AND PARAMETRIC DUALITY MODELS IN MINMAX
FRACTIONAL SUBSET PROGRAMMING
G. J. ZALMAI
Received 28 December 2003 and in revised form 3 October 2004
We construct three parametric duality models and establish a fairly large number of dual-ity results 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 formulate three parametric duality models and prove a variety of duality results under various generalized (Ᏺ,b,φ,ρ,θ)-univexity hypotheses for the fol-lowing discrete minmax fractional subset programming problem:
Minimize max
1≤i≤p Fi(S)
Gi(S)subject toHj(S)≤0, j∈q,S∈A
n, (P)
where An is the n-fold product of the σ-algebra Aof 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.
This problem was considered previously in the companion paper [7] where a survey of currently available optimality and duality results for minmax fractional susbset pro-gramming problems was presented, a fairly comprehensive list of references dealing with different aspects of these problems was provided, several new classes of generalized con-vexn-set functions were defined, and numerous sets of global parametric sufficient op-timality conditions under various generalized (Ᏺ,b,φ,ρ,θ)-univexity assumptions were established. In the present study, we construct some dual problems for (P) and prove ap-propriate duality theorems utilizing most of the new classes of generalizedn-set convex functions that were introduced in [7].
The rest of this paper is organized as follows. InSection 2, we recall the definitions of differentiability and certain types of generalized convexity forn-set functions which will be used frequently throughout the sequel. We begin our discussion of duality for (P) in Section 3 where we formulate a simple dual problem and prove weak, strong, and strict converse duality theorems. In Section 4, we consider another dual problem
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with a relatively flexible constraint structure that allows for a greater variety of general-ized (Ᏺ,b,φ,ρ,θ)-univexity hypotheses under which duality can be established. Finally, in Section 5, we state and discuss a general duality model which is, in fact, a family of dual problems for (P) whose members can readily be identified by appropriate choices of cer-tain sets and functions. The nonparametric counterparts of the duality results established in this paper are investigated in [5].
Evidently, all these duality results are also applicable, when appropriately specialized, to the following three classes of problems with discrete max, fractional, and conventional objective functions, which are particular cases of (P):
Minimize
S∈F 1max≤i≤pFi(S), (P1)
Minimize S∈F
F1(S)
G1(S), (P2)
Minimize
S∈F F1(S), (P3)
whereF(assumed to be nonempty) is the feasible set of (P), that is,
F=S∈An:H
j(S)≤0, j∈q
. (1.1)
Since, in most cases, the duality 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 few basic definitions and auxil-iary results which will be used often throughout the paper.
Let (X,A,µ) be a finite atomless measure space withL1(X,A,µ) separable, and letdbe
the pseudometric onAndefined by
d(R,S)=
n
i=1
µ2R
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 integralThdµ will be denoted byh,χT.
We next define the notion of differentiability forn-set functions. It was originally in-troduced by Morris [3] for a set function, and subsequently extended by Corley [1] for n-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)=F(S∗) + DF(S∗),χS−χS∗+VF(S,S∗), (2.2)
Definition 2.2. A functionG:An→Ris said to have apartial derivativeatS∗=(S∗
1,...,S∗n) ∈Anwith respect to itsith argument if the functionF(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→Ris said to bedifferentiableatS∗ if all the partial derivativesDiG(S∗),i∈n, exist and
G(S)=G(S∗) + n
i=1
DiG(S∗),χSi−χS∗i
+WG(S,S∗), (2.3)
whereWG(S,S∗) iso(d(S,S∗)) for allS∈An.
We next recall the definitions of the generalized (Ᏺ,b,φ,ρ,θ)-univexn-set functions which will be used in the statements of our duality theorems. For more information about these and a number of other related classes ofn-set functions, the reader is referred to [7]. We begin by defining asublinear functionwhich is an integral part of all the subsequent definitions.
Definition 2.4. A functionᏲ:Rn→Ris said to besublinear (superlinear)ifᏲ(x+y)≤ (≥)Ᏺ(x) +Ᏺ(y) for allx,y∈Rn, andᏲ(ax)=aᏲ(x) for allx∈Rnanda∈R+≡[0,∞).
LetS,S∗∈An, and assume that the functionF:An→Ris differentiable atS∗.
Definition 2.5. The functionF is said to be(strictly)(Ᏺ,b,φ,ρ,θ)-univex atS∗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×Ansuch thatS=S∗⇒θ(S,S∗)=(0, 0), a functionφ:R→R, and a real numberρsuch that for eachS∈An,
φF(S)−F(S∗)(>)≥ᏲS,S∗;b(S,S∗)DF(S∗)+ρd2θ(S,S∗). (2.4)
Definition 2.6. The functionFis said to be(strictly)(Ᏺ,b,φ,ρ,θ)-pseudounivexatS∗if there exist a sublinear functionᏲ(S,S∗;·) :Ln1(X,A,µ)→R, a function b:An×An→
Rwith positive values, a functionθ:An×An→An×Ansuch thatS=S∗⇒θ(S,S∗)= (0, 0), a functionφ:R→R, and a real numberρsuch that for eachS∈An(S=S∗),
ᏲS,S∗;b(S,S∗)DF(S∗)≥ −ρd2θ(S,S∗)=⇒φF(S)−F(S∗)(>)≥0. (2.5)
Definition 2.7. 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×Ansuch thatS=S∗⇒θ(S,S∗)=(0, 0), a function φ:R→R, and a real numberρsuch that for eachS∈An,
Definition 2.8. The functionF is said to beprestrictly(Ᏺ,b,φ,ρ,θ)-quasiunivexatS∗if there exist a sublinear functionᏲ(S,S∗;·) :Ln1(X,A,µ)→R, a function b:An×An→
Rwith positive values, a functionθ:An×An→An×Ansuch thatS=S∗⇒θ(S,S∗)= (0, 0), a functionφ:R→R, and a real numberρsuch that for eachS∈An,S=S∗,
φF(S)−F(S∗)<0=⇒ᏲS,S∗;b(S,S∗)DF(S∗)≤ −ρd2θ(S,S∗). (2.7)
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 duality 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∗;b(S,S∗)DF(S∗)>−ρd2θ(S,S∗)=⇒φF(S)−F(S∗)>0. (2.8)
Needless to say that the new classes of generalized convex n-set functions specified in Definitions2.5–2.8contain a variety of special cases; in particular, they subsume all the previously defined types of generalizedn-set functions. This can easily be seen by appro-priate choices ofᏲ,b,φ,ρ, andθ.
