Generalized (

(1)

GENERALIZED

(

Ᏺ

,

b

,

φ

,

ρ

,

θ

)-UNIVEX

n

-SET FUNCTIONS

AND GLOBAL PARAMETRIC SUFFICIENT OPTIMALITY

CONDITIONS IN MINMAX FRACTIONAL

SUBSET PROGRAMMING

G. J. ZALMAI

Received 28 December 2003 and in revised form 3 October 2004

We first define a new class of generalized convexn-set functions, called (Ᏺ,b,φ,ρ,θ

)-univex functions, and then establish a fairly large number of global parametric suﬃcient

optimality conditions under a variety of generalized (Ᏺ,b,φ,ρ,θ)-univexity assumptions for a discrete minmax fractional subset programming problem.

1. Introduction

In this paper, we will present a number of global parametric suﬃcient optimality con-ditions under various generalized (Ᏺ,b,φ,ρ,θ)-univexity hypotheses for the following discrete minmax fractional subset programming problem:

Minimize max 1≤i≤p

Fi(S)

Gi(S) subject to

Hj(S)≤0, j∈q,S∈An, (1.1)

whereAn _{is the} _n_{-fold product of the}_σ_-algebra _A_{of subsets of a given set}_X_,_F i,Gi,

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.

Optimization problems of this type in which the functionsFi,Gi,i∈p, andHj,j∈q, are defined on a subset ofRn₍_n_{-dimensional Euclidean space) are called}_generalized

frac-tional programming problems. These problems have arisen in multiobjective

program-ming [1], approximation theory [2,3,20,34], goal programming [8,19], and economics [33].

The notion of duality for a generalized linear fractional programming problem with point functions was originally considered by von Neumann [33] in the context of an economic equilibrium problem. More recently, various optimality conditions, duality re-sults, and computational algorithms for several classes of generalized fractional programs with point functions have appeared in the related literature. A fairly extensive list of ref-erences pertaining to diﬀerent aspects of these problems is given in [40].

In the area of subset programming problems, minmax fractional programs like (1.1) were first discussed in [37,38]. In [37], necessary and suﬃcient optimality conditions and several duality results were established under generalizedρ-convexity assumptions.

(2)

This was accomplished by combining the necessary optimality conditions of [9] for a nonlinear program involving differentiablen-set functions, which are then-set versions of the seminal results of Morris [28], with a Dinkelbach-type parametric approach [11]. Subsequently, a Lagrangian-type dual problem was constructed for (1.1) in [38] via a Gordan-type theorem of the alternative, and appropriate duality theorems were proved without imposing any differentiability requirements. Later, some results of [37] were gen-eralized in [30] by replacing the notion ofρ-convexity with (Ᏺ,ρ)-convexity, and in [7] by placing generalizedρ-convexity hypotheses on different combinations of the prob-lem functions; different derivations of the dual problem of [38] were given in [4,18]. In addition, in [18] then-set counterpart of a Lagrangian-type dual problem originally for-mulated by Xu [35] was presented. Recently, parameter-free versions of the results of [37] were established in [21], some optimality and duality results for (1.1) were obtained in [6] under generalizedb-vexity assumptions, several optimality results and duality relations for (1.1) with nonsmooth generalized (Ᏺ,ρ,θ)-convex functions were discussed in [22], and a number of generalized sufficient optimality criteria and duality theorems for (1.1) were proved in [42] under various (Ᏺ,α,ρ,θ)-V-convexity hypotheses. The optimality re-sults developed here and the complementary duality rere-sults obtained in the companion paper [36] under various generalized (Ᏺ,b,φ,ρ,θ)-univexity hypotheses subsume a great variety of optimality and duality results obtained previously for several classes of subset programming problems, including those of [6,7,9,25,28,30,37,39].

For brief surveys and lists of references pertaining to various aspects of subset pro-gramming problems, including areas of applications, optimality conditions, and duality models, the reader is referred to [21,30,32,39].

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 sufficiency criteria for (1.1) inSection 3where we state and prove a number of suffi -ciency results. More general sets of sufficiency conditions are formulated and discussed inSection 4 with the help of two partitioning schemes. The first of these schemes was originally used in [27] for constructing generalized dual problems for nonlinear pro-grams with point functions, whereas the second appears to be new and leads to a number of different sufficiency criteria for generalized fractional programming problems.

Evidently, all these optimality results are also applicable, when appropriately special-ized, to the following three classes of problems with discrete max, fractional, and conven-tional objective functions, which are particular cases of (1.1):

Minimize

S∈F 1max≤i≤pFi(S),

Minimize S∈F

F1(S)

G1(S) ,

Minimize S∈F F1(S),

(1.2)

whereF(assumed to be nonempty) is the feasible set of (1.1), that is,

F=S∈An_:_H

j(S)≤0, j∈q

(3)

Since in most cases, the optimality results established for (1.1) 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

µ2RiSi

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 integral

Th dµ will be denoted byh,χT.

