Generalized (ℱ,b,φ,ρ,θ) univex n set functions and global semiparametric sufficient efficiency conditions in multiobjective fractional subset programming

(1)

AND GLOBAL SEMIPARAMETRIC SUFFICIENT

EFFICIENCY CONDITIONS IN MULTIOBJECTIVE

FRACTIONAL SUBSET PROGRAMMING

G. J. ZALMAI

Received 28 December 2003 and in revised form 3 October 2004

A fairly large number of global semiparametric suﬃcient eﬃciency results are established under various generalized (Ᏺ,b,φ,ρ,θ)-univexity assumptions for a multiobjective frac-tional subset programming problem.

1. Introduction

In this paper, we will present a multitude of global semiparametric suﬃcient eﬃciency conditions under various generalized (Ᏺ,b,φ,ρ,θ)-univexity hypotheses for the following multiobjective fractional subset programming problem:

(P) Minimize(F1(S)/G1(S),F2(S)/G2(S),. . .,Fp(S)/Gp(S)) subject toHj(S)0,j∈q, S∈An_,

where An _{is the} _n_{-fold product of the} _σ_-algebra _A_{of subsets of a given set} _X_,_F_i,_G_i, i∈p≡ {1, 2,. . .,p}, andHj, j∈q, are real-valued functions defined onAn_{, and for each} i∈p,Gi(S)>0 for allS∈Ansuch thatHj(S)0, j∈q.

The point-function counterparts of (P) are known in the area of mathematical pro-gramming asmultiobjective fractional programming problems. These problems have been the focus of intense interest in the past few years, which has resulted in numerous publi-cations. A fairly extensive list of references concerning various aspects of these problems is given in [43]. For more information about general multiobjective problems with point functions, the reader may consult the recently published monograph by Miettinen [30].

In the area of subset programming, multiobjective problems have been investigated in [2,4,7,14,15,16,18,23,24,26,27,29,38,42], and multiobjectivefractionalproblems in [1,5,6,19,21,22,24,36,44,46]. We next give a brief overview of the available results pertaining to the latter class of problems.

A parametric dual problem for (P) was constructed in [6] and a number of weak and strong duality theorems involving generalizedρ-convexity assumptions were proved. In [18], two parametric dual problems, which are slightly diﬀerent from the one considered in [6], were formulated and some weak, strong, and strict converse duality results were es-tablished using generalizedρ-convexity hypotheses. Some of these results are further ex-tended in [22] by using generalizedᏲ-convexn-set functions. A multiobjective fractional

(2)

problem like (P) in which the functionsFi,−Gi,i∈p, andHj,j∈q, are assumed to be convex was considered in [2] where parametric, semiparametric, and Lagrangian-type dual problems were formulated and weak, strong, and strict converse duality theorems were proved; in addition, a set of sufficient conditions characterizing properly efficient solutions of the problem under consideration was given. A problem similar to the one studied in [2], but with one additional restriction, was discussed in [21]. In this paper, it was assumed that the functionsFi,−Gi,i∈p, andHj, j∈q, are convex and that the denominators of the objective functions are equal. With these assumptions, the authors established necessary and sufficient proper efficiency results, formulated a dual prob-lem that has the same objective function as the primal probprob-lem, and proved weak and strong duality theorems. In [36], Preda defined a (ρ,b)-vexn-set function, discussed some of its properties, and then established weak, strong, and converse duality results for a parametric dual problem for (P) under appropriate (ρ,b)-vexity conditions.B-vexn-set functions were utilized in [4] for obtaining sufficient proper efficiency criteria and some duality relations for a nonfractional multiobjective subset programming problem. The relevance and applicability of these results to a problem like (P) in which the functions Fi,−Gi,i∈p, andHj,j∈q, are convex, and for eachi∈p,Fi(S)≥0 andGi(S)>0 for all S∈An_{were also discussed. Recently, saddle-point-type proper e}_ffi_{ciency conditions and} Lagrangian-type duality results were obtained in [5] under cone-convexity assumptions for a cone-constrained multiobjective subset programming problem. In [44], a number of sufficient efficiency criteria and duality theorems were established for (P) under vari-ous (Ᏺ,α,ρ,θ)-V-convexity assumptions.

For brief surveys and additional references dealing with diﬀerent aspects of subset pro-gramming problems, including areas of applications, optimality conditions, and duality models, the reader is referred to [4,8,24,33,37,40,41].

The rest of this paper is organized as follows. InSection 2, we recall the definitions of differentiability, convexity, and certain types of generalized convexity forn-set func-tions, which will be used frequently throughout the sequel. We begin our discussion of sufficient efficiency criteria for (P) inSection 3where we state and prove a number of sufficiency results. More general sets of sufficiency conditions are formulated and dis-cussed inSection 4with the help of two partitioning schemes. The first of these schemes was originally used in [32] for constructing generalized dual problems for nonlinear pro-grams with point functions, whereas the second was utilized in [39] for formulating a dual problem for a multiobjective fractional program involving point functions.

Evidently, all these eﬃciency results are also applicable, when appropriately special-ized, to the following three classes of problems with multiple, fractional, and conventional objective functions, which are particular cases of (P):

(P1) MinimizeS∈F(F1(S),F2(S),. . .,Fp(S)); (P2) MinimizeS∈FF1(S)/G1(S);

(P3) MinimizeS∈FF1(S),

whereF(assumed to be nonempty) is the feasible set of (P), that is,

(3)

Since in most cases, the eﬃciency results established for (P) can easily be modified and restated for each one of the above problems, we will not explicitly state these results.

2. Preliminaries

In this section, we gather, for convenience of reference, a number of basic definitions along with a few auxiliary results, which will be used often throughout the sequel.

Let (X,A,µ) be a finite atomless measure space withL1(X,A,µ) separable, and letdbe the pseudometric onAn_{defined by}

d(R,S)=

_n

i=1 µ2_R

iSi

1/2

, R=R1,. . .,Rn,S=S1,. . .,Sn∈An, (2.1)

wheredenotes symmetric diﬀerence; thus (An_,_d_{) is a pseudometric space. For} _h_∈ L1(X,A,µ) andT∈Awith characteristic functionχT∈L∞(X,A,µ), the integralTh dµ will be denoted byh,χT.

We next define the notions of diﬀerentiability and convexity forn-set functions. They were originally introduced by Morris [33] for set functions, and subsequently extended by Corley [8] forn-set functions.

Definition 2.1. A function F:A→R is said to be diﬀerentiable at S∗ if there exists

DF(S∗)∈L1(X,A,µ), called thederivativeofFatS∗, such that for eachS∈A,

F(S)=FS∗+ DFS∗,χS−χS∗+V_FS,S∗, (2.2)

whereVF(S,S∗) iso(d(S,S∗)), that is, limd(S,S∗)→0VF(S,S∗)/d(S,S∗)=0.