We next recall a set of parametric necessary optimality conditions which will be needed for proving strong and strict converse duality theorems for (P).
Theorem2.9 [6]. Assume thatFi,Gi,i∈p, andHj, j∈q, are differentiable atS∗∈An, and there existsSˆ∈Ansuch that
Hj(S∗) + n
k=1
DkHj(S∗),χSˆk−χS∗ k
<0, j∈q. (2.9)
If S∗is an optimal solution of (P), 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,
(2.10)
u∗iFi(S∗)−λ∗Gi(S∗)=0, i∈p, (2.11)
v∗jHj(S∗)=0, j∈q, (2.12)
whereU= {u∈R+p:
p
i=1ui=1}andR+pdenotes the nonnegative orthant ofRp.
For brevity, we will henceforth refer to anS∗∈Fsatisfying (2.9) as aregularfeasible solution of (P).
Lemma2.10 [6]. For eachS∈An,
ϕ(S)≡max
1≤i≤p Fi(S) Gi(S)=maxu∈U
p
i=1uiFi(S)
p
i=1uiGi(S)
. (2.13)
3. Duality model I
In this section, we discuss a duality model for (P) with a somewhat restricted constraint structure that allows only certain types of generalized (Ᏺ,b,φ,ρ,θ)-univexity conditions for establishing duality. More general duality models will be presented in Sections4and5. In the remainder of this paper, we assume that the functionsFi,Gi,i∈p, andHj,j∈q, are differentiable onAn.
Let the functionsᏭi(·,λ), Ꮽ(·,u,λ), and Ꮾ(·,v) :An→Rbe defined, for fixed λ,u, andv, by
Ꮽi(S,λ)=Fi(S)−λGi(S), i∈p,
Ꮽ(S,u,λ)= p
i=1
ui[Fi(S)−λGi(S)],
Ꮾ(S,v)= q
j=1
vjHj(S).
(3.1)
For givenu∈Uandv∈R+q, letI+(u)= {i∈p:ui>0}andJ+(v)= {j∈q:vj>0}. Consider the following problem:
Maximizeλ (DI)
subject to
Ᏺ
S,T; p
i=1
uiDFi(T)−λDGi(T)+ q
j=1
vjDHj(T)
≥0 ∀S∈An, (3.2)
p
i=1
ui
Fi(T)−λGi(T)
≥0, (3.3)
q
j=1
vjHj(T)≥0, (3.4)
T∈An, u∈U, v∈Rq
+, λ∈R+, (3.5)
whereᏲ(S,T;·) :Ln
1(X,A,µ)→Ris a sublinear function.
The following two theorems show that (DI) is a dual problem for (P).
Theorem3.1 (weak duality). LetSand(T,u,v,λ)be arbitrary feasible solutions of (P) and (DI), respectively, and assume that any of the following three sets of hypotheses is satisfied:
(ii)for eachj∈q,Hjis(Ᏺ,b, ˜φj, ˜ρj,θ)-univex atT,φ˜jis increasing, andφ˜j(0)= 0;
(iii)ρ∗+j∈J+vjρ˜j≥0, whereρ∗=i=p1ui( ¯ρi+λρˆi);
(b) (i)for each i∈p,Fi is (Ᏺ,b, ¯φ, ¯ρi,θ)-univex at T, and −Gi is (Ᏺ,b, ¯φ, ˆρi,θ) -univex atT,φ¯is superlinear, andφ(a)¯ ≥0⇒a≥0;
(ii)Ꮾ(·,v)is(Ᏺ,b, ˜φ, ˜ρ,θ)-quasiunivex atT,φ˜is increasing, andφ(0)˜ =0; (iii)ρ∗+ ˜ρ≥0;
(c) (i)T→i=p1ui[Fi(T)−λGi(T,u)] +qj=1vjHj(T)is(Ᏺ,b, ¯φ, 0,θ)-pseudounivex atT, andφ(a)¯ ≥0⇒a≥0.
Thenϕ(S)≥λ.
Proof. (a) From (i) and (ii), it follows that
¯
φFi(S)−Fi(T)
≥ᏲS,T;b(S,T)DFi(T)
+ ¯ρid2
θ(S,T), i∈p, (3.6) ¯
φ−Gi(S) +Gi(T)
≥ᏲS,T;−b(S,T)DGi(T)
+ ˆρid2
θ(S,T), i∈p, (3.7) ˜
φHj(S)−Hj(T)
≥ᏲS,T;b(S,T)DHj(T)
+ ˜ρjd2
θ(S,T), j∈q. (3.8)
Multiplying (3.6) byuiand (3.7) byλui,i∈p, adding the resulting inequalities, and then using the superlinearity of ¯φand sublinearity ofᏲ(S,T;·), we obtain
¯ φ p i=1 ui
Fi(S)−λGi(S)
− p i=1 ui
Fi(T)−λGi(T)
≥Ᏺ S,T;b(S,T) p i=1 ui
DFi(T)−λDGi(T)
+ p
i=1
uiρ¯i+λρˆid2θ(S,T).
(3.9)
Likewise, from (3.8), we deduce that
˜ φ
q
j=1
vj
Hj(S)−Hj(T)
≥Ᏺ
S,T;b(S,T)q
j=1
vjDHj(T)
+q
j=1
vjρ˜jd2
θ(S,T).
(3.10)
Sincev≥0,S∈F, and (3.4) holds, it is clear that
q
j=1
vj
Hj(S)−Hj(T)
≤0, (3.11)
which implies, in view of the properties of ˜φ, that the left-hand side of (3.10) is less than or equal to zero, that is,
0≥Ᏺ
S,T;b(S,T)q
j=1
vjDHj(T)
+
q
j=1
vjρ˜jd2
From the nonnegativity ofb(S,T), sublinearity ofᏲ(S,T;·), and (3.2), it follows that
Ᏺ
S,T;b(S,T)p
i=1
ui
DFi(T)−λDGi(T)
+Ᏺ
S,T;b(S,T)q
j=1
vjDHj(T)
≥0.