We next define the notions of diﬀerentiability and convexity forn-set functions. They were originally introduced by Morris [28] for set functions, and subsequently extended by Corley [9] forn-set functions.

Definition 2.1. A functionF:A→Ris said to bediﬀerentiableatS∗if there existsDF(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∗) is o(d(S,S∗)), that is, limd(S,S∗₎_→₀V_F(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

DiGS∗,χSi−χS∗i

+WGS,S∗, (2.3)

whereWG(S,S∗) is o(d(S,S∗)) for allS∈An.

It was shown by Morris [28] 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)

wherew→∗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)}inAnsuch that

lim sup k→∞

FVk(λ)

(<)≤λF(S) + (1−λ)F(T). (2.6)

It was shown in [9,28] 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 [28] and its extension forn-set functions by Corley [9], various generalizations of convexity for set andn-set functions were proposed in [6,22,23,24,30,31,36,37,41,42]. More specifically, quasiconvexity and pseudoconvexity for set functions were defined in [23], and forn-set functions in [24]; generalizedρ-convexity forn-set functions was defined in [37], (Ᏺ,ρ)-convexity in [30],b-vexity in [6], (ρ,b)-vexity in [31], (Ᏺ,ρ,θ)-convexity for nondiﬀerentiable set functions in [22], and (Ᏺ,α,ρ,θ)-V-convexity in [41,42]. 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 [29].

For the purpose of formulating and proving various collections of suﬃciency crite-ria for (1.1), 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 gen-eralized convex functions. Its main ingredients areᏲ-convex functions and univex func-tions, which were introduced in [15] and [5], respectively. These functions were proposed as generalizations of the class of invex functions.

(5)

Definition 2.5(see [14]). 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)−f(x∗)≥ ∇f(x∗)Tη(x,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 definitions, 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 [14] who showed that for a nonlinear programming problem of the form

Minimize f(x) subject to gi(x)≤0, i∈m,x∈Rn, (2.10)

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 [10] 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 [12,13,16, 17,26,29]. 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_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,∞).

The functionᏲis said to besuperlinearif the conditions specified in the above defini-tion hold with the inequality reversed, that is, with≤replaced by≥.

Now combining the definitions ofᏲ-convex and (ρ,η)-invex functions given in [15, 16], respectively, we can define (Ᏺ,ρ)-convex, (Ᏺ,ρ)-pseudoconvex, and (Ᏺ,ρ )-quasicon-vex 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₎_. _(2.12)

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_. _(2.13)

Evidently, if in Definitions2.7,2.8, and2.9we chooseᏲ(x,y;∇g(y))= ∇g(y)T_η₍_x_,_y_), whereη: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 [17,29].

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 [5]. 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(see [5]). The functionhis said to beunivexatywith respect toη,Φ, and

bif for eachx∈S,

b(x,y)Φh(x)−h(y)≥ ∇h(y)Tη(x,y). (2.14)

Definition 2.11(see [5]). 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. (2.15)

Definition 2.12(see [5]). The functionhis said to bequasiunivexatywith respect toη,

Φ, andbif for eachx∈S,

Φh(x)−h(y)≤0=⇒b(x,y)∇h(y)T_η₍_x_,_y₎_≤₀_. _(2.16)

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, 2.6,2.7,2.8,2.9,2.10,2.11, and2.12.

LetS,S∗∈An_{, and let the function}_F_:_An_→_R_{be di}_ﬀ_{erentiable at}_S∗_.

(7)

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∗_. _(2.17)

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

if there exist a sublinear functionᏲ(S,S∗;·) :Ln

1(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∗≥ −ρd2_θ_S_,_S∗_=⇒_φ_F₍_S₎₋_F_S∗₍_>₎_≥₀_. _(2.18)

Definition 2.15. The functionFis said to be (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 function}

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

φF(S)−FS∗≤0=⇒ᏲS,S∗;bS,S∗DFS∗≤ −ρd2θS,S∗. (2.19)

Definition 2.16. The functionF is said to beprestrictly (Ᏺ,b,φ,ρ,θ)-quasiunivex atS∗

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∗_,

φF(S)−FS∗<0=⇒ᏲS,S∗;bS,S∗DFS∗≤ −ρd2_θ_S_,_S∗_. _(2.20)

From the above definitions, it is clear that ifF is (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_. _(2.21)

Needless to say, the new classes of generalized convexn-set functions specified in Def-initions2.13,2.14,2.15, and2.16contain 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.

