈p, and assume thatF_i, G_i, i∈p, and H_j, j∈q, are differentiable at S^∗, and that there existu^∗ ∈U and v^∗ ∈R^q₊ such that

F

S, S^∗;b(S, S^∗) p

X

i=1

u^∗_i h

Gi(S^∗)DFi(S^∗)−Fi(S^∗)DGi(S^∗) i

+ (3.1)

+

q

X

j=1

v^∗_jDHj(S^∗)

=0, S∈F, v^∗_jHj(S^∗) = 0, j∈q, (3.2)

where F(S, S^∗;·) :Lⁿ₁(X, A, µ)→Ris a convex function.

Theorem 3.1. LetS^∗ ∈Fand assume that F_i, G_i, i∈p,andH_j, j∈q, are differentiable at S^∗, and that there exist u^∗ ∈U and v^∗ ∈ R^q₊ such that (3.1) and (3.2) hold. Assume furthermore that any one of the following sets of hypotheses is satisfied:

(a) (i) 2Ψ(·, S^∗, u^∗, v^∗) is F, b, φ, ρ, θ

-pseudounivex at S^∗ and φ(a) =0 ⇒ a=0;

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(ii) 2mΛt(·, v^∗) is F, b,φet,ρet, θ

-quasiunivex at S^∗, φet is increasing and φe_t(0) = 0for each t∈m;

(iii) ρ+ 1 m

m

P

t=1ρe_t=0;

(b) (i) 2Ψ(·, S^∗, u^∗, v^∗)is prestrictly F, b, φ, ρ, θ

-quasiunivex atS^∗and φ(a)

=0⇒a=0;

(ii) 2mΛt(·, v^∗) is strictly F, b,φet,ρet, θ

-pseudounivex at S^∗, φet is increasing and φe_t(0) = 0for each t∈m;

(iii) ρ+ 1 m

m

P

t=1ρe_t=0;

(c) (i) 2Ψ(·, S^∗, u^∗, v^∗)is prestrictly F, b, φ, ρ, θ

-quasiunivex at S^∗and φ(a)

=0⇒a=0;

(ii) 2mΛt(·, v^∗) is F, b,φet,ρet, θ

-quasiunivex at S^∗,φet is increasing and φe_t(0) = 0for each t∈m;

(iii) ρ+ 1 m

Pm

t=1ρe_t>0;

(d) (i) 3Ψ(·, S^∗, u^∗, v^∗)is prestrictly F, b, φ, ρ, θ

-quasiunivex atS^∗and φ(a)

=0⇒a=0;

(ii) 3m1Λt(·, v^∗) is F, b,φet,ρet, θ

-quasiunivex at S^∗ for each t ∈ m1; 3m₂Λ_t(·, v^∗)is strictly (F, b,φe_t,ρe_t, θ)-pseudounivex at S^∗ for each t∈ m2, φet is increasing and φet(0) = 0 for each t∈m, where {m₁, m2} is a partition of m;

(iii) ρ+ 1 m1

P

t∈m₁ρet+ 1 m2

P

t∈m₂ρet=0.

Then S^∗ is an efficient solution to (P).

Proof. (a) Let S be an arbitrary feasible solution to (P). It then follows from the convexity of F(S, S^∗;·) and (3.1) that

F

S, S^∗; 2b(S, S^∗)

p

X

i=1

u^∗_i[Gi(S^∗)DFi(S^∗)−Fi(S^∗)DGi(S^∗)]+

(3.3)

+X

j∈J0

2v^∗_jDH_j(S^∗)

+F

S, S^∗;b(S, S^∗)

m

X

t=1

X

j∈Jt

2v_j^∗DH_j(S^∗)

=0.

From the nonnegativity of v^∗, feasibility of S and (3.2), it is clear that for each t∈m we have

2mΛt(S, v^∗) =X

j∈Jt

2mv^∗_jHj(S)50 = X

j∈Jt

2mv_j^∗Hj(S^∗) = 2mΛt(S^∗, v^∗),

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i.e.,

2mΛt(S, v^∗)−2mΛt(S^∗, v^∗)50,

which, on account of the properties of the function φe_t, reduces to φet(2mΛt(S, v^∗)−2mΛt(S^∗, v^∗))50

for each t∈m.

By (ii), this inequality implies that

(3.4) F

S, S^∗;b(S, S^∗)X

j∈J_t

2mv_j^∗DH_j(S^∗)

5−ρe_td²(θ(S, S^∗)). for each t∈m.