(3.13)
Now adding (3.9) and (3.12), and then using (3.13) and (iii), we obtain
¯ φ
p
i=1
uiFi(S)−λGi(S)− p
i=1
uiFi(T)−λGi(T)
≥0. (3.14)
But ¯φ(a)≥0⇒a≥0, and so (3.14) yields
p
i=1
uiFi(S)−λGi(S)− p
i=1
uiFi(T)−λGi(T)≥0, (3.15)
which in view of (3.3) reduces to
p
i=1
ui
Fi(S)−λGi(S)
≥0. (3.16)
Making use ofLemma 2.10and (3.16), we obtain the desired inequality as follows:
ϕ(S)=max
1≤i≤p Fi(S) Gi(S)=maxa∈U
p
i=1aiFi(S)
p
i=1aiGi(S)
(byLemma 2.10)
≥
p
i=1uiFi(S)
p
i=1uiGi(S)
≥λ by (3.16).
(3.17)
(b) Since, for each j∈q,vjHj(S)≤0, it follows from (3.4) that
q
j=1
vjHj(S)≤0≤ q
j=1
vjHj(T), (3.18)
and so using the properties of ˜φ, we obtain
˜ φ
q
j=1
vjHj(S)− q
j=1
vjHj(T)
≤0, (3.19)
which in view of (ii) implies that
Ᏺ
S,T;b(S,T)q
j=1
vjDHj(T)
≤ −ρd˜ 2θ(S,T). (3.20)
(c) From the nonnegativity of b(S,T), sublinearity of Ᏺ(S,T;·), (Ᏺ, ¯b, ¯φ, 0, θ)-pseudounivexity assumption, and (3.2), it follows that
¯ φ
p
i=1
ui
Fi(S)−λGi(S)
+ q
j=1
vjHj(S)−
p i=1 ui
Fi(T)−λGi(T)
+ q
j=1
vjHj(T)
≥0.
(3.21)
In view of the properties of ¯φ, we deduce from this inequality that
p
i=1
ui
Fi(S)−λGi(S)
+ q
j=1
vjHj(S)−
p i=1 ui
Fi(T)−λGi(T)
+ q
j=1
vjHj(T)
≥0,
(3.22)
which, because of (3.3), (3.4), primal feasibility ofS, and nonnegativity ofv, reduces to (3.16), and so the rest of the proof is identical to that of part (a).
Theorem 3.2 (strong duality). LetS∗ be a regular optimal solution of (P), letᏲ(S,S∗; DF(S∗))=n
k=1DkF(S∗),χSk−χS∗k for any differentiable functionF:An→RandS∈ An, and assume that any of the three sets of hypotheses specified inTheorem 3.1holds for all feasible solutions of (DI). Then there exist u∗∈U,v∗∈Rq+, andλ∗∈R+such that
(S∗,u∗,v∗,λ∗)is an optimal solution of (DI) andϕ(S∗)=λ∗.
Proof. ByTheorem 2.9, there existu∗,v∗, andλ∗(=ϕ(S∗)), as specified above, such that (S∗,u∗,v∗,λ∗) is a feasible solution of (DI). Sinceϕ(S∗)=λ∗, optimality of (S∗,u∗,v∗, λ∗) for (DI) follows fromTheorem 3.1.
We also have the following converse duality result for (P)–(DI).
Theorem3.3 (strict converse duality). Let S∗ andᏲ(S,S∗;·)be as inTheorem 3.2, let ( ˜S, ˜u, ˜v, ˜λ)be an optimal solution of (DI), and assume that any of the following three sets of hypotheses is satisfied:
(a)the assumptions specified inTheorem 3.1(a) are satisfied for all feasible solutions of (DI);Fi is strictly(Ᏺ,b, ¯φ, ¯ρi,θ)-univex atS˜for at least one indexi∈p with the corresponding componentu˜iofu˜positive, andφ(a)¯ >0⇒a >0, or−Giis strictly (Ᏺ,b, ¯φ, ˆρi,θ)-univex atS˜for at least one indexi∈pwithu˜i(andλ˜) positive, and
¯
φ(a)>0⇒a >0, orHjis strictly(Ᏺ,b, ˜φ, ˜ρj,θ)-univex atS˜for at least one index j∈qwithv˜jpositive, andφ(a)˜ >0⇒a >0, ori=p1u˜i( ¯ρi+ ˜λρˆi) +qj=1v˜jρ˜j>0; (b)the assumptions specified inTheorem 3.1(b) are satisfied for all feasible solutions
of (DI), Fi andφ¯ or −Gi and φ¯ satisfy the requirements described in part (a), or the function R→qj=1v˜jHj(R)is strictly (Ᏺ,b, ˜φ, ˜ρ,θ)-pseudounivex atS˜, or
p
i=1u˜i( ¯ρi+ ˜λρˆi) + ˜ρ >0;
(c)the assumptions specified in Theorem 3.1(c) are satisfied for all feasible solutions of (DI), and the functionR→i=p1u˜i[Fi(R)−λG˜ i(R)] +qj=1v˜jHj(R)is strictly (Ᏺ,b, ¯φ, 0,θ)-pseudounivex atS˜, andφ(a)¯ >0⇒a >0.
Proof. (a) Suppose to the contrary that ˜S=S∗. FromTheorem 3.2, we know that there existu∗∈U,v∗∈Rq+, andλ∗∈R+such that (S∗,u∗,v∗,λ∗) is an optimal solution of
(DI) andϕ(S∗)=λ∗. Proceeding as in the proof of Theorem 3.1(a) (withS replaced byS∗ and (T,u,v,λ) by ( ˜S, ˜u, ˜v, ˜λ)), we arrive at the strict inequalityϕ(S∗)>λ, which˜ contradicts the fact thatϕ(S∗)=λ∗=λ. Hence, we conclude that ˜˜ S=S∗andϕ(S∗)=λ.˜
(b)-(c) The proofs are similar to that of part (a).