(8)

Theorem2.17 (see [37]). Assume thatFi,Gi,i∈p, andHj, j∈q, are diﬀerentiable at

S∗∈An_{, and there exists}_S_ˆ_∈_An_{such that}

Hj

S∗+

n

k=1

DkHj

S∗,χ_Sˆ_k−χS∗k

<0, j∈q. (2.22)

IfS∗is an optimal solution of (1.1), then there existu∗∈U,v∗∈Rq+, andλ∗∈Rsuch that

p

i=1

u∗_i DkFi

S∗−λ∗DkGi

S∗+

q

j=1

v∗_jDkHj

S∗,χSk−χS∗k

≥0 ∀Sk∈A,k∈n,

u∗_iFiS∗−λ∗GiS∗=0, i∈p,

v∗jHj

S∗=0, j∈q,

(2.23)

whereU= {u∈R+p:ip=1ui=1}andR+pdenotes the nonnegative orthant ofRp.

We will also need the following result which provides an alternative expression for the objective function of (1.1).

Lemma2.18 (see [37]). For eachS∈An_,

ϕ(S)≡max 1≤i≤p

Fi(S)

Gi(S)=maxu∈U

p

i=1uiFi(S)

p

i=1uiGi(S)

. (2.24)

3. Suﬃcient optimality conditions

In this section, we formulate several sets of suﬃcient optimality conditions for (1.1) with a variety of generalized (Ᏺ,b,φ,ρ,θ)-univexity assumptions. We begin by introducing some notation.

Let the functionsᏭi(·,λ∗),Ꮽ(·,u∗,λ∗), andᏮ(·,v∗) :An→Rbe defined, for fixed

λ∗,u∗, andv∗, by

Ꮽi(S,λ∗)=Fi(S)−λ∗Gi(S), i∈p,

Ꮽ(S,u∗,λ∗)=

p

i=1

u∗_iFi(S)−λ∗Gi(S)

,

Ꮾ(S,v∗)=

q

j=1

v∗_jHj(S).

(3.1)

For givenu∗∈Uandv∗∈R+q, letI+(u∗)= {i∈p:u∗i >0}andJ+(v∗)= {j∈q:v∗j > 0}.

Theorem3.1. LetS∗∈FwithFi(S∗)≥0,i∈p, letλ∗=ϕ(S∗), and assume thatFi,Gi,

(9)

such that

Ᏺ

S,S∗;

p

i=1

u∗_iDFi

S∗−λ∗DGi

S∗+

q

j=1

v∗_jDHj

S∗

≥0 ∀S∈F, (3.2)

u∗i

Fi

S∗−λ∗Gi

S∗=0, i∈p, (3.3)

v∗_jHj

S∗=0, j∈q, (3.4)

whereᏲ(S,S∗;·) :Ln1(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⇒a≥0;

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

increas-ing, andφj(0)=0;

(iii)ρ∗+j∈J+v∗jρj≥0, whereρ∗=

p

i=1u∗i( ¯ρi+λ∗ρˆi);

(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⇒a≥0;

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

(iii)ρ∗+ρ≥0;

(c) (i)the Lagrangian-type functionL(·,u∗,v∗,λ∗) :An_→_R_{defined, for fixed}_u∗_,_v∗_,

andλ∗, by

L(S,u∗,v∗,λ∗)= p

i=1

u∗_iFi(S)−λ∗Gi(S)

+ q

j=1

v∗_jHj(S) (3.5)

is(Ᏺ,b, ¯φ, 0,θ)-pseudounivex atS∗, andφ¯(a)≥0⇒a≥0. ThenS∗is an optimal solution of (1.1).

Proof. (a) LetSbe an arbitrary feasible solution of (1.1). Using the hypotheses specified

in (i), we have for eachi∈p,

¯

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

φ−Gi(S) +GiS∗≥ᏲS,S∗;−bS,S∗DGiS∗+ ˆρid2θS,S∗.

(3.6)

Inasmuch asλ∗≥0,u∗≥0,_ip₌1u∗i =1, ¯φis superlinear, andᏲ(S,S∗;·) is sublinear, we deduce from the above inequalities that

¯

φ

p

i=1

u∗i

Fi(S)−λ∗Gi(S)

−

p

i=1

u∗i

Fi

S∗−λ∗Gi

S∗

≥Ᏺ

S,S∗;bS,S∗

p

i=1

u∗_i DFi

S∗−λ∗DGi

S∗ + p

i=1

u∗_iρ¯i+λ∗ρˆi

d2_θ_S_,_S∗_.