Definition 2.2. A functionG:An_→_R_{is said to have a}_{partial derivative}_at_S∗₌₍_S∗

1,. . .,S∗n) ∈An_{with respect to its}_i_{th argument if the function}_F₍_S_i)₌_G₍_S∗

1,. . .,S∗i−1,Si,S∗i+1,. . .,S∗n) has derivativeDF(S∗_i),i∈n; in that case, theith partial derivative ofGatS∗is defined to beDiG(S∗)=DF(S∗i),i∈n.

Definition 2.3. A functionG:An_→_R_{is said to be}_di_ﬀ_erentiable_at_S∗ _{if all the partial}

derivativesDiG(S∗),i∈n, exist and

G(S)=GS∗+ n

i=1 DiG

S∗,χSi−χS∗i

+WG

S,S∗, (2.3)

whereWG(S,S∗) iso(d(S,S∗)) for allS∈An_.

It was shown by Morris [33] that for any triple (S,T,λ)∈A×A×[0, 1], there exist sequences{Sk}and{Tk}inAsuch that

χSk w∗

−−→λχS\T, χTk w∗

−−→(1−λ)χT\S (2.4)

imply

χSk∪Tk∪(S∩T) w∗

(4)

where−−→w∗ denotes weak∗convergence of elements inL∞(X,A,µ), andS\Tis the

comple-ment ofTrelative toS. The sequence{Vk(λ)} = {Sk∪Tk∪(S∩T)}satisfying (2.4) and (2.5) is called theMorris sequenceassociated with (S,T,λ).

Definition 2.4. A functionF:An_→_R_{is said to be (strictly) convex if for every (}_S_,_T_,_λ₎_∈ An_×_An_×_{[0, 1], there exists a Morris sequence}_{_V_k(_λ₎_}_in_An_{such that}

lim sup k→∞

FVk(λ)

(<)λF(S) + (1−λ)F(T). (2.6)

It was shown in [8,33] that if a diﬀerentiable functionF:S→Ris (strictly) convex, then

F(S)(>)F(T) + n

i=1

DiF(T),χSi−χTi

(2.7)

for allS,T∈An_.

Following the introduction of the notion of convexity for set functions by Morris [33] and its extension forn-set functions by Corley [8], various generalizations of convex-ity for set andn-set functions were proposed in [4,24,25,28,35,36,40,44,45]. More specifically, quasiconvexity and pseudoconvexity for set functions were defined in [25], and forn-set functions in [28]; generalizedρ-convexity forn-set functions was defined in [40], (Ᏺ,ρ)-convexity in [35],b-vexity in [4], (ρ,b)-vexity in [36], (Ᏺ,ρ,θ)-convexity for nondiﬀerentiable set functions in [24], and (Ᏺ,α,ρ,θ)-V-convexity in [44,45]. For predecessors and point-function counterparts of these convexity concepts, the reader is referred to the original papers where the extensions to set andn-set functions are dis-cussed. A survey of recent advances in the area of generalized convex functions and their role in developing optimality conditions and duality relations for optimization problems is given in [34].

For the purpose of formulating and proving various collections of suﬃciency criteria for (P), in this study, we will use a new class of generalized convexn-set functions, called (Ᏺ,b,φ,ρ,θ)-univex functions, which will be defined later in this section. This class of functions may be viewed as a combination of several previously defined types of general-ized convex functions. Its main ingredients areᏲ-convex functions and univex functions, which were introduced in [3,13], respectively. These functions were proposed as gener-alizations of the class of invex functions.

(5)

Definition 2.5[12]. Let f be a real-valued diﬀerentiable function defined on an open subsetSofRn_{. Then}_f _{is said to be}_η_-_invex₍_{invex with respect to}_η_{) at}_x∗_{if there exists a}

functionη:S×S→Rn_{such that for each}_x_∈_S_,

f(x)−fx∗∇fx∗Tηx,x∗, (2.8)

where∇f(x∗) is the gradient of f atx∗, andT denotes transposition; f is said to be η-invex(invex with respect toη) onSif there exists a functionη:S×S→Rn_{such that for} allx,y∈S,

f(x)−f(y)∇f(y)Tη(x,y). (2.9)

From the above definition, it is clear that every real-valued diﬀerentiable function is invex with respect toη(x,y)=x−y. This generalization of the concept of convexity was originally proposed by Hanson [12] who showed that for a nonlinear programming prob-lem of the form

(P0) Minimizef(x) subject togi(x)0,i∈m,x∈Rn,

where the diﬀerentiable functions f,gi:Rn→Rare invex with respect to the same func-tionη, the Karush-Kuhn-Tucker necessary optimality conditions are also suﬃcient. The

terminvex (forinvariant convex) was coined by Craven [9] to signify the fact that the

invexity property, unlike convexity, remains invariant under bijective coordinate trans-formations.

In a similar manner, one can readily defineη-pseudoinvex andη-quasi-invex functions as generalizations of diﬀerentiable pseudoconvex and quasiconvex functions.

The notion of invexity has been extended in several directions. Some recent surveys and syntheses of results pertaining to various generalizations of invex functions and their applications along with extensive lists of relevant references are available in [10,11,20, 31,34]. Two of the earliest generalizations of invex functions areᏲ-convex and (ρ,η )-invex functions. AnᏲ-convex function is defined in terms of a sublinear function, that is, a function that is subadditive and positively homogeneous.

Definition 2.6. A functionᏲ:Rn_→_R_{is said to be}_{sublinear (superlinear)}_if_Ᏺ(_x₊_y₎₍₎

Ᏺ(x) +Ᏺ(y) for allx,y∈Rn_{, and}_Ᏺ₍_ax₎₌_a_Ᏺ₍_x_{) for all}_x_∈_Rn_and_a_∈_R+_≡_[0,_∞_). Now combining the definitions ofᏲ-convex and (ρ,η)-invex functions given in [13, 17], respectively, we can define (Ᏺ,ρ)-convex, (Ᏺ,ρ)-pseudoconvex, and (Ᏺ,ρ )-quasi-convex functions.

Letgbe a real-valued diﬀerentiable function defined on the open subsetSofRn_{, and} assume that for eachx,y∈S, the functionᏲ(x,y;·) :Rn_→_R_{is sublinear.}

Definition 2.7. The functiongis said to be (Ᏺ,ρ)-convexatyif there exists a real number

ρsuch that for eachx∈S,

(6)

Definition 2.8. The functiong is said to be (Ᏺ,ρ)-pseudoconvexat yif there exists a real numberρsuch that for eachx∈S,

Ᏺx,y;∇g(y)−ρx−y2_=⇒_g₍_x₎_g₍_y₎_. _(2.11)

Definition 2.9. The functiong is said to be (Ᏺ,ρ)-quasiconvexat yif there exists a real

numberρsuch that for eachx∈S,

g(x)g(y)=⇒Ᏺx,y;∇g(y)−ρx−y2_. _(2.12)

Evidently, if in Definitions2.7–2.9we chooseᏲ(x,y;∇g(y))= ∇g(y)T_η₍_x_,_y_{), where} η:S×S→Rn _{is a given function, and set}_ρ₌_{0, then we see that they reduce to the} definitions ofη-invexity,η-pseudoinvexity, andη-quasi-invexity for the functiong.