4. Duality model II
In this section, we consider a slightly different version of (DI) that allows for a greater variety of generalized (Ᏺ,b,φ,ρ,θ)-univexity conditions under which duality can be es-tablished. This duality model has the form
Maximizeλ (DII)
subject to
Ᏺ
S,T;p
i=1
ui
DFi(T)−λDGi(T)
+ q
j=1
vjDHj(T)
≥0 ∀S∈An, (4.1)
ui
Fi(T)−λGi(T)
≥0, i∈p, (4.2) vjHj(T)≥0, j∈q, (4.3) T∈An, u∈U, v∈Rq
+, λ∈R+, (4.4)
where Ᏺ(S,T;·) :Ln1(X,A,µ)→R is a sublinear function. We next show that (DII) is
a dual problem for (P) by establishing weak and strong duality theorems. As demon-strated below, this can be accomplished under numerous sets of generalized (Ᏺ,b,φ,ρ,θ)-univexity conditions. In the statements and proofs of our duality theorems in this section, we use the functionsᏭi(·,λ),Ꮽ(·,u,λ), andᏮ(·,v), which were defined inSection 3.
Theorem4.1 (weak duality). LetSand(T,u,v,λ)be arbitrary feasible solutions of (P) and (DII), respectively, and assume that any of the following six sets of hypotheses is satisfied:
(a) (i)Ꮽ(·,u,λ)is(Ᏺ,b, ¯φ, ¯ρ,θ)-pseudounivex atT, andφ(a)¯ ≥0⇒a≥0;
(ii)for each j∈J+≡J+(v),Hjis(Ᏺ,b, ˜φj, ˜ρj,θ)-quasiunivex atT,φ˜jis increas-ing, andφ˜j(0)=0;
(iii) ¯ρ+j∈J+vjρ˜j≥0;
(b) (i)Ꮽ(·,u,λ)is(Ᏺ,b, ¯φ, ¯ρ,θ)-pseudounivex atT, andφ(a)¯ ≥0⇒a≥0; (ii)Ꮾ(·,v)is(Ᏺ,b, ˜φ, ˜ρ,θ)-quasiunivex atT,φ˜is increasing, andφ(0)˜ =0; (iii) ¯ρ+ ˜ρ≥0;
(c) (i)Ꮽ(·,u,λ)is prestrictly(Ᏺ,b, ¯φ, ¯ρ,θ)-quasiunivex atT, andφ(a)¯ ≥0⇒a≥0; (ii)for each j∈J+,Hj is(Ᏺ,b, ˜φj, ˜ρj,θ)-quasiunivex atT,φ˜j is increasing, and
˜
φj(0)=0; (iii) ¯ρ+j∈J+vjρ˜j>0;
(e) (i)Ꮽ(·,u,λ)is prestrictly(Ᏺ,b, ¯φ, ¯ρ,θ)-quasiunivex atT, andφ(a)¯ ≥0⇒a≥0; (ii)for each j∈J+,Hjis strictly(Ᏺ,b, ˜φj, ˜ρj,θ)-pseudounivex atT,φ˜jis
increas-ing, andφ˜j(0)=0; (iii) ¯ρ+j∈J+vjρ˜j≥0;
(f) (i)Ꮽ(·,u,λ)is prestrictly(Ᏺ,b, ¯φ, ¯ρ,θ)-quasiunivex atT, andφ(a)¯ ≥0⇒a≥0; (ii)Ꮾ(·,v)is strictly(Ᏺ,b, ˜φ, ˜ρ,θ)-pseudounivex atT,φ˜is increasing, andφ(0)˜ =
0; (iii) ¯ρ+ ˜ρ≥0. Thenϕ(S)≥λ.
Proof. (a) From the primal feasibility ofSand (4.3), it is clear that for eachj∈J+,Hj(S)≤ Hj(T) and so using the properties of ˜φj, we obtain ˜φj(Hj(S)−Hj(T))≤0, which by virtue of (ii) implies that for eachj∈J+,
ᏲS,T;b(S,T)DHj(T)
≤ −ρ˜jd2
θ(S,T). (4.5)
Sincev≥0,vj=0 for each j∈q\J+, andᏲ(S,T;·) is sublinear, the above inequalities
can be combined as follows:
Ᏺ
S,T;b(S,T)q
j=1
vjDHj(T)
≤ −
j∈J+ vjρ˜jd2
θ(S,T). (4.6)
From (3.13) (which is valid for the present case becauseb(S,T)>0,Ᏺ(S,T;·) is sublinear, and (4.1) holds) and (4.6), we see that
Ᏺ
S,T;b(S,T) p
i=1
ui
DFi(T)−λDGi(T)
≥
j∈J+ vjρ˜jd2
θ(S,T)≥ −ρd¯ 2θ(S,T),
(4.7)
where the second inequality follows from (iii). In view of (i), (4.7) implies that
¯
φᏭ(S,u,λ)−Ꮽ(T,u,λ)≥0, (4.8)
which in view of the properties of ¯φreduces toᏭ(S,u,λ)−Ꮽ(T,u,λ)≥0. ButᏭ(T,u,λ) ≥0 because of (4.2) and hence we have thatᏭ(S,u,λ)≥0. This is, of course, (3.16). Therefore, the rest of the proof is identical to that ofTheorem 3.1(a).
(b) The proof is similar to that of part (a).
(c) Proceeding as in the proof of part (a), we arrive at the strict inequality
Ᏺ
S,T;b(S,T)p
i=1
uiDFi(T)−λDGi(T)
>−ρd¯ 2θ(S,T), (4.9)
which in view of (i) implies that ¯φ(Ꮽ(S,u,λ)−Ꮽ(T,u,λ))≥0. Since ¯φ(a)≥0⇒a≥0 and (4.2) holds, this inequality reduces toᏭ(S,u,λ)≥0, which leads, as seen in the proof ofTheorem 3.1, to the desired conclusion thatϕ(S)≥λ.