(10)

SinceS∈F, it follows from (3.4) that for each j∈J+,Hj(S)≤0=Hj(S∗), and so using the properties ofφj, we get for eachj∈J+,

φj

Hj(S)−Hj

S∗≤0, (3.8)

which in view of (ii) implies that

ᏲS,S∗;bS,S∗DHjS∗≤ −ρjd2θS,S∗. (3.9)

Becausev∗≥0,v∗j =0 for eachj∈q\J+, andᏲ(S,S∗;·) is sublinear, the above inequal-ities yield

Ᏺ

S,S∗;bS,S∗

q

j=1

v∗_jDHj

S∗

≤ −

j∈J+

v∗_jρjd2

θS,S∗. (3.10)

From the positivity ofb(S,S∗), sublinearity ofᏲ(S,S∗;·), and (3.2) it is clear that

Ᏺ

S,S∗;bS,S∗

p

i=1

u∗_iDFiS∗−λ∗DGiS∗

+Ᏺ

S,S∗;bS,S∗

q

j=1

v∗_jDHjS∗

≥0.

(3.11) Combining (3.7), (3.10), and (3.11) and using (3.3) and (iii), we obtain

¯

φ

p

i=1

u∗_i Fi(S)−λ∗Gi(S)

≥

ρ∗+

j∈J+

v∗_jρj

d2_θ_S_,_S∗_≥₀_. _(3.12)

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

p

i=1

u∗_i Fi(S)−λ∗Gi(S)≥0. (3.13)

Now usingLemma 2.18and (3.13), we see that

ϕ(S)=max 1≤i≤p

Fi(S)

Gi(S)=maxu∈U

p

i=1uiFi(S)

p

i=1uiGi(S)

(byLemma 2.18)

≥

p

i=1u∗iFi(S)

p

i=1u∗i Gi(S)

≥λ∗ by (3.13).

(3.14)

Sinceλ∗=ϕ(S∗) andS∈Fwas arbitrary, we conclude from the above inequality thatS∗

is an optimal solution of (1.1).

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

(c) Sinceb(S,S∗)>0,Ᏺ(S,S∗;·) is sublinear, andL(·,u∗,v∗,λ∗) is (Ᏺ,b, ¯φ, 0,θ )-pseu-dounivex atS∗, it follows from (3.2) that ¯φ(L(S,u∗,v∗,λ∗)−L(S∗,u∗,v∗,λ∗))≥0. But

¯

(11)

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∗∈F, letλ∗=ϕ(S∗), and assume thatFi,Gi,i∈p, andHj, j∈q,

are diﬀerentiable atS∗, and that there existu∗∈Uandv∗∈Rq+such that (3.2), (3.3), and

(3.4) hold. Assume, furthermore, that any of the following six sets of hypotheses is satisfied:

(a) (i)Ꮽ(·,u∗,λ∗)is(Ᏺ,b, ¯φ, ¯ρ,θ)-pseudounivex atS∗, andφ¯(a)≥0⇒a≥0;

increas-ing, andφj(0)=0; (iii) ¯ρ+j∈J+v

∗ jρj≥0;

(b) (i)Ꮽ(·,u∗,λ∗)is(Ᏺ,b, ¯φ, ¯ρ,θ)-pseudounivex atS∗, andφ¯(a)≥0⇒a≥0;

(iii) ¯ρ+ρ≥0;

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

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

φj(0)=0; (iii) ¯ρ+j∈J+v

∗ jρj>0;

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

(iii) ¯ρ+ρ > 0;

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

(ii)for each j∈J+,Hj is strictly(Ᏺ,b,φj,ρj,θ)-pseudounivex atS∗,φjis

increas-ing, andφj(0)=0; (iii) ¯ρ+j∈J+v

∗ jρj≥0;

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

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

(iii) ¯ρ+ρ≥0.

ThenS∗is an optimal solution of (1.1).

Proof. (a) LetSbe an arbitrary feasible solution of (1.1). Then, as seen in the proof of

Theorem 3.1, our hypotheses in (ii) lead to (3.10), which when combined with (3.11) yields

Ᏺ

S,S∗;bS,S∗

p

i=1

≥

j∈J+

v∗_jρjd2

(12)

where the second inequality follows from (iii). By virtue of (i), (3.15) implies that

¯

φᏭS,u∗,λ∗−ᏭS∗,u∗,λ∗≥0, (3.16)

which because of the property of the function ¯φ, reduces toᏭ(S,u∗,λ∗)≥Ꮽ(S∗,u∗,λ∗). But by (3.3),Ꮽ(S∗,u∗,λ∗)=0, and hence we have thatᏭ(S,u∗,λ∗)≥0, which is pre-cisely (3.13). Therefore, we conclude, as in the proof ofTheorem 3.1, thatS∗is an optimal solution of (1.1).