The convexity ofF(S, S^∗;·) and (3.4) imply that F

S, S^∗;b(S, S^∗)

m

X

t=1

X

j∈Jt

2v_j^∗DH_j(S^∗)

≤ (3.5)

≤ 1 m

m

X

t=1

F

S, S^∗;b(S, S^∗)X

j∈Jt

2mv_j^∗DHj(S^∗)

5−1 m

m

X

t=1

ρetd²(θ(S, S^∗)). It follows from (3.3), (3.4) and (iii) that

F

S, S^∗;b(S, S^∗)

p

X

i=1

2u^∗_i[Gi(S^∗)DFi(S^∗)−Fi(S^∗)DGi(S^∗)]+

(3.6)

+X

j∈J0

2v_j^∗DH_j(S^∗)

− F

S, S^∗;b(S, S^∗)

m

X

t=1

X

j∈Jt

2v_j^∗H_j(S^∗)

=

m

X

t=1

ρetd²(θ(S, S^∗))

m =−ρd²(θ(S, S^∗)). By (i), this inequality implies that

φ(2Ψ(S, S^∗, u^∗, v^∗)−2Ψ(S^∗, S^∗, u^∗, v^∗))=0, which, on account of the properties of the function φreduces to

Ψ(S, S^∗, u^∗, v^∗)−Ψ(S^∗, S^∗, u^∗, v^∗)=0.

But Ψ(S^∗, S^∗, u^∗, v^∗) = 0, hence Ψ(S, S^∗, u^∗, v^∗) =0,which, because S is feasible solution to (P) and v^∗ is nonnegative, reduces to

p

X

i=1

u^∗_i [G_i(S^∗)F_i(S)−F_i(S^∗)G_i(S)]=0.

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Sinceu^∗>0,the above inequality implies that ϕ(S) =

F₁(S)

G1(S),F₂(S)

G2(S), . . . , F_p(S) Gp(S)

F₁(S^∗)

G1(S^∗),F₂(S^∗)

G2(S^∗), . . . ,F_p(S^∗) Gp(S^∗)

=ϕ(S^∗).

BecauseS∈Fwas arbitrary, we conclude thatS^∗ is an efficient solution to (P).

(b) Now, proceeding as in the proof of part (a) and using the strict F, b, φ, ρ, θ

-pseudounivexity atS^∗ of 2m∧_t(·;v^∗) and the properties of the function φ,we get

(3.7) F

S, S^∗;b(S, S^∗)X

j∈Jt

<−ρe_td²(θ(S, S^∗)).

The convexity ofF(S, S^∗;·) and (3.7) imply that

F

S, S^∗;b(S, S^∗)

m

X

t=1

X

j∈Jt

2v_j^∗DHj(S^∗) (3.8) ≤

≤ 1 m

m

X

t=1

F

S, S^∗;b(S, S^∗)X

j∈J_t

<−1 m

m

X

t=1

ρe_td²(θ(S, S^∗)). It follows from (3.3), (3.7) and (iii) that

F

S, S^∗;b(S, S^∗)

p

X

i=1

2u^∗_i[G_i(S^∗)DF_i(S^∗)−F_i(S^∗)DG_i(S^∗)]+

(3.9)

+X

j∈J₀

2v^∗_jDHj(S^∗)

>

m

X

t=1

m >−ρd²(θ(S, S^∗)). The rest of the proof is identical to that of part (a).

(c) The proof is similar to that of part (a) in which inequality (3.6) is replaced by inequality (3.9).

(d) Let S be an arbitrary feasible solution to (P). It then follows from the convexity of F(S, S^∗;·) and (3.1) that

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F

S, S^∗; 3b(S, S^∗)

p

X

i=1

u^∗_i[Gi(S^∗)DFi(S^∗)−Fi(S^∗)DGi(S^∗)]+

(3.10)

+X

j∈J₀

3v_j^∗DHj(S^∗)

+ F

S, S^∗;b(S, S^∗)X

t∈m₁

X

j∈Jt

3v_j^∗DHj(S^∗)

+ +F

S, S^∗;b(S, S^∗)X

t∈m2

X

j∈J_t

3v^∗_jDH_j(S^∗)

=0.