The foregoing classes of generalized convex functions have been utilized for establish-ing numerous sets of suﬃcient optimality conditions and a variety of duality results for several categories of static and dynamic optimization problems. For a wealth of informa-tion as well as long lists of references concerning these results, the reader is referred to [20,34].

Another significant generalization of the notion of invexity, called univexity, which subsumes a number of previously proposed classes of generalized convex functions, was recently given in [3]. We recall the definitions of univex, pseudounivex, and quasiunivex functions.

Lethbe a real-valued diﬀerentiable function defined on an open subsetSofRn_{, let}_η be a function fromS×StoRn_{, let}_Φ_{be a real-valued function defined on}_R_{, and let}_b_be a function fromS×StoR+\ {0} ≡(0,∞).

Definition 2.10[3]. The functionhis said to beunivexatywith respect toη,Φ, andbif

for eachx∈S,

b(x,y)Φh(x)−h(y)∇h(y)Tη(x,y). (2.13)

Definition 2.11[3]. The functionhis said to bepseudounivexatywith respect toη,Φ,

andbif for eachx∈S,

∇h(y)Tη(x,y)0=⇒b(x,y)Φh(x)−h(y)0. (2.14)

Definition 2.12[3]. The functionhis said to bequasiunivexatywith respect toη,Φ, and

bif for eachx∈S,

Φh(x)−h(y)0=⇒b(x,y)∇h(y)Tη(x,y)0. (2.15)

Finally, we are in a position to give our definitions of generalized (Ᏺ,b,φ,ρ,θ)-univex n-set functions. They are formulated by combining then-set versions of Definitions2.5– 2.12.

(7)

Definition 2.13. The functionF is said to be(strictly)(Ᏺ,b,φ,ρ,θ)-univexatS∗if there exist a sublinear functionᏲ(S,S∗;·) :Ln1(X,A,µ)→R, a functionb:An×An→Rwith positive values, a functionθ:An_×_An_→_An_×_An_{such that}_S₌_S∗_⇒_θ₍_S_,_S∗₎₌_{(0, 0), a}

functionφ:R→R, and a real numberρsuch that for eachS∈An_,

φF(S)−FS∗(>)ᏲS,S∗;bS,S∗DFS∗+ρd2_θ_S_,_S∗_. _(2.16)

Definition 2.14. The functionF is said to be(strictly) (Ᏺ,b,φ,ρ,θ)-pseudounivexatS∗

if there exist a sublinear functionᏲ(S,S∗;·) :Ln1(X,A,µ)→R, a functionb:An×An→ Rwith positive values, a functionθ:An_×_An_→_An_×_An_{such that}_S₌_S∗_⇒_θ₍_S_,_S∗₎₌

(0, 0), a functionφ:R→R, and a real numberρsuch that for eachS∈An₍_S₌_S∗_),

ᏲS,S∗;bS,S∗DFS∗−ρd2θS,S∗=⇒φF(S)−FS∗(>)0. (2.17)

Definition 2.15. The functionFis said to be(prestrictly)(Ᏺ,b,φ,ρ,θ)-quasiunivexatS∗ if there exist a sublinear functionᏲ(S,S∗;·) :Ln1(X,A,µ)→R, a functionb:An×An→ Rwith positive values, a functionθ:An_×_An_→_An_×_An_{such that}_S₌_S∗_⇒_θ₍_S_,_S∗₎₌

(0, 0), a functionφ:R→R, and a real numberρsuch that for eachS∈An_,

φF(S)−FS∗(<)0=⇒ᏲS,S∗;bS,S∗DFS∗−ρd2_θ_S_,_S∗_. _(2.18)

From the above definitions it is clear that ifFis (Ᏺ,b,φ,ρ,θ)-univex atS∗, then it is both (Ᏺ,b,φ,ρ,θ)-pseudounivex and (Ᏺ,b,φ,ρ,θ)-quasiunivex atS∗, ifFis (Ᏺ,b,φ,ρ,θ )-quasiunivex atS∗, then it is prestrictly (Ᏺ,b,φ,ρ,θ)-quasiunivex atS∗, and ifFis strictly (Ᏺ,b,φ,ρ,θ)-pseudounivex atS∗, then it is (Ᏺ,b,φ,ρ,θ)-quasiunivex atS∗.

In the proofs of the suﬃciency theorems, sometimes it may be more convenient to use certain alternative but equivalent forms of the above definitions. These are obtained by considering the contrapositive statements. For example, (Ᏺ,b,φ,ρ,θ)-quasiunivexity can be defined in the following equivalent way:

Fis said to be (Ᏺ,b,φ,ρ,θ)-quasiunivex atS∗if for eachS∈An_,

ᏲS,S∗;bS,S∗DFS∗>−ρd2_θ_S_,_S∗_=⇒_φ_F₍_S₎₋_F_S∗_>₀_. _(2.19)

Needless to say that the new classes of generalized convex n-set functions specified in Definitions2.13–2.15contain a variety of special cases; in particular, they subsume all the previously defined types of generalizedn-set functions. This can easily be seen by appropriate choices ofᏲ,b,φ,ρ, andθ.

In the sequel, we will also need a consistent notation for vector inequalities. For all a,b∈Rm_{, the following order notation will be used:}_a_b _{if and only if}_a

ibi for all i∈m;abif and only ifaibifor alli∈m, buta=b;a > bif and only ifai> bifor all i∈m; andabis the negation ofab.

Throughout the sequel, we will deal exclusively with the eﬃcient solutions of (P). An x∗∈ᐄis said to be aneﬃcient solutionof (P) if there is no otherx∈ᐄsuch thatϕ(x) ϕ(x∗), whereϕis the objective function of (P).

(8)

Theorem2.16 [44]. Assume thatFi,Gi,i∈p, andHj,j∈q, are diﬀerentiable atS∗∈An,

and that for eachi∈p, there existSˆi_∈_An_{such that}

Hj

S∗+ n

k=1 DkHj

S∗,χˆ_S_k−χS∗k

<0, j∈q, (2.20)

and for each∈p\ {i},

n

k=1 DkF

S∗−λ∗DkG

<0. (2.21)

IfS∗is an eﬃcient solution of (P) andλ∗_i =ϕ(S∗),i∈p, then there existu∗∈U= {u∈

Rp_:_{u >}_0,p

i=1ui=1}andv∗∈Rq+such that

n

k=1

p

i=1 u∗i

DkFi

S∗−λ∗iDkGi

S∗+ q

j=1 v∗jDkHj

S∗,χSk−χS∗k

0 ∀S∈An_,

v∗_jHj

S∗=0, j∈q.