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

(c) Proceeding as in the proof of part (a), we obtain the first inequality of (3.15). Thus in view of (iii) we have

Ᏺ

S,S∗;bS,S∗

p

i=1

>−ρd¯ 2_θ_S_,_S∗_, _(3.17)

which by virtue of (i) implies that

¯

φᏭS,u∗,λ∗−ᏭS∗,u∗,λ∗≥0. (3.18)

But ¯φ(a)≥0⇒a≥0, and hence we getᏭ(S,u∗,λ∗)≥Ꮽ(S∗,u∗,λ∗). From (3.3), it is clear thatᏭ(S∗,u∗,λ∗)=0 and so we have

ᏭS,u∗,λ∗=

p

i=1

u∗i

Fi

S∗−λ∗Gi(S∗

≥0. (3.19)

Now proceeding as in the proof ofTheorem 3.1and usingLemma 2.18along with this inequality, we find thatϕ(S)≥λ∗=ϕ(S∗), showing thatS∗ is an optimal solution of (1.1).

(d)–(f) The proofs are similar to that of part (c).

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

increasing, andφ¯i(0)=0;

increas-ing, andφj(0)=0; (iii)ρ◦+j∈J+v

∗

jρj≥0, whereρ◦=

i∈I+u

∗ iρ¯i;

(b) (i)for eachi∈I+,Ꮽi(·,λ∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atS∗,φ¯iis increasing,

andφ¯i(0)=0;

(iii)ρ◦+ρ≥0;

(c) (i)for eachi∈I+,Ꮽi(·,λ∗)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗,φ¯i is

increasing, andφ¯(0)=0;

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

φj(0)=0; (iii)ρ◦+j∈J+v

(13)

(d) (i)for eachi∈I+,Ꮽi(·,λ∗)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗,φ¯i is

(iii)ρ◦+ρ > 0;

(e) (i)for eachi∈I+,Ꮽi(·,λ∗)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗,φ¯i is

(ii)for each j∈J+,Hjis strictly(Ᏺ,b,φj,ρj,θ)-pseudounivex atS∗,φjis

∗ jρj≥0;

(f) (i)for eachi∈I+,Ꮽi(·,λ∗)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗,φ¯i is

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

(iii)ρ◦+ρ≥0;

Proof. (a) Suppose to the contrary thatS∗is not an optimal solution of (1.1). Then there

is a feasible solution ¯Sof (1.1) such thatϕ( ¯S)< ϕ(S∗)=λ∗, and so for eachi∈p,Fi( ¯S)<

λ∗Gi( ¯S). From these inequalities and (3.3) it is clear that for eachi∈I+,

Fi

_¯

S−λ∗Gi

_¯

S<0=Fi

S∗−λ∗Gi

S∗, (3.20)

which in view of the properties of ¯φi, can be expressed as

¯

φi

Fi

_¯

S−λ∗Gi

_¯

S−Fi

S∗−λ∗Gi

S∗<0. (3.21)

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

Ᏺ_S¯_,_S∗_;_b_S_¯_,_S∗_DF

iS∗−λ∗DGiS∗<−ρ¯id2θS¯,S∗. (3.22)

Since u∗≥0,u∗_i =0 for eachi∈p\I+,

i∈I+u

∗

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

Ᏺ

¯

S,S∗;bS¯,S∗

p

i=1

u∗_iDFi

S∗−λ∗DGi

S∗

<−

i∈I+

u∗_iρ¯id2

θS¯,S∗. (3.23)

From (3.11), (3.23), and (iii), it is clear that

Ᏺ

¯

S,S∗;bS¯,S∗

q

j=1

v∗_jDHj

S∗

>

i∈I+

u∗_iρ¯id2

θS¯,S∗≥ −

j∈J+

v∗_jρjd2

θS¯,S∗.

(3.24) But this contradicts (3.10) (withS replaced by ¯S), which is valid for the present case because of the assumptions specified in (ii). Hence we conclude thatS∗is an optimal solution of (1.1).

(14)

(c) Suppose to the contrary thatS∗is not an optimal solution of (1.1). Now proceeding as in the proof of part (a), we obtain, for some ¯S∈F, the inequality

Ᏺ

¯

S,S∗;bS¯,S∗

p

i=1

u∗_iDFi

S∗−λ∗DGi

S∗

≤ −ρ◦d2_θ_S_¯_,_S∗_, _(3.25)

which when combined with (3.11) (withSreplaced by ¯S) gives

Ᏺ

¯

S,S∗;bS,S∗

q

j=1

v∗_jDHjS∗

≥ρ◦d2_θ_S_¯_,_S∗_>₋ j∈J+

v∗_jρjd2θS¯,S∗, (3.26)

where the strict inequality follows from (iii). This obviously contradicts (3.10) (withS

replaced by ¯S), which is valid for the present case because of our assumptions specified in (ii), and so we conclude thatS∗is an optimal solution of (1.1).