Theorem4.2 (weak duality). LetSand(T,u,v,λ)be arbitrary feasible solutions of (P) and (DII), respectively, and assume that any of the following six sets of hypotheses is satisfied:
(a) (i)for eachi∈I+≡I+(u),Ꮽi(·,λ)is(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atT,φ¯iis in-creasing, andφ¯i(0)=0;
(ii)for each j∈J+≡J+(v),Hjis(Ᏺ,b, ˜φj, ˜ρj,θ)-quasiunivex atT,φ˜jis increas-ing, andφ˜j(0)=0;
(iii)ρ◦+j∈J+vjρ˜j≥0, whereρ◦=i∈I+uiρ¯i;
(b) (i)for eachi∈I+,Ꮽi(·,λ)is(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atT,φ¯iis increasing, andφ¯i(0)=0;
(ii)Ꮾ(·,v)is(Ᏺ,b, ˜φ, ˜ρ,θ)-quasiunivex atT,φ˜is increasing, andφ(0)˜ =0; (iii)ρ◦+ ˜ρ≥0;
(c) (i)for each i∈I+,Ꮽi(·,λ)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex at T,φ¯i is increasing, andφ¯i(0)=0;
(ii)for each j∈J+,Hj is(Ᏺ,b, ˜φj, ˜ρj,θ)-quasiunivex atT,φ˜j is increasing, and ˜
φj(0)=0;
(iii)ρ◦+j∈J+vjρ˜j>0;
(d) (i)for each i∈I+,Ꮽi(·,λ)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex at T,φ¯i is increasing, andφ¯i(0)=0;
(ii)Ꮾ(·,v)is(Ᏺ,b, ˜φ, ˜ρ,θ)-quasiunivex atT,φ˜is increasing, andφ(0)˜ =0; (iii)ρ◦+ ˜ρ >0;
(e) (i)for each i∈I+,Ꮽi(·,λ)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex at T,φ¯i is increasing, andφ¯i(0)=0;
(ii)for each j∈J+,Hjis strictly(Ᏺ,b, ˜φj, ˜ρj,θ)-pseudounivex atT,φ˜jis increas-ing, andφ˜j(0)=0;
(iii)ρ◦+j∈J+vjρ˜j≥0;
(f) (i)for each i∈I+,Ꮽi(·,λ)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex at T,φ¯i is increasing, andφ¯i(0)=0;
(ii)Ꮾ(·,v)is strictly(Ᏺ,b, ˜φ, ˜ρ,θ)-pseudounivex atT,φ˜is increasing, andφ(0)˜ = 0;
(iii)ρ◦+ ˜ρ≥0. Thenϕ(S)≥λ.
Proof. (a) Suppose to the contrary thatϕ(S)< λ. This implies that for eachi∈p,Fi(S)− λGi(S)<0. From this and (4.2), we deduce that for eachi∈I+,
Fi(S)−λGi(S)<0≤Fi(T)−λGi(T), (4.10)
which in view of the properties of ¯φican be expressed as follows:
¯ φi
Ꮽi(S,λ)−Ꮽi(T,λ)
<0. (4.11)
By virtue of (i), these inequalities imply that for eachi∈I+,
ᏲS,T;b(S,T)DFi(T)−λDGi(T)
<−ρ¯id2
Inasmuch asu≥0,ui=0 for eachi∈p\I+,
i∈I+ui=1, andᏲ(S,T;·) is sublinear, the above inequalities yield
Ᏺ
S,T;b(S,T)p
i=1
ui
DFi(T)−λDGi(T)
<− i∈I+
uiρ¯id2
θ(S,T). (4.13)
From (3.13) (which is valid for the present case becauseb(S,T)>0,Ᏺ(S,T;·) is sublinear, and (4.1) holds), (4.13), and (iii), we infer that
Ᏺ
S,T;b(S,T)q
j=1
vjDHj(T)
> ρ◦d2θ(S,T)≥ −
j∈J+
vjρ˜jd2θ(S,T). (4.14)
But this contradicts (4.6), which is valid for the present case because of our hypotheses set forth in (ii). Hence,ϕ(S)≥λ.
(b) The proof is similar to that of part (a).
(c) Suppose to the contrary thatϕ(S)< λ. As seen in the proof of part (a), this suppo-sition leads to the inequalities
¯ φi
Ꮽi(S,λ)−Ꮽi(T,λ)
<0, i∈I+, (4.15)
which by (i) imply that for eachi∈I+,
ᏲS,T;b(S,T)DFi(T)−λDGi(T)
≤ −ρ¯id2
θ(S,T). (4.16)
Now proceeding as in the proof ofTheorem 4.1and using these inequalities and (iii), we obtain
Ᏺ
S,T;b(S,T)q
j=1
vjDHj(T)
>−
j∈J+
vjρ˜jd2θ(S,T), (4.17)
in contradiction to (4.6), which is valid for the present case because of the assumptions made in (ii). Therefore,ϕ(S)≥λ.
(d)–(f) The proofs are similar to that of part (c).
The following theorem may be viewed as a variant ofTheorem 4.2; its proof is almost identical to that ofTheorem 4.2and hence omitted.
Theorem4.3 (weak duality). LetSand(T,u,v,λ)be arbitrary feasible solutions of (P) and (DII), respectively, and 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 atT, for each i∈I2+,Ꮽi(·,λ)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atT, and for each i∈I+≡I+(u),φ¯iis increasing andφ¯i(0)=0, where{I1+,I2+}is a partition of
I+;
(ii)for each j∈J+≡J+(v),Hjis(Ᏺ,b, ˜φj, ˜ρj,θ)-quasiunivex atT,φ˜jis increas-ing, andφ˜j(0)=0;
(iii)ρ◦+j∈J+vjρ˜j≥0, whereρ◦=
(b) (i)for eachi∈I1+= ∅,Ꮽi(·,λ)is(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atT, for each i∈I2+,Ꮽi(·,λ)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atT, and for each i∈I+,φ¯iis increasing andφ¯i(0)=0, where{I1+,I2+}is a partition ofI+;
(ii)Ꮾ(·,v)is(Ᏺ,b, ˜φ, ˜ρ,θ)-quasiunivex atT,φ˜is increasing, andφ(0)˜ =0; (iii)ρ◦+ ˜ρ≥0;
(c) (i)for each i∈I+,Ꮽi(·,λ)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex at T,φ¯i is increasing, andφ¯i(0)=0;
(ii)for each j∈J1+,Hj is(Ᏺ,b, ˜φj, ˜ρj,θ)-quasiunivex atT, for each j∈J2+=
∅,Hjis strictly(Ᏺ,b, ˜φj, ˜ρj,θ)-pseudounivex atT, and for each j∈J+,φ˜jis increasing andφ˜j(0)=0, where{J1+,J2+}is a partition of J+;
(iii)ρ◦+j∈J+vjρ˜j≥0;
(d) (i)for eachi∈I1+,Ꮽi(·,λ)is(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atT, for eachi∈I2+,
Ꮽi(·,λ)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atT, and for eachi∈I+,φ¯i is increasing andφ¯i(0)=0, where{I1+,I2+}is a partition ofI+;
(ii)for eachj∈J1+,Hjis(Ᏺ,b, ˜φj, ˜ρj,θ)-quasiunivex atT, for eachj∈J2+,Hjis strictly(Ᏺ,b, ˜φj, ˜ρj,θ)-pseudounivex atT, and for eachj∈J+,φ˜jis increasing andφ˜j(0)=0, where{J1+,J2+}is a partition ofJ+;
(iii)ρ◦+j∈J+vjρ˜j≥0;
(iv)I1+= ∅,J2+= ∅, orρ◦+j∈J+vjρ˜j>0. Thenϕ(S)≥λ.