(2.22)

The above theorem contains two sets of parameters u∗_i and λ∗_i, i∈p, which were introduced as a consequence of an indirect approach in [44] requiring two intermediate auxiliary problems. It is possible to eliminate one of these two sets of parameters and thus obtain a semiparametric version ofTheorem 2.16. Indeed, this can be accomplished by simply replacingλ∗_i byFi(S∗)/Gi(S∗),i∈p, and redefiningu∗andv∗. This result is stated in the next theorem.

Theorem2.17. Assume thatFi,Gi,i∈p, andHj,j∈q, are diﬀerentiable atS∗∈An, and

that for eachi∈p, there existSˆi_∈_An_{such that}

HjS∗+ n

k=1

DkHjS∗,χˆSk−χS∗k

<0, j∈q, (2.23)

and for each∈p\ {i},

n

k=1 Gi

S∗DkF

S∗−Fi

S∗DkG

<0. (2.24)

IfS∗is an eﬃcient solution of (P), then there existu∗∈Uandv∗∈Rq+such that

n

k=1

p

i=1 u∗_iGi

S∗DkFi

S∗−Fi

S∗DkGi

S∗

+ q

j=1

v∗_jDkHjS∗,χSk−χS∗k

0 ∀S∈An_,

v∗jHj

S∗=0, j∈q.

(9)

The form and contents of the necessary efficiency conditions given inTheorem 2.17 provide clear guidelines for devising numerous sets of semiparametric sufficient effi -ciency criteria as well as for constructing various types of semiparametric duality models for (P).

3. Suﬃcient eﬃciency conditions

In this section, we present several sets of suﬃcient eﬃciency conditions for (P) under a variety of generalized (Ᏺ,b,φ,ρ,θ)-univexity hypotheses. We begin by introducing some notation.

Let the functions fi(·,S∗), i∈p, f(·,S∗,u∗), andh(·,v∗) :An_→_R_{be defined, for} fixedS∗,u∗, andv∗, by

fi

T,S∗=Gi

S∗Fi(T)−Fi

S∗Gi(T), i∈p,

fT,S∗,u∗= p

i=1 u∗i

Gi

S∗Fi(T)−Fi

S∗Gi(T),

hT,v∗= q

j=1

v∗jHj(T).

(3.1)

For givenu∗∈U0= {u∈Rq+:

p

i=1ui=1}andv∗∈Rq+, letI+(u∗)= {i∈p:u∗i >0} andJ+(v∗)= {j∈q:v∗j >0}.

Theorem3.1. Let S∗∈FwithFi(S∗)0,i∈p, and assume thatFi,Gi,i∈p, andHj, j∈q, are diﬀerentiable atS∗, and that there existu∗∈Uandv∗∈Rq+such that

Ᏺ

S,S∗; p

i=1 u∗i

Gi

S∗DFi

S∗−Fi

S∗DGi

S∗+ q

j=1 v∗jDHj

S∗

0 ∀S∈F,

(3.2)

v∗jHj

S∗=0, j∈q, (3.3)

whereᏲ(S,S∗;·) :L1n(X,A,µ)→Ris a sublinear function. Assume, furthermore, that any of

the following three sets of hypotheses is satisfied:

(a) (i)for eachi∈p,Fiis(Ᏺ,b, ¯φ, ¯ρi,θ)-univex atS∗, and−Giis(Ᏺ,b, ¯φ, ˆρi,θ)-univex

atS∗,φ¯is superlinear, andφ¯(a)0⇒a0;

(ii)for eachj∈J+≡J+(v∗),Hjis(Ᏺ,b, ˜φj, ˜ρj,θ)-quasiunivex atS∗,φ˜jis increasing,

andφ˜j(0)=0;

(iii)ρ∗+j∈J+v

∗

jρ˜j0, whereρ∗=pi=1ui∗[Gi(S∗) ¯ρi+Fi(S∗) ˆρi];

(b) (i)for eachi∈p,Fiis(Ᏺ,b, ¯φ, ¯ρi,θ)-univex atS∗, and−Giis(Ᏺ,b, ¯φ, ˆρi,θ)-univex

atS∗,φ¯is superlinear, andφ¯(a)0⇒a0;

(ii)h(·,v∗)is(Ᏺ,b, ˜φ, ˜ρ,θ)-quasiunivex atS∗,φ˜is increasing, andφ˜(0)=0;

(10)

(c) (i)the Lagrangian-type function

T−→LT,S∗,u∗,v∗= p

i=1 u∗i

Gi

S∗Fi(T)−Fi

S∗Gi(T)+ q

j=1

v∗jHj(T) (3.4)

is(Ᏺ,b,φ, 0,θ)-pseudounivex atS∗andφ(a)0⇒a0. ThenS∗is an eﬃcient solution of (P).

Proof. (a) LetSbe an arbitrary feasible solution of (P). Using the hypotheses specified in

(i), we see that for eachi∈p,

¯

φFi(S)−FiS∗ᏲS,S∗;bS,S∗DFiS∗+ ¯ρid2θS,S∗, ¯

φ−Gi(S) +Gi

S∗ᏲS,S∗;−bS,S∗DGi

S∗+ ˆρid2

θS,S∗. (3.5)

Sinceu∗>0,_ip₌1u∗i =1,Fi(S∗)0,Gi(S∗)>0, ¯φis superlinear, andᏲ(S,S∗;·) is sub-linear, we deduce from these inequalities that

¯ φ

p

i=1 u∗i

Gi

S∗Fi(S)−Fi

S∗Gi(S)− p

i=1 u∗i

Gi

S∗Fi

S∗−Fi

S∗Gi

S∗

Ᏺ

S,S∗;bS,S∗ p

i=1 u∗_i Gi

S∗DFi

S∗−Fi

S∗DGi

S∗ + p

i=1

u∗_i GiS∗ρ¯i+FiS∗ρˆid2θS,S∗.