(d)–(f) The proofs are similar to that of part (c).

InTheorem 3.3, it was required that eachᏭi(·,λ∗), i∈I+, possess the same gener-alized (Ᏺ,b,φi,ρi,θ)-univexity property, namely, (Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivexity in parts (a) and (b), and prestrict (Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivexity in parts (c)–(f). It is also pos-sible to partitionI+into disjoint subsets and then assume that different collections of theᏭi(·,λ∗)’s corresponding to these subsets have different generalized (Ᏺ,b,φi,ρi,θ )-univexity properties. Of course, similar partitioning can be applied toJ+. This process can generate some additional sets of sufficiency results for (1.1). We next formulate a variant ofTheorem 3.3whose proof is similar to that ofTheorem 3.3, and hence omitted.

(3.4) hold. Furthermore, assume that any of the following four sets of hypotheses is satisfied:

(a) (i)for eachi∈I1+= ∅,Ꮽi(·,λ∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atS∗, for each

i∈I2+,Ꮽi(·,λ∗)is (Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex at S∗, and for each i∈I+≡

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

∗

jρj≥0, whereρ◦=i∈I+u

∗ iρ¯i;

(b) (i)for eachi∈I1+= ∅,Ꮽi(·,λ∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atS∗, for each

i∈I2+,Ꮽi(·,λ∗)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+;

(iii)ρ◦+ρ≥0;

(c) (i)for eachi∈I+,Ꮽi(·,λ∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atS∗,φ¯iis increasing,

andφ¯i(0)=0;

(ii)for eachj∈J1+,Hjis(Ᏺ,b,φj,ρj,θ)-quasiunivex atS∗, for each j∈J2+,Hjis

strictly(Ᏺ,b,φj,ρj,θ)-pseudounivex atS∗, and for eachj∈J+,φjis increasing,

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

(15)

(d) (i)for eachi∈I1+,Ꮽi(·,λ∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atS∗, for eachi∈I2+, Ꮽi(·,λ∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗, and for eachi∈I+,φ¯iis

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

(ii)for eachj∈J1+,Hjis(Ᏺ,b,φj,ρj,θ)-quasiunivex atS∗, for each j∈J2+,Hjis

strictly(Ᏺ,b,φj,ρj,θ)-pseudounivex atS∗, and for each j∈J+,φjis increasing

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

∗ jρj≥0;

(iv)I1+= ∅,J2+= ∅, orρ◦+j∈J+v

∗ jρj>0.

4. Generalized suﬃciency criteria

In this section, we formulate and discuss several families of generalized suﬃciency results for (1.1) with the help of a partitioning scheme that was originally proposed in [27] 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 functionsAi(·,v∗,λ∗),A(·,u∗,v∗,λ∗), andBt(·,v∗) :An→Rdefined, for fixedλ∗,u∗, andv∗, by

Ai

T,v∗,λ∗=Fi(T)−λ∗Gi(T) +

j∈J0

v∗_jHj(T), i∈p,

AT,u∗,v∗,λ∗=

p

i=1

u∗i

Fi(T)−λ∗Gi(T)

+

j∈J0

v∗jHj(T),

Bt

T,v∗=

j∈Jt

v∗_jHj(T), t∈m.

(4.1)

Using these sets and functions, we next state and prove a number of generalized suﬃ -ciency results for (1.1).

(3.4) hold. Assume, furthermore, that any of the following four sets of hypotheses is satisfied:

(a) (i)A(·,u∗,v∗,λ∗)is(Ᏺ,b, ¯φ, ¯ρ,θ)-pseudounivex atS∗, andφ¯(a)≥0⇒a≥0;

(ii)for eacht∈m,Bt(·,v∗)is(Ᏺ,b,φt,ρt,θ)-quasiunivex atS∗,φt is increasing,

andφt(0)=0; (iii) ¯ρ+m_t₌1ρt≥0;

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

(ii)for eacht∈m,Bt(·,v∗)is strictly(Ᏺ,b,φt,ρt,θ)-pseudounivex atS∗,φtis

in-creasing, andφt(0)=0; (iii) ¯ρ+m_t₌1ρt≥0;

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

(16)

andφt(0)=0; (iii) ¯ρ+m_t₌1ρt>0;

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

(ii)for each t∈m1, Bt(·,v∗) is(Ᏺ,b,φt,ρt,θ)-quasiunivex at S∗, φt is

increas-ing, andφt(0)=0, for eacht∈m2= ∅,Bt(·,v∗)is strictly(Ᏺ,b,φt,ρt,θ)

-pseudounivex atS∗,φtis increasing, andφt(0)=0, where{m1,m2}is a

parti-tion ofm;

(iii) ¯ρ+mt=1ρt≥0.