Theorem 4.4 (strong duality). LetS∗ be a regular optimal solution of (P), letᏲ(S,S∗; DF(S∗))=n
k=1DkF(S∗),χSk−χS∗k for any differentiable functionF:A
n→RandS∈ An, and assume that any of the fifteen sets of hypotheses specified in Theorems4.1–4.3holds for all feasible solutions of (DII). Then there existu∗∈U,v∗∈Rq+, andλ∗∈R+such that
(S∗,u∗,v∗,λ∗)is an optimal solution of (DII) andφ(S∗)=λ∗.
Proof. ByTheorem 2.9, there existu∗,v∗, andλ∗, as specified above, such that (S∗,u∗,v∗, λ∗) is a feasible solution of (DII) andφ(S∗)=λ∗. That (S∗,u∗,v∗,λ∗) is optimal for (DII) follows from (the corresponding part of) Theorems4.1–4.3.
Theorem4.5 (strict converse duality). Let S∗ andᏲ(S,S∗;·)be as inTheorem 4.4, let ( ˜S, ˜u, ˜v, ˜λ)be an optimal solution of (DII), and assume that either one of the two sets of hy-potheses specified inTheorem 4.1(a) and (b) is satisfied for all feasible solutions of (DII). Assume, furthermore, that Ꮽ(·, ˜u, ˜λ) is strictly(Ᏺ,b, ¯φ, ¯ρ,θ)-pseudounivex atS˜and that
¯
φ(a)>0⇒a >0. ThenS˜=S∗, that is,S˜is an optimal solution of (P), andϕ(S∗)=λ˜.
Proof. Suppose to the contrary that ˜S=S∗. FromTheorem 4.4, we know that there exist u∗∈U,v∗∈Rq+, andλ∗∈R+such that (S∗,u∗,v∗,λ∗) is an optimal solution of (DII)
andϕ(S∗)=λ∗. Proceeding as in the proof ofTheorem 4.1(a) (withSreplaced byS∗and (T,u,v,λ) by ( ˜S, ˜u, ˜v, ˜λ)), we arrive at the strict inequalityϕ(S∗)>λ, which contradicts the˜ fact thatϕ(S∗)=λ∗=λ. Hence, we conclude that ˜˜ S=S∗andϕ(S∗)=λ.˜
is(Ᏺ,b, ¯φ, ¯ρ,θ)-quasiunivex atS˜and thatφ(a)¯ >0⇒a >0. ThenS˜=S∗, that is,S˜is an op-timal solution of (P), andϕ(S∗)=λ˜.
Proof. The proof is similar to that ofTheorem 4.5.
5. Duality model III
In this section, we formulate and discuss a more general duality model for (P) with the help of a partitioning scheme that was originally proposed in [2] for constructing gen-eralized dual problems for nonlinear programs with point functions. The flexible struc-ture of this duality model will allow us to establish duality under various generalized (Ᏺ,b,φ,ρ,θ)-univexity hypotheses that can be imposed on certain combinations of the problem 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
m
r=0Jr=q. In addition, we will make use of the functionsAi(·,v,λ),A(·,u,v,λ), andBt(·,v) :An→Rdefined, for fixed λ,u, andv, by
Ai(T,v,λ)=Fi(T)−λGi(T) +
j∈J0
vjHj(T), i∈p,
A(T,u,v,λ)= p
i=1
ui
Fi(T)−λGi(T)
+ j∈J0
vjHj(T),
Bt(T,v)=
j∈Jt
vjHj(T), t∈m.
(5.1)
Consider the following problem:
Maximizeλ (DIII)
subject to
Ᏺ
S,T;p
i=1
uiDFi(T)−λDGi(T)+ q
j=1
vjDHj(T)
≥0 ∀S∈An, (5.2)
Fi(T)−λGi(T) +
j∈J0
vjHj(T)≥0, i∈p, (5.3)
j∈Jt
vjHj(T)≥0, t∈m, (5.4)
T∈An, u∈U, v∈Rq
+, λ∈R+, (5.5)
whereᏲ(S,T;·) :Ln1(X,A,µ)→Ris a sublinear function.
Theorem5.1 (weak duality). LetSand(T,u,v,λ)be arbitrary feasible solutions of (P) and (DIII), respectively, and assume that any of the following four sets of hypotheses is satisfied:
(a) (i)A(·,u,v,λ)is(Ᏺ,b, ¯φ, ¯ρ,θ)-pseudounivex atT, andφ(a)¯ ≥0⇒a≥0; (ii)for eacht∈m,Bt(·,v)is(Ᏺ,b, ˜φt, ˜ρt,θ)-quasiunivex atT,φ˜t is increasing,
andφ˜t(0)=0; (iii) ¯ρ+mt=1ρ˜t≥0;
(b) (i)A(·,u,v,λ)is prestrictly(Ᏺ,b, ¯φ, ¯ρ,θ)-quasiunivex atT, andφ(a)¯ ≥0⇒a≥ 0;
(ii)for eacht∈m,Bt(·,v)is strictly(Ᏺ,b, ˜φt, ˜ρt,θ)-pseudounivex atT,φ˜t is in-creasing, andφ˜t(0)=0;
(iii) ¯ρ+mt=1ρ˜t≥0;
(c) (i)A(·,u,v,λ)is prestrictly(Ᏺ,b, ¯φ, ¯ρ,θ)-quasiunivex atT, andφ(a)¯ ≥0⇒a≥ 0;
(ii)for eacht∈m,Bt(·,v)is(Ᏺ,b, ˜φt, ˜ρt,θ)-quasiunivex atT,φ˜t is increasing, andφ˜t(0)=0;
(iii) ¯ρ+mt=1ρ˜t>0;
(d) (i)A(·,u,v,λ)is prestrictly(Ᏺ,b, ¯φ, ¯ρ,θ)-quasiunivex atT, andφ(a)¯ ≥0⇒a≥ 0;
(ii)for each t∈m1,Bt(·,v)is (Ᏺ,b, ˜φt, ˜ρt,θ)-quasiunivex atT,φ˜t is increas-ing, and φ˜t(0)=0, for each t∈m2= ∅,Bt(·,v)is strictly (Ᏺ,b, ˜φt, ˜ρt,θ) -pseudounivex atT,φ˜tis increasing, andφ˜t(0)=0, where{m1,m2}is a
parti-tion ofm; (iii) ¯ρ+mt=1ρ˜t≥0. Thenϕ(S)≥λ.