(3.6)

AsS∈F, it follows from (3.3) that for each j∈J+,Hj(S)0=Hj(S∗), and so using the properties of ˜φj, we get for each j∈J+, ˜φj(Hj(S)−Hj(S∗))0, which in view of (ii) implies thatᏲ(S,S∗;b(S,S∗)DHj(S∗))−ρ˜jd2(θ(S,S∗)). Becausev∗0,v∗j =0 for eachj∈q\J+, andᏲ(S,S∗;·) is sublinear, these inequalities yield

Ᏺ

S,S∗;bS,S∗ q

j=1 v∗_jDHj

S∗

− j∈J+

v∗_jρ˜jd2

θS,S∗. (3.7)

From the sublinearity ofᏲ(S,S∗;·), (3.2), and the fact thatb(S,S∗)>0, it is clear that

Ᏺ

S,S∗;bS,S∗ p

i=1 u∗_iGi

S∗DFi

S∗−Fi

S∗DGi

S∗

+Ᏺ

S,S∗;bS,S∗ q

j=1 v∗_jDHj

S∗

0.

(11)

Combining (3.6)–(3.8) and using (iii), we obtain

¯ φ

p

i=1

u∗_iGiS∗Fi(S)−FiS∗Gi(S)

ρ∗+ j∈J+

v∗_jρ˜j

d2_θ_S_,_S∗₀_. _(3.9)

Since ¯φ(a)0⇒a0, the above inequality reduces to

p

i=1 u∗_iGi

S∗Fi(S)−Fi

S∗Gi(S)0. (3.10)

Sinceu∗>0, (3.10) implies that

G1S∗F1(S)−F1S∗G1(S),. . .,Gp

S∗Fp(S)−Fp

S∗Gp(S)0,. . ., 0, (3.11)

which in turn implies that

ϕ(S)=

F1(S) G1(S),. . .,

Fp(S) Gp(S)

F1S∗ G1S∗,. . .,

Fp

S∗ Gp

S∗

=ϕS∗.

(3.12)

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

(b) Sincev∗0, it follows that for anyS∈F,v∗_jHj(S)0 for each j∈q, and so by (3.3), we have thath(S,v∗)h(S∗,v∗), which in view of the properties of ˜φcan be expressed as ˜φh(S,v∗)−h(S∗,v∗)0. Because of (ii) the last inequality implies that

Ᏺ

S,S∗;bS,S∗ q

j=1 v∗jDHj

S∗

−ρd˜ 2θS,S∗. (3.13)

Now proceeding as in the proof of part (a) and using this inequality instead of (3.7), we will obtain (3.10), which leads to the desired conclusion thatS∗is an eﬃcient solution of (P).

(c) Since (3.2) holds,Ᏺ(S,S∗;·) is sublinear, andb(S,S∗)>0, we have

Ᏺ

S,S∗;bS,S∗

p

i=1

u∗_i GiS∗DFiS∗−FiS∗DGiS∗+ q

j=1

v∗_jDHjS∗

0,

(3.14)

which because of our (Ᏺ,b,φ, 0,θ)-pseudounivexity assumption implies that

φLS,u∗,v∗−LS∗,u∗,v∗0. (3.15)

Butφ(a)0⇒a0, and henceL(S,u∗,v∗)L(S∗,u∗,v∗)=0, where the equality fol-lows from (3.3). Sincev∗jHj(S)0 for each j∈q, the inequality reduces to (3.10), which leads, as seen in the proof of part (a), to the conclusion thatS∗is an eﬃcient solution

(12)

InTheorem 3.1, separate (Ᏺ,b,φ,ρ,θ)-univexity conditions were imposed on the func-tionsFiand −Gi,i∈p. In the remainder of this section, we will present a number of suﬃciency results in which various generalized (Ᏺ,b,φ,ρ,θ)-univexity requirements will be placed on certain combinations of these functions.

Theorem3.2. LetS∗∈Fand assume thatFi,Gi,i∈p, andHj, j∈q, are diﬀerentiable

atS∗, and that there existu∗∈U andv∗∈Rq+such that (3.2) and (3.3) hold. Assume,

furthermore, that any of the following six sets of hypotheses is satisfied:

(a) (i) f(·,S∗,u∗)is(Ᏺ,b, ¯φ, ¯ρ,θ)-pseudounivex atS∗, andφ¯(a)0⇒a0;

(ii)for eachj∈J+≡J+(v∗),Hjis(Ᏺ,b, ˜φj, ˜ρj,θ)-quasiunivex atS∗,φ˜jis increasing,

andφ˜j(0)=0;

(iii) ¯ρ+j∈J+v

∗

jρ˜j0;

(b) (i) f(·,S∗,u∗)is(Ᏺ,b, ¯φ, ¯ρ,θ)-pseudounivex atS∗, andφ¯(a)0⇒a0;

(iii) ¯ρ+ ˜ρ0;

(c) (i) f(·,S∗,u∗)is prestrictly(Ᏺ,b, ¯φ, ¯ρ,θ)-quasiunivex atS∗, andφ¯(a)0⇒a0;

(ii)for each j∈J+, Hj is (Ᏺ,b, ˜φj, ˜ρj,θ)-quasiunivex atS∗, φ˜j is increasing, and ˜

φj(0)=0; (iii) ¯ρ+j∈J+v

∗

jρ˜j>0;

(d) (i) f(·,S∗,u∗)is prestrictly(Ᏺ,b, ¯φ, ¯ρ,θ)-quasiunivex atS∗, andφ¯(a)0⇒a0;

(iii) ¯ρ+ ˜ρ >0;

(e) (i) f(·,S∗,u∗)is prestrictly(Ᏺ,b, ¯φ, ¯ρ,θ)-quasiunivex atS∗, andφ¯(a)0⇒a0;

(ii)for eachj∈J+,Hjis strictly(Ᏺ,b, ˜φj, ˜ρj,θ)-pseudounivex atS∗,φ˜jis increasing,

andφ˜j(0)=0; (iii) ¯ρ+j∈J+v

∗

jρ˜j0;

(f) (i) f(·,S∗,u∗)is prestrictly(Ᏺ,b, ¯φ, ¯ρ,θ)-quasiunivex atS∗, andφ¯(a)0⇒a0;

(ii)h(·,v∗)is strictly(Ᏺ,b, ˜φ, ˜ρ,θ)-pseudounivex atS∗,φ˜is increasing, andφ˜(0)=0;

(iii) ¯ρ+ ˜ρ0.

ThenS∗is an eﬃcient solution of (P).

Proof. (a) LetS be an arbitrary feasible solution of (P). Then, as seen in the proof of

Theorem 3.1, our hypotheses in (ii) lead to (3.7), which when combined with (3.8) yields

Ᏺ

S,S∗;bS,S∗ p

i=1 u∗_iGi

S∗DFi

S∗−Fi

S∗DGi

S∗

¯ ρ+

j∈J+

v∗_jρ˜j

d2_θ_S_,_S∗₋_ρd_¯ 2_θ_S_,_S∗_,

(3.16)

where the second inequality follows from (iii). By virtue of (i), this inequality implies that ¯φ(f(S,S∗,u∗)−f(S∗,S∗,u∗))0, which because of the properties of the function

¯

(13)

(b) The proof is similar to that of part (a).