Proof. (a) LetSbe an arbitrary feasible solution of (1.1). Asv∗≥0, it is clear from (3.4) that for eacht∈m,

Bt

S,v∗=

j∈Jt

v∗_jHj(S)≤0=

j∈Jt

v∗_jHj

S∗=Bt

S∗,v∗, (4.2)

and hence using the properties ofφt, we get

φt

Bt

S,v∗−Bt

S∗,v∗≤0, (4.3)

which by (ii) implies that for eacht∈m,

Ᏺ

S,S∗;bS,S∗

j∈Jt

v∗_jDHj

S∗

≤ −ρtd2

θ(S,S∗. (4.4)

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∗. (4.5)

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

Ᏺ

S,S∗;bS,S∗

p

i=1

u∗_i DFi

S∗−λ∗DGi

S∗+ j∈J0

v∗_jDHj

S∗

+Ᏺ

S,S∗;bS,S∗

m

t=1

j∈Jt

v∗_jDHjS∗

≥0.

(4.6)

Combining (4.5) and (4.6) and using (iii), we obtain the inequality

Ᏺ

S,S∗;bS,S∗

p

i=1

u∗_i DFiS∗−λ∗DGiS∗+

j∈J0

v∗_jDHjS∗

≥

m

t=1

ρtd2θ(S,S∗≥ −ρd¯ 2θ(S,S∗,

(17)

which in view of (i) implies that

¯

φAS,u∗,v∗,λ∗−AS∗,u∗,v∗,λ∗≥0. (4.8)

Because of the property of the function ¯φ, the above inequality yields

AS,u∗,v∗,λ∗−AS∗,u∗,v∗,λ∗≥0. (4.9)

But in view of (3.3) and (3.4),A(S∗,u∗,v∗,λ∗)=0, and so we have thatA(S,u∗,v∗,λ∗)≥ 0. Sincev∗jHj(S)≤0 for eachj∈q, this inequality reduces toip=1u∗i [Fi(S∗)−λ∗Gi(S∗)]≥ 0. Now usingLemma 2.18and this inequality, as in the proof ofTheorem 3.1, we obtain the desired conclusion thatS∗is an optimal solution of (1.1).

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

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

is increasing, andφ¯i(0)=0;

andφt(0)=0; (iii)_i_∈_I+u

∗ i ρ¯i+

m

t=1ρt≥0;

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

(ii)for eacht∈m,Bt(·,v∗)is strictly(Ᏺ,b,φt,ρt,θ)-pseudounivex atS∗,φtis

in-creasing, andφt(0)=0; (iii)i∈I+u∗i ρ¯i+

_m

t=1ρt≥0;

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

andφt(0)=0; (iii)i∈I+u

∗

i ρ¯i+mt=1ρt>0;

(d) (i)for eachi∈I1+= ∅,Ai(·,v∗,λ∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atS∗, for each

i∈I2+,Ai(·,v∗,λ∗)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atS∗, for eachi∈

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

andφt(0)=0; (iii)i∈I+u

∗ i ρ¯i+

_m

t=1ρt≥0;

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

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

is increasing, andφt(0)=0, and for each t∈m2,Bt(·,v∗)is(Ᏺ,b,φt,ρt,θ)

-quasiunivex atS∗,φtis increasing, andφt(0)=0, where{m1,m2}is a partition

ofm;

(iii)i∈I+u

∗ i ρ¯i+

m

(18)

(f) (i)for eachi∈I1+,Ai(·,v∗,λ∗)is(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atS∗,φ¯iis

increas-ing, andφ¯i(0)=0, and for eachi∈I2+,Ai(·,v∗,λ∗)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)

-quasiunivex atS∗,φ¯iis increasing, andφ¯i(0)=0, where{I1+,I2+}is a partition

ofI+;

(ii)for eacht∈m1,Bt(·,v∗)is strictly(Ᏺ,b,φt,ρt,θ)-pseudounivex atS∗,φtis

in-creasing, andφt(0)=0, and for eacht∈m2,Bt(·,v∗)is(Ᏺ,b,φt,ρt,θ)

-quasiu-nivex atS∗,φt is increasing, andφt(0)=0, where{m1,m2}is a partition of

m;

(iii)i∈I+u

∗ i ρ¯i+

_m

t=1ρt≥0; (iv)I1+= ∅,m1= ∅, or

i∈I+u

∗ iρ¯i+

m

t=1ρt>0.