It follows from the nonnegativity ofv^∗,feasibility ofS and (3.2) that for each t∈mi,i∈ {1,2} we have

3m_iΛ_t(S, v^∗) =X

j∈J_t

3m_iv^∗_jH_j(S)50 = X

j∈J_t

3m_iv_j^∗H_j(S^∗) = 3m_iΛ_t(S^∗, v^∗), i.e.,

3m_iΛ_t(S, v^∗)−3m_iΛ_t(S^∗, v^∗)50

which, on account of the properties of the function ˜φ_t reduces to φ˜t(3miΛt(S, v^∗)−3miΛt(S^∗, v^∗))50

for each t∈m_i and i∈ {1,2}.

By (ii), this inequality implies that (3.11)

F

S, S^∗;b(S, S^∗)X

j∈J_t

3m₁v_j^∗DH_j(S^∗)

5−ρe_td²(θ(S, S^∗)), for each t∈m₁ and

(3.12) F

S, S^∗;b(S, S^∗)X

j∈J_t

3m₂v_j^∗DH_j(S^∗)

<−ρe_td²(θ(S, S^∗)), for each t∈m₂. We also have

F

S, S^∗;b(S, S^∗)X

t∈m1

X

j∈Jt

3v^∗_jDH_j(S^∗)

= (3.13)

=F

S, S^∗;b(S, S^∗) 1 m1

X

t∈m1

X

j∈Jt

m₁3v_j^∗DH_j(S^∗)

5 5 1

m1

X

t∈m1

F

S, S^∗;b(S, S^∗)X

j∈Jt

m13v_j^∗DHj(S^∗)

5− 1 m1

X

t∈m1

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and

F

S, S^∗;b(S, S^∗)X

t∈m2

X

j∈J_t

3v^∗_jDH_j(S^∗)

= (3.14)

=F

S, S^∗;b(S, S^∗) 1 m₂

X

t∈m2

X

j∈Jt

m₂3v_j^∗DH_j(S^∗)

5 5 1

m2

X

t∈m2

F

S, S^∗;b(S, S^∗)X

j∈Jt

m₂3v^∗_jDH_j(S^∗)

<− 1 m2

X

t∈m2

ρe_td²(θ(S, S^∗)). It follows from (3.10), (3.13), (3.14) and (iii) that

F

S, S^∗;b(S, S^∗)

p

X

i=1

3u^∗_i[G_i(S^∗)DF_i(S^∗)−F_i(S^∗)DG_i(S^∗)]+

+X

j∈J0

3v_j^∗DHj(S^∗)

=−F

S, S^∗;b(S, S^∗)X

t∈m₁

X

j∈Jt

3v_j^∗DHj(S^∗)

−

−F

S, S^∗;b(S, S^∗)X

t∈m₂

X

j∈Jt

3v_j^∗DHj(S^∗)

>

> 1 m₁

X

t∈m₁

ρetd²(θ(S, S^∗)) + 1 m₂

X

t∈m₂

ρetd²(θ(S, S^∗))=−ρd²(θ(S, S^∗)), which on account of (i) implies that

φ(3Ψ(S, S^∗, u^∗, v^∗)−3Ψ(S^∗, S^∗, u^∗, v^∗))=0, that is,

Ψ(S, S^∗, u^∗, v^∗)−Ψ(S^∗, S^∗, u^∗, v^∗)=0.

But Ψ(S^∗, S^∗, u^∗, v^∗) = 0,so we have

p

X

i=1

u^∗_i [Gi(S^∗)Fi(S^∗)−Fi(S^∗)Gi(S^∗)]=0.

The rest of the proof is identical to that of parts a)–c).

Note that Theorem 3.1 contains a number of special cases that can be easily identified by appropiate choices of the partitionning sets J₀, J₁, . . . , J_m. We shall state some of these as corrolaries.

Corrolary 3.1. Let S^∗ ∈ F and assume that Fi, Gi, i ∈ p and Hj, j ∈q, are differentiable at S^∗ and there exist u^∗ ∈U0 and v^∗ ∈R^q₊ such that (3.1) and (3.2) hold. Furthermore, we assume that any one of the following sets of hypothesis is satisfied

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(a) (i) the functionT →2

p

P

i=1

u^∗_i [Gi(S^∗)Fi(T)−Fi(S^∗)Gi(T)]is(F, b, φ, ρ, θ)- pseudounivex at S^∗ and φ(a)=0⇒a=0;

(ii) the function T → 2mv_t^∗H_t(T) is (F, b,φe_t,ρe_t, θ)-quasiunivex at S^∗,φe_t is increasing and φet(0) = 0for each t∈m;

(iii) ρ+_m¹

m

P

t=1ρet=0;