(c) As seen in the proof ofTheorem 3.1, our hypotheses in (ii) lead to (3.7), which when combined with (3.8) yields

Ᏺ

S,S∗;bS,S∗ p

i=1 u∗_iGi

S∗DFi

S∗−Fi

S∗DGi

S∗

ρ∗+ j∈J+

v∗_jρ˜j

d2_θ_S_,_S∗_>₋_ρd_¯ 2_θ_S_,_S∗_,

(3.17)

where the second inequality follows from (iii). By (i), this inequality implies that ¯

φ(f(S,S∗,u∗)−f(S∗,S∗,u∗))0. But ¯φ(a)0⇒a0, and hence we getf(S,S∗,u∗) f(S∗,S∗,u∗)=0, which is (3.10), and therefore we conclude, as inTheorem 3.1, thatS∗ is an eﬃcient solution of (P).

(d)–(f) The proofs are similar to those of parts (a)–(c).

atS∗, and that there existu∗∈U0 andv∗∈Rq+such that (3.2) and (3.3) hold. Assume,

(a) (i)for eachi∈I+≡I+(u∗), fi(·,S∗)is strictly(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atS∗,φ¯i

is increasing, andφ¯i(0)=0;

(ii)for eachj∈J+≡J+(v∗),Hjis(Ᏺ,b, ˜φj, ˜ρj,θ)-quasiunivex atS∗, ˜φjis increasing,

andφ˜j(0)=0;

(iii)ρ◦+j∈J+v

∗

jρ˜j0, whereρ◦=

i∈I+u

∗

i ρ¯i;

(b) (i)for eachi∈I+,fi(·,S∗)is strictly(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atS∗, ¯φiis

increas-ing, andφ¯i(0)=0;

(iii)ρ◦+ ˜ρ0;

(c) (i)for eachi∈I+, fi(·,S∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗,φ¯iis increasing, and ¯

φi(0)=0;

(ii)for eachj∈J+,Hjis strictly(Ᏺ,b, ˜φj, ˜ρj,θ)-pseudounivex atS∗,φ˜jis increasing,

andφ˜j(0)=0;

(iii)ρ◦+_j_∈_J+v

∗

jρ˜j0;

(d) (i)for eachi∈I+, fi(·,S∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗,φ¯iis increasing, and ¯

φi(0)=0;

(ii)h(·,v∗)is strictly(Ᏺ,b, ˜φ, ˜ρ,θ)-pseudounivex atS∗,φ˜is increasing, andφ˜(0)=0;

(iii)ρ◦+ ˜ρ0;

(e) (i)for eachi∈I+, fi(·,S∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗,φ¯iis increasing, and ¯

φi(0)=0;

(ii)for each j∈J+, Hj is (Ᏺ,b, ˜φj, ˜ρj,θ)-quasiunivex atS∗, φ˜j is increasing, and ˜

φj(0)=0; (iii)ρ◦+j∈J+v

∗

jρ˜j>0;

(f) (i)for eachi∈I+, fi(·,S∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗,φ¯iis increasing, and ¯

(14)

(iii)ρ◦+ ˜ρ >0.

Proof. (a) Suppose to the contrary thatS∗is not an eﬃcient solution of (P). Then there

exists ¯S∈Fsuch thatϕi( ¯S)ϕi(S∗) for eachi∈p, andϕ( ¯S)< ϕ(S∗) for some∈p. From these inequalities it can easily be seen that for eachi∈I+,

Gi

S∗Fi

_¯

S−Fi

S∗Gi( ¯S)0=Gi

S∗Fi

S∗−Fi

S∗Gi

S∗, (3.18)

which in view of the properties of ¯φican be expressed as

¯

φiGiS∗Fi( ¯S)−FiS∗Gi( ¯S)−GiS∗FiS∗−FiS∗GiS∗0. (3.19)

By (i), this implies that for eachi∈I+,

Ᏺ_S¯_,_S∗_;_b_S_¯_,_S∗_G

iS∗DFiS∗−FiS∗DGiS∗<−ρ¯id2θS¯,S∗. (3.20)

Since u∗0,u∗_i =0 for each i∈p\I+,i∈I+u

∗

i =1, andᏲ( ¯S,S∗;·) is sublinear, the above inequalities yield

Ᏺ

¯

S,S∗;bS¯,S∗ p

i=1 u∗i

Gi

S∗DFi

S∗−Fi

S∗DGi

S∗

<− i∈I+

u∗i ρ¯id2

θS¯,S∗.

(3.21)

Now combining (3.7) (which is valid for the present case because of (ii)), (3.8), and (3.21), and using the sublinearity ofᏲ( ¯S,S∗;·), we obtain

Ᏺ

¯

S,S∗;bS¯,S∗ p

i=1

u∗_i GiS∗DFiS∗−FiS∗DGiS∗+ q

j=1

v∗_jDHjS∗

<−

ρ◦+ i∈J+

v∗_jρ˜j

d2_θ_S_¯_,_S∗_,

(3.22)

which in view of (iii) contradicts (3.2). Hence,S∗is an eﬃcient solution of (P). (b)–(d) The proofs are similar to that of part (a).

(e) Proceeding as in the proof of part (a) and using the conditions set forth in (i), we arrive at the inequality

Ᏺ

¯

S,S∗;bS¯,S∗ p

i=1 u∗_iGi

S∗DFi

S∗−Fi

S∗DGi

S∗

− i∈I+

u∗i ρ¯id2

θS¯,S∗.

(15)

Combining this inequality with (3.8) and using (iii), we obtain

Ᏺ

¯

S,S∗;bS¯,S∗ q

j=1 v∗_jDHj

S∗

i∈I+

u∗iρ¯id2

θS¯,S∗>− j∈J+

v∗jρ˜jd2

θS¯,S∗,

(3.24)

which contradicts (3.7). Therefore, we conclude thatS∗is an eﬃcient solution of (P).

(f) The proof is similar to that of part (e).