Proof. (a) Suppose to the contrary thatS∗is not an optimal solution of (1.1). As seen in

the proof ofTheorem 3.3, this supposition leads to the inequalitiesFi( ¯S)−λ∗Gi( ¯S)<0,

i∈p, for some ¯S∈F. Since for eachi∈I+,

Ai

_¯

S,v∗,λ∗=Fi

_¯

S−λ∗Gi

_¯

S+ j∈J0

v∗_jHj

_¯

S

≤Fi

_¯

S−λ∗Gi

_¯

S sincev∗jHj

_¯

S≤0 for eachj∈q <0

=Fi

S∗−λ∗Gi

S∗ by (3.3)

=Fi

S∗−λ∗Gi

S∗+ j∈J0

v∗jHj

S∗ by (3.4)

=AiS∗,v∗,λ∗,

(4.10)

it follows from the properties of ¯φithat for eachi∈I+,

¯

φAiS¯,v∗,λ∗−AiS∗,v∗,λ∗<0, (4.11)

which in view of (i) implies that

Ᏺ

¯

S,S∗;bS¯,S∗

DFi

S∗−λ∗DGi

S∗+ j∈J0

v∗_jDHj

S∗

<−ρ¯id2

θS¯,S∗.

(4.12)

Inasmuch asu∗≥0,u∗_i =0 for eachi∈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

DFi

S∗−λ∗DGi

S∗+ j∈J0

v∗jDHj

S∗

<−

i∈I+

u∗i ρ¯id2

θS¯,S∗.

(19)

Now combining this inequality with (4.6) and using (iii), we obtain

Ᏺ

¯

S,S∗;bS¯,S∗

m

t=1

j∈Jt

v∗_jDHj

S∗

>

i∈I+

u∗_iρ¯id2

θS¯,S∗≥ −

m

t=1

ρtd2

θS¯,S∗,

(4.14)

contradicting (4.5) (withSreplaced by ¯S), which is valid for the present case because of our hypotheses in (ii). Hence,S∗is an optimal solution of (1.1).

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

Each of the ten sets of conditions specified in Theorems4.1and4.2can be viewed as a collection of suﬃciency results for (1.1). Their special cases can easily be identified by appropriate choices of the partitioning setsJr,r=0, 1,. . .,m. We illustrate this possibility by stating explicitly some important special cases ofTheorem 4.2(a). They are collected in the following corollary.

Corollary4.3. LetS∗∈F, letλ∗=ϕ(S∗), and assume thatFi,Gi,i∈p, andHj,j∈q,

(3.4) hold. Assume, furthermore, that any of the following five sets of hypotheses is satisfied:

(a)for eachi∈I+≡I+(u∗), the functionT→Fi(T)−λ∗Gi(T)is(Ᏺ,b, ¯φi, ¯ρi,θ)

-pseu-dounivex atS∗,φ¯iis increasing, andφ¯i(0)=0; the functionT→qj=1v∗jHj(T)is (Ᏺ,b,φ,ρ,θ)-quasiunivex atS∗,φis increasing, andφ(0)=0; andi∈I+u

∗ iρ¯i+ρ≥ 0;

(b)for eachi∈I+, the functionT→Fi(T)−λ∗Gi(T) +

q

j=1v∗jHj(T)is(Ᏺ,b, ¯φi, ¯ρi,θ)

-pseudounivex atS∗,φ¯iis increasing, andφ¯i(0)=0; andi∈I+u

∗ i ρ¯i≥0;

(c)for eachi∈I+,T→Fi(T)−λ∗Gi(T)is(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atS∗,φ¯iis

in-creasing, andφ¯i(0)=0; for eachj∈q,T→v∗jHj(T)is(Ᏺ,b,φj,ρj,θ)−quasiunivex

atS∗,φjis increasing, andφj(0)=0; andi∈I+u

∗

iρ¯i+qj=1ρj≥0;

(d)for eachi∈I+,T→Fi(T)−λ∗Gi(T)is(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atS∗,φ¯iis

in-creasing, andφ¯i(0)=0; for eacht∈m,T→

j∈Jtv ∗

jHj(T)is(Ᏺ,b,φt,ρt,θ)

-quasi-univex atS∗,φtis increasing, andφt(0)=0; and

i∈I+u

∗ iρ¯i+

_m

t=1ρt≥0; (e)for eachi∈I+,T→Fi(T)−λ∗Gi(T) +

j∈J0v

∗

jHj(T)is(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex

atS∗,φ¯iis increasing, andφ¯i(0)=0;T→

j∈J1v

∗

jHj(T)is(Ᏺ,b,φ,ρ,θ)-quasiunivex

atS∗,φis increasing, andφ(0)=0; andi∈I+u

∗

i ρ¯i+ρ≥0.

Proof. InTheorem 4.2(a), let (a)J1=q, (b) J0=q, (c)m=q and Jt= {t},t∈q, (d)

J0= ∅, and (e)Jt= ∅fort=2, 3,. . .,m.