Proof. (a) From the nonnegativity ofb(S,T), sublinearity ofᏲ(S,T;·), and (5.2), it fol-lows that
Ᏺ
S,T;b(S,T)
p
i=1
uiDFi(T)−λDGi(T)+
j∈J0
vjDHj(T)
+Ᏺ
S,T;b(S,T)m
t=1
j∈Jt
vjDHj(T)
≥0.
(5.6)
SinceS∈Fandv≥0, it is clear from (5.4) that for eacht∈m,
Bt(S,v)= j∈Jt
vjHj(S)≤
j∈Jt
vjHj(T)=Bt(T,v), (5.7)
and so using the properties of ˜φt, we get
˜ φt
Bt(S,v)−Bt(T,v)
which in view of (ii) implies that for eacht∈m,
Ᏺ
S,T;b(S,T)
j∈Jt
vjDHj(T)
≤ −ρ˜td2
θ(S,T). (5.9)
Adding these inequalities and using the sublinearity ofᏲ(S,T;·), we obtain
Ᏺ
S,T;b(S,T)m
t=1
j∈Jt
vjDHj(T)
≤ −m
t=1
˜ ρtd2
θ(S,T). (5.10)
From (5.6) and (5.10), we deduce that
Ᏺ
S,T;b(S,T)
p
i=1
ui
DFi(T)−λDGi(T)
+ j∈J0
vjDHj(T)
≥m t=1
˜ ρtd2
θ(S,T)≥ −ρd¯ 2θ(S,T),
(5.11)
where the second inequality follows from (iii). Because of (i), this inequality implies that
¯
φA(S,u,v,λ)−A(T,u,v,λ)≥0. (5.12)
But ¯φ(a)≥0⇒a≥0, and so we getA(S,u,v,λ)−A(T,u,v,λ)≥0, which in view of the nonnegativity ofuand (5.3) reduces toA(S,u,v,λ)≥0. Asv≥0 andS∈F, we have
p
i=1
uiFi(S)−λGi(S)≥A(S,u,v,λ)≥0. (5.13)
Now usingLemma 2.10and this inequality, as in the proof ofTheorem 3.1, we obtain the desired conclusion thatϕ(S)≥λ.
(b)–(d) The proofs are similar to that of part (a).
Theorem5.2 (weak duality). LetSand(T,u,v,λ)be arbitrary feasible solutions of (P) and (DIII), respectively, and assume that any of the following six sets of hypotheses is satisfied:
(a) (i)for eachi∈I+≡I+(u),Ai(·,v,λ)is(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atT,φ¯i is increasing, andφ¯i(0)=0;
(ii)for eacht∈m,Bt(·,v)is(Ᏺ,b, ˜φt, ˜ρt,θ)-quasiunivex atT,φ˜t is increasing, andφ˜t(0)=0;
(iii)i∈I+uiρ¯i+mt=1ρ˜t≥0;
(b) (i)for eachi∈I+,Ai(·,v,λ)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atT,φ¯iis increasing, andφ¯i(0)=0;
(ii)for eacht∈m,Bt(·,v)is strictly(Ᏺ,b, ˜φt, ˜ρt,θ)-pseudounivex atT,φ˜t is in-creasing, andφ˜t(0)=0;
(iii)i∈I+uiρ¯i+
m
(c) (i)for eachi∈I+,Ai(·,v,λ)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atT,φ¯iis increasing, andφ¯i(0)=0;
(ii)for eacht∈m,Bt(·,v)is(Ᏺ,b, ˜φt, ˜ρt,θ)-quasiunivex atT,φ˜t is increasing, andφ˜t(0)=0;
(iii)i∈I+uiρ¯i+
m
t=1ρ˜t>0;
(d) (i)for eachi∈I1+= ∅,Ai(·,v,λ)is(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atT, for each i∈I2+,Ai(·,v,λ)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atT, for eachi∈I+,
¯
φiis increasing, andφ¯i(0)=0, where{I1+,I2+}is a partition ofI+;
(ii)for eacht∈m,Bt(·,v)is(Ᏺ,b, ˜φt, ˜ρt,θ)-quasiunivex atT,φ˜t is increasing, andφ˜t(0)=0;
(iii)i∈I+uiρ¯i+
m
t=1ρ˜t≥0;
(e) (i)for eachi∈I+,Ai(·,v,λ)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ)-quasiunivex atT,φ¯iis increasing, andφ¯i(0)=0;
(ii)for eacht∈m1= ∅,Bt(·,v)is strictly(Ᏺ,b, ˜φt, ˜ρt,θ)-pseudounivex atT,φ˜t is increasing, andφ˜t(0)=0, and for eacht∈m2,Bt(·,v)is(Ᏺ,b, ˜φt, ˜ρt,θ) -quasiunivex atT,φ˜tis increasing, andφ˜t(0)=0, where{m1,m2}is a partition
ofm;
(iii)i∈I+uiρ¯i+mt=1ρ˜t≥0;
(f) (i)for eachi∈I1+,Ai(·,v,λ)is(Ᏺ,b, ¯φi, ¯ρi,θ)-pseudounivex atT,φ¯iis increas-ing, andφ¯i(0)=0, and for eachi∈I2+,Ai(·,v,λ)is prestrictly(Ᏺ,b, ¯φi, ¯ρi,θ) -quasiunivex atT,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 atT,φ˜t is increasing, and φ˜t(0)=0, and for each t∈m2, Bt(·,v) is (Ᏺ,b, ˜φt, ˜ρt,θ) -quasiunivex atT,φ˜t is increasing, andφ˜t(0)=0, where{m1,m2} is a
par-tition ofm; (iii)i∈I+uiρ¯i+
m
t=1ρ˜t≥0;
(iv)I1+= ∅,m1= ∅, ori∈I+uiρ¯i+mt=1ρ˜t>0. Thenϕ(S)≥λ.