We close this section by stating a variant ofTheorem 4.4; its proof is similar to that of Theorem 4.4and hence omitted.

furthermore, that any of the following four sets of hypotheses is satisfied:

(a) (i)for eachi∈I1+= ∅, fi(·,S∗)is strictly (Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex at S∗, for

eachi∈I2+, fi(·,S∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗, and for eachi∈I+≡

I+(u∗),φ¯iis increasing andφ¯i(0)=0, where{I1+,I2+}is a partition ofI+; (ii)for eachj∈J+≡J+(v∗),Hjis(Ᏺ,b, ˜φj, ˜ρj,θ)-quasiunivex atS∗,φ˜jis increasing,

andφ˜j(0)=0;

(iii)ρ◦+j∈J+v

∗

jρ˜j0, whereρ◦=

i∈I+u

∗

i ρ¯i;

(b) (i)for eachi∈I1+= ∅, fi(·,S∗)is strictly (Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex at S∗, for

eachi∈I2+,fi(·,S∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗, and for eachi∈I+,φ¯iis

increasing andφ¯i(0)=0, where{I1+,I2+}is a partition ofI+;

(ii)h(·,v∗)is(Ᏺ, ˜b, ˜φ, ˜ρ,θ)-quasiunivex atS∗,φ˜is increasing, andφ˜(0)=0;

(iii)ρ◦+ ˜ρ0;

(c) (i)for eachi∈I+, fi(·,S∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗,φ¯iis increasing, and ¯

φ(0)=0;

(ii)for each j∈J1+= ∅,Hj is strictly(Ᏺ,b, ˜φj, ˜ρj,θ)-pseudounivex atS∗, for each j∈J2+,Hjis(Ᏺ,b, ˜φj, ˜ρj,θ)-quasiunivex atS∗, and for eachj∈J+,φ˜jis

increas-ing andφ˜j(0)=0, where{J1+,J2+}is a partition ofJ+;

(iii)ρ◦+j∈J+v∗jρ˜j0;

(d) (i)for eachi∈I1+, fi(·,S∗)is strictly (Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atS∗, for each

i∈I2+, fi(·,S∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗, and for eachi∈I+≡I+(u∗), ¯

φiis increasing andφ¯i(0)=0, where{I1+,I2+}is a partition ofI+;

(ii)for eachj∈J1+,Hjis strictly(Ᏺ,b, ˜φj, ˜ρj,θ)-pseudounivex atS∗, for eachj∈J2+, Hj is(Ᏺ,b, ˜φj, ˜ρj,θ)-quasiunivex atS∗, and for eachj∈J+,φ˜jis increasing and

˜

φj(0)=0, where{J1+,J2+}is a partition ofJ+; (iii)ρ◦+j∈J+v

∗

jρ˜j0;

(iv)I1+= ∅,J1+= ∅, orρ◦+j∈J+v

∗

jρ˜j>0.

(16)

4. Generalized suﬃcient eﬃciency criteria

In this section, we formulate and discuss several families of generalized suﬃciency results for (P) with the help of a partitioning scheme that was originally proposed in [32] for constructing generalized dual problems for nonlinear programs with point functions.

Let{J0,J1,. . .,Jm}be a partition of the index setq; thusJr⊂qfor eachr∈ {0, 1,. . .,m}, Jr∩Js= ∅for eachr,s∈ {0, 1,. . .,m} withr=s, and ∪mr=0Jr=q. In addition, we will make use of the functionsΨi(·,S∗,v∗),Ψ(·,S∗,u∗,v∗), andΛt(·,v∗) :An_→_R_defined, for fixedu∗,v∗, andS∗, by

ΨiT,S∗,v∗=GiS∗Fi(T)−FiS∗Gi(T) +

j∈J0

v∗_jHj(T), i∈p,

ΨT,S∗,u∗,v∗= p

i=1

u∗_iGiS∗Fi(T)−FiS∗Gi(T)+

j∈J0

v∗_jHj(T),

ΛtT,v∗= j∈Jt

v∗_jHj(T), t∈m∪ {0}.

(4.1)

Using these sets and functions, we next state and prove a number of generalized suﬃ -ciency results for (P).

atS∗, and that there existu∗∈U andv∗∈Rq+such that (3.2) and (3.3) hold. Assume,

furthermore, that any of the following four sets of hypotheses is satisfied:

(a) (i)Ψ(·,S∗,u∗,v∗)is(Ᏺ,b, ¯φ, ¯ρ,θ)-pseudounivex atS∗, andφ¯(a)0⇒a0;

(ii)for eacht∈m,Λt(·,v∗)is(Ᏺ,b, ˜φt, ˜ρt,θ)-quasiunivex atS∗,φ˜tis increasing, and ˜

φt(0)=0; (iii) ¯ρ+m_t₌1ρ˜t0;

(b) (i)Ψ(·,S∗,u∗,v∗)is prestrictly(Ᏺ,b, ¯φ, ¯ρ,θ)-quasiunivex atS∗, andφ¯(a)0⇒a 0;

(ii)for eacht∈m,Λt(·,v∗)is strictly(Ᏺ,b, ˜φt, ˜ρt,θ)-pseudounivex atS∗,φ˜t is

in-creasing, andφ˜t(0)=0; (iii) ¯ρ+m_t₌1ρ˜t0;

(c) (i)Ψ(·,S∗,u∗,v∗)is prestrictly(Ᏺ,b, ¯φ, ¯ρ,θ)-quasiunivex atS∗, andφ¯(a)0⇒a 0;

(ii)for eacht∈m,Λt(·,v∗)is(Ᏺ,b, ˜φt, ˜ρt,θ)-quasiunivex atS∗,φ˜tis increasing, and ˜

φt(0)=0;

(iii) ¯ρ+mt=1ρ˜t>0;

(d) (i)Ψ(·,S∗,u∗,v∗)is prestrictly(Ᏺ,b, ¯φ, ¯ρ,θ)-quasiunivex atS∗, andφ¯(a)0⇒a 0;

(ii)for eacht∈m1,Λt(·,v∗)is(Ᏺ,b, ˜φt, ˜ρt,θ)-quasiunivex atS∗,φ˜tis increasing, and ˜

φt(0)=0, for eacht∈m2= ∅,Λt(·,v∗)is strictly(Ᏺ,b, ˜φt, ˜ρt,θ)-pseudounivex atS∗,φ˜tis increasing, andφ˜t(0)=0, where{m1,m2}is a partition ofm; (iii) ¯ρ+m_t₌1ρ˜t0.

(17)

Proof. (a) LetSbe an arbitrary feasible solution of (P). Asv∗0, it is clear from (3.3) that for eacht∈m,

ΛS,v∗= j∈Jt

v∗jHj(S)0=

j∈Jt

v∗jHj

S∗=ΛS∗,v∗, (4.2)

and hence using the properties of ˜φt, we get ˜φt(Λt(S,v∗)−Λt(S∗,v∗))0, which by (ii) implies that for eacht∈m,

Ᏺ

S,S∗;bS,S∗ j∈Jt

v∗_jDHj

S∗

−ρ˜td2

θS,S∗. (4.3)

Adding these inequalities and using the sublinearity ofᏲ(S,S∗;·), we obtain

Ᏺ

S,S∗;bS,S∗ m

t=1

j∈Jt

v∗_jDHjS∗

− m

t=1 ˜

ρtd2θS,S∗. (4.4)

From the sublinearity ofᏲ(S,S∗;·) and (3.2), it follows that

Ᏺ

S,S∗;bS,S∗

p

i=1

u∗_iGiS∗DFiS∗−FiS∗DGiS∗+

j∈J0

v∗_jDHjS∗

+Ᏺ

S,S∗;bS,S∗ m

t=1

j∈Jt

v∗_jDHj

S∗

0.