(b) (i) the function T → 2

p

P

i=1

u^∗_i [Gi(S^∗)Fi(T)−Fi(S^∗)Gi(T)] is prestrictly (F, b, φ, ρ, θ)-quasiunivex at S^∗ and φ(a)=0⇒a=0;

(ii) the function T → 2mv_t^∗Ht(T) is strictly (F, b,φet,ρet, θ)-pseudounivex at S^∗,φetis increasing and φet(0) = 0 for each t∈m;

(iii) ρ+_m¹

m

P

t=1ρet=0;

(c) (i) the function T → 2

p

P

i=1

u^∗_i [G_i(S^∗)F_i(T)−F_i(S^∗)G_i(T)] is prestrictly F, b, φ, ρ, θ

-quasiunivex at S^∗ and φ(a)=0⇒a=0;

(ii) the function T →2mv_t^∗H_t(T) is (F, b,φe_t,ρe_t, θ)-quasiunivex at S^∗,φe_t is increasing and φe_t(0) = 0for each t∈m;

(iii) ρ+_m¹

m

P

t=1ρet>0;

(d) (i) the function T → 3

p

P

i=1

u^∗_i [Gi(S^∗)Fi(T)−Fi(S^∗)Gi(T)] is prestrictly F, b, φ, ρ, θ

(ii) the function T →3m1v^∗_tHt(T)is (F, b,φet,ρet, θ)-quasiunivex at S^∗ for each t∈m₁;the function T →3m₂v_t^∗H_t(T) is strictly (F, b,φe_t,ρe_t, θ)- pseudounivex at S^∗ for each t∈m2,φetis increasing and φet(0) = 0for each t∈m, where {m₁, m₂} is a partition of m;

(iii) ρ+_m¹

1

P

t∈m1

ρe_t+ _m¹

2

P

t∈m2

ρe_t=0;

Proof. Letm=q, J₀ =φ and J_t={t},t= 1,2, . . . , qin Theorem 3.1.

Corrolary 3.2. Let S^∗ ∈ F and assume that Fi, Gi, i ∈ p and Hj, j ∈q, are differentiable at S^∗ and there exist u^∗ ∈U0 and v^∗∈R^q₊ such that (3.1) and (3.2) hold. Furthermore, we assume that any one of the following sets of hypothesis is satisfied

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(a) (i) the functionT →2

p

P

i=1

u^∗_i [Gi(S^∗)Fi(T)−Fi(S^∗)Gi(T)]is(F, b, φ, ρ, θ)- pseudounivex at S^∗ and φ(a)=0⇒a=0;

(ii) 2mΛ_t(·, v^∗) is (F, b,φe_t,ρe_t, θ)-quasiunivex at S^∗, φe_t is increasing and φet(0) = 0for each t∈m;

(iii) ρ+_m¹

m

P

t=1ρet=0;

(b) (i) the function T → 2

p

P

i=1

(ii) 2mΛt(·, v^∗) is strictly (F, b,φet,ρet, θ)-pseudounivex at S^∗, φet is increasing and φe_t(0) = 0for each t∈m;

(iii) ρ+_m¹

m

P

t=1ρe_t=0;

(c) (i) the function T → 2

p

P

i=1

u^∗_i [G_i(S^∗)F_i(T)−F_i(S^∗)G_i(T)] is prestrictly (F, b, φ, ρ, θ)-quasiunivex at S^∗ and φ(a)=0⇒a=0;

(ii) 2mΛ_t(·, v^∗) is (F, b,φe_t,ρe_t, θ)-quasiunivex at S^∗, φe_t is increasing and φe_t(0) = 0for each t∈m;

(iii) ρ+_m¹

m

P

t=1ρet>0;

(d) (i) the function T → 3

p

P

i=1

(ii) 3m1Λt(·, v^∗) is (F, b,φet,ρet, θ)-quasiunivex at S^∗ for each t ∈ m1; 3m₂Λ_t(·, v^∗) is strictly (F, b,φe_t,ρe_t, θ)-pseudounivex at S^∗ for each t∈m2,φet is increasing and φet(0) = 0for each t∈m,where {m₁, m2} is a partition of m;

(iii) ρ+_m¹

1

P

t∈m₁ρet+_m¹

2

P

t∈m₂ρet=0;

Proof. LetJ₀ =∅ in Theorem 3.1.

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Received 18 May 2009 Romanian Academy

“Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics

Calea 13 Septembrie 13 050711 Bucharest 5, Romania

stancu [email protected] and

National College “Elena Cuza”

Str. Pe¸stera Sc˘ari¸soara 1 062071 Bucharest 6, Romania