Proof. (a) Suppose to the contrary that
ϕ(S)=max
1≤i≤p Fi(S)
Gi(S)< λ. (5.14)
This implies that for eachi∈p,
Fi(S)−λGi(S)<0. (5.15)
Since for eachi∈p,
Ai(S,v,λ)=Fi(S)−λGi(S) +
j∈J0 vjHj(S)
≤Fi(S)−λGi(S) (by the nonnegativity ofvand primal feasibility ofS) <0 by (5.15)
≤Ai(T,v,λ) by (5.3),
it follows from the properties of ¯φi that for eachi∈p, ¯φi(Ai(S,v,λ)−Ai(T,v,λ))<0, which by (i) implies that for eachi∈p,
Ᏺ
S,T;b(S,T)
DFi(T)−λDGi(T) +
j∈J0
vjDHj(T)
<−ρ¯id2
θ(S,T). (5.17)
Inasmuch asu≥0,ui=0 for eachi∈p\I+,
i∈I+ui=1, andᏲ(S,T;·) is sublinear, the above inequalities yield
Ᏺ
S,T;b(S,T)
p
i=1
ui
DFi(T)−λDGi(T)
+ j∈J0
vjDHj(T)
<−
i∈I+ uiρ¯id2
θ(S,T).
(5.18)
Comparing (5.6) and (5.18), we observe that
Ᏺ!S,T;b(S,T) m
t=1
j∈Jt
vjDHj(T)
"
> p
i=1
uiρ¯id2θ(S,T)>− m
t=1
˜
ρtd2θ(S,T), (5.19)
where the second inequality follows from (iii). Obviously, this inequality contradicts (5.10), which is valid for the present case because of (ii). Hence, we must haveϕ(S)≥λ.
(b)–(f) The proofs are similar to that of part (a).
Theorem 5.3 (strong duality). LetS∗ be a regular optimal solution of (P), letᏲ(S,S∗; DF(S∗))=nk=1DkF(S∗),χSk−χS∗k for any differentiable functionF:An→RandS∈ An, and assume that any of the ten sets of hypotheses specified in Theorems5.1and5.2holds for all feasible solutions of (DIII). Then there existu∗∈U,v∗∈Rq+, andλ∗∈R+such that
(S∗,u∗,v∗,λ∗)is an optimal solution of (DIII) andϕ(S∗)=λ∗.
Proof. The proof is similar to that ofTheorem 4.4.
We next show that certain modifications inTheorem 5.1lead to a number of strict converse duality results for (P)–(DIII).
Theorem5.4 (strict converse duality). LetS∗andᏲbe as inTheorem 5.3, let( ˜S, ˜u, ˜v, ˜λ) be an optimal solution of (DIII), and assume that any of the four sets of hypotheses specified inTheorem 5.1is satisfied for all feasible solutions of (DIII), and thatA(·, ˜u, ˜v, ˜λ)is strictly (Ᏺ,b, ¯φ, ¯ρ,θ)-pseudounivex atS˜, andφ(a)˜ >0⇒a >0. ThenS˜=S∗andϕ(S∗)=λ˜. Proof. We prove the theorem under the assumption that the conditions specified in Theorem 5.1(a) hold; the proofs with the conditions given in parts (b)–(d) are similar. Suppose to the contrary that ˜S=S∗. Since S∗ is a regular optimal solution of (P), by Theorem 5.3, there existu∗∈U,v∗∈Rq+, andλ∗∈R+such that (S∗,u∗,v∗,λ∗) is an
(withSreplaced byS∗and (T,u,v,λ) by ( ˜S, ˜u, ˜v, ˜λ)), we arrive at the inequality
Ᏺ
S∗, ˜S;b(S∗, ˜S)
p
i=1
˜ ui
DFi( ˜S)−λDG˜ i( ˜S)
+ j∈J0 ˜ vjHj( ˜S)
≥ −ρd¯ 2θ(S∗, ˜S).
(5.20)
Since by hypothesis, ˜S=S∗andA(·, ˜u, ˜v, ˜λ) is strictly (Ᏺ, ¯b, ¯φ, ¯ρ,θ)-pseudounivex at ˜S, the above inequality implies that
¯
φA(S∗, ˜u, ˜v, ˜λ)−A( ˜S, ˜u, ˜v, ˜λ)>0. (5.21)
But ¯φ(a)>0⇒a >0, and so the above inequality yieldsA(S∗, ˜u, ˜v, ˜λ)−A( ˜S, ˜u, ˜v, ˜λ)>0, which in view of the dual feasibility of ( ˜S, ˜u, ˜v, ˜λ) reduces toA(S∗, ˜u, ˜v, ˜λ)>0. Inasmuch as ˜
v≥0 andS∗∈F, this inequality yields
p
i=1
˜ ui
Fi(S∗)−λG˜ i(S∗)
>0. (5.22)
Now usingLemma 2.10and this inequality, we find that
ϕ(S∗)=max
1≤i≤p Fi(S∗) Gi(S∗)=maxu∈U
p
i=1uiFi(S∗)
p
i=1uiGi(S∗) ≥
p
i=1u˜iFi(S∗)
p
i=1u˜iGi(S∗)
>λ,˜ (5.23)
which contradicts the fact that ϕ(S∗)=λ∗=λ. Hence, we conclude that ˜˜ S=S∗ and
ϕ(S∗)=λ.˜
We conclude this section by briefly looking at the following slightly modified version of (DIII):
Maximizeλ (DIV)
subject to (5.2), (5.4), (5.5), and
p
i=1
uiFi(T)−λGi(T)+
j∈J0
vjHj(T)≥0. (5.24)
An examination of the statements and proofs ofTheorem 5.1,Theorem 5.3(restricted toTheorem 5.1), andTheorem 5.4will readily reveal the fact that these theorems remain valid for the pair (P)–(DIV).
This dual problem was originally proposed and investigated in [6], where some duality results were established underρ-convexity conditions. Subsequently, these results were extended by using (Ᏺ,ρ)-convexity assumptions in [4].
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G. J. Zalmai: Department of Mathematics and Computer Science, Northern Michigan University, Marquette, MI 49855, USA