(4.5)

Combining (4.4) and (4.5) and using (iii), we obtain the inequality

Ᏺ

S,S∗;bS,S∗

p

i=1 u∗_iGi

S∗DFi

S∗−Fi

S∗DGi

S∗+ j∈J0

v∗_jDHj

S∗

m

t=1 ˜ ρtd2

θS,S∗−ρd¯ 2_θ_S_,_S∗_,

(4.6)

which in view of (i) implies that ¯φ(Ψ(S,S∗,u∗,v∗)−Ψ(S∗,S∗,u∗,v∗))0. Due to the properties of the function ¯φ, this inequality yieldsΨ(S,S∗,u∗,v∗)−Ψ(S∗,S∗,u∗,v∗)0. But in view of (3.2) and (3.3),Ψ(S∗,S∗,u∗,v∗)=0, and so we have thatΨ(S,S∗,u∗,v∗) 0. Sincev∗jHj(S)0 for each j∈q, the last inequality reduces to

p

i=1

u∗_iGiS∗Fi(S)−FiS∗Gi(S)0, (4.7)

which is precisely (3.10). Hence, we conclude, as in the proof ofTheorem 3.1, thatS∗is an eﬃcient solution of (P).

(18)

(a) (i)for eachi∈I+≡I+(u∗),Ψi(·,S∗,v∗)is strictly (Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex at

S∗,φ¯iis increasing, andφ¯i(0)=0;

φt(0)=0;

(iii)i∈I+u

∗

iρ¯i+mt=1ρ˜t0;

(b) (i)for eachi∈I+,Ψi(·,S∗,v∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗,φ¯iis increasing,

andφ¯i(0)=0;

(ii)for eacht∈m,Λt(·,v∗)is strictly(Ᏺ,b, ˜φt, ˜ρt,θ)-pseudounivex atS∗,φ˜t is

in-creasing, andφ˜t(0)=0; (iii)i∈I+u

∗

iρ¯i+

_m

t=1ρ˜t0;

(c) (i)for eachi∈I+,Ψi(·,S∗,v∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗,φ¯iis increasing,

andφ¯i(0)=0;

φt(0)=0; (iii)i∈I+u

∗

iρ¯i+mt=1ρ˜t>0;

(d) (i)for eachi∈I1+= ∅,Ψi(·,S∗,v∗)is strictly(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atS∗, for eachi∈I2+,Ψi(·,S∗,v∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗, and for eachi∈I+,

¯

φiis increasing andφ¯i(0)=0, where{I1+,I2+}is a partition ofI+;

φt(0)=0; (iii)i∈I+u

∗

(e) (i)for eachi∈I+,Ψi(·,S∗,v∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗,φ¯iis increasing,

andφ¯i(0)=0;

(ii)for eacht∈m1= ∅,Λt(·,v∗)is strictly(Ᏺ,b, ˜φt, ˜ρt,θ)-pseudounivex atS∗, for

eacht∈m2,Λt(·,v∗)is(Ᏺ,b, ˜φt, ˜ρt,θ)-quasiunivex atS∗, and for eacht∈m,φ˜t

is increasing andφ˜t(0)=0, where{m1,m2}is a partition ofm; (iii)i∈I+u

∗

(f) (i)for eachi∈I1+,Ψi(·,S∗,v∗)is strictly(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atS∗, for each

i∈I2+,Ψi(·,S∗,v∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗, and for eachi∈I+,φ¯iis

increasing andφ¯i(0)=0, where{I1+,I2+}is a partition ofI+;

(ii)for eacht∈m1,Λt(·,v∗)is strictly(Ᏺ,b, ˜φt, ˜ρt,θ)-pseudounivex atS∗, for each t∈m2,Λt(·,v∗)is(Ᏺ,b, ˜φt, ˜ρt,θ)-quasiunivex atS∗, and for eacht∈m,φ˜t is

increasing andφ˜t(0)=0, where{m1,m2}is a partition ofm; (iii)i∈I+u

∗

iρ¯i+mt=1ρ˜t0; (iv)I1+= ∅,m1= ∅, ori∈I+u

∗

i ρ¯i+

_m

t=1ρ˜t>0.

Proof. (a) Suppose to the contrary thatS∗ is not an eﬃcient solution of (P). As seen

in the proof of Theorem 3.3, this supposition leads to the inequalities Gi(S∗)Fi( ¯S)−

Fi(S∗)Gi( ¯S)0 for eachi∈p, andG(S∗)F( ¯S)−F(S∗)G( ¯S)<0 for some∈p, where ¯

(19)

(3.3), we see that for eachi∈p,

Ψi_S¯_,_S∗_,_v∗₌_G

iS∗Fi( ¯S)−FiS∗Gi( ¯S) +

j∈J0

v∗_jHj( ¯S)

Gi

S∗Fi( ¯S)−ΦS∗,u∗Gi( ¯S)0

=Gi

S∗Fi

S∗−Fi

S∗Gi

S∗+ j∈J0

v∗jHj

S∗

=ΨiS∗,S∗,v∗,

(4.8)

with strict inequality holding for at least one indexi∈p. From the properties of ¯φi, we see that for eachi∈p, ¯φi(Ψi( ¯S,S∗,u∗,v∗)−Ψi(S∗,S∗,u∗,v∗))0, which in view of (i) implies that

Ᏺ

¯

S,S∗;bS¯,S∗

Gi

S∗DFi

S∗−Fi

S∗DGi

S∗+ j∈J0

v∗_jDHj

S∗

<−ρ¯id2θS¯,S∗.

(4.9)

Inasmuch asu∗0,u∗_i =0 for eachi∈p\I+,i∈I+u∗i =1, andᏲ( ¯S,S∗;·) is sublinear,

these inequalities yield

Ᏺ

¯

S,S∗;bS¯,S∗

p

i=1 u∗_i Gi

S∗DFi

S∗−Fi

S∗DGi

S∗+ j∈J0

v∗_jDHj

S∗

<− i∈I+

u∗_iρ¯id2θS¯,S∗.

(4.10)

Now combining this inequality with (4.5) and using (iii), we obtain

Ᏺ

S,S∗;bS¯,S∗ m

t=1

j∈Jt

v∗_jDHj

S∗

> i∈I+

u∗i ρ¯id2

θS¯,S∗− m

t=1 ˜ ρtd2

θS¯,S∗,

(4.11)

contradicting (4.4), which is valid for the present case because of our hypotheses in (ii). Hence,S∗is an eﬃcient solution of (P).

(b)–(f) The proofs are similar to that of part (a).