IN SEMIINFINITE FRACTIONAL PROGRAMMING.
PART III: SECOND-ORDER DUALITY MODELS
G.J. ZALMAI
Communicated by the former editorial board
In this paper, we construct and discuss several second-order parametric and nonparametric duality models for a semiinnite fractional programming problem and establish appropriate duality results under various generalized second-order (α, β, γ, η, ρ, θ)-invexity assumptions.
AMS 2010 Subject Classication: 49N15, 90C30, 90C32, 90C34, 90C46.
Key words: semiinnite programming, fractional programming, generalized sec- ond-order (α, β, γ, η, ρ, θ)-invex functions, innitely many equality and inequality constraints, second-order dual problems, duality the- orems.
1. INTRODUCTION
In this paper, we rst introduce the second-order versions of the general- ized invex functions which were recently dened in [9], and then, utilizing these new classes of functions, we state and prove numerous second-order parametric and nonparametric duality results under various generalized (α, β, γ, η, ρ, θ)- sonvexity assumptions for the following semiinnite fractional programming problem:
(P) Minimizeϕ(x) = f(x) g(x) subject to
Gj(x, t)≤0 for allt∈Tj, j ∈q, Hk(x, s) = 0 for alls∈Sk, k∈r,
x∈X,
whereqand rare positive integers,X is a nonempty open convex subset ofRn (n-dimensional Euclidean space), for each j ∈q ≡ {1,2, . . . , q} and k ∈r, Tj andSkare compact subsets of complete metric spaces,f andg are real-valued functions dened on X, for each j ∈q, z → Gj(z, t) is a real-valued function
REV. ROUMAINE MATH. PURES APPL., 58 (2013), 2, 105138
dened on X, for all t ∈ Tj, for each k ∈ r, z → Hk(z, s) is a real-valued function dened onX, for alls∈Sk, for eachj∈q andk∈r, t→Gj(x, t)and s→ Hk(x, s) are continuous real-valued functions dened, respectively, on Tj
and Sk, for all x∈X, and g(x)>0, for allx satisfying the constraints of(P). The present study is essentially a continuation of the investigation ini- tiated in the companion papers [9] and [10]. In [9] some information about fractional programming is presented, the current status of semiinnite program- ming is briey discussed and numerous key references are cited, an overview of the concept of (α, β, γ, η, ρ, θ)-invexity and some of its extensions is given, and a fairly large number of sets of global sucient optimality results under various generalized(α, β, γ, η, ρ, θ)-invexity assumptions are established.In [10]
a number of rst-order duality models are formulated and various weak, strong, and strict converse duality theorems are proved using appropriate generalized (α, β, γ, η, ρ, θ)-invexity conditions. For the necessary background material and preliminaries, the reader is referred to [9, 10]. Here, we shall make use of the optimality and duality results developed in [9, 10], and a set of necessary op- timality conditions for (P) given in [10], and construct several second-order duality models for (P) and prove appropriate duality theorems. The higher- order counterparts of these duality models and results are discussed in [11].
Second-order duality for a conventional nonlinear programming problem of the form
(P0) Minimize f(x) subject to gi(x)≤0, i∈m, x∈Rn,
where f and gi, i ∈ m, are real-valued functions dened on Rn, was initially considered by Mangasarian [4]. The idea underlying his approach to construct- ing a second-order dual problem was based on taking linear and quadratic approximations of the objective and constraint functions about an arbitrary but xed point, forming the Wolfe dual of the approximated problem, and then letting the xed point to vary. More specically, he formulated the following second-order dual problem for(P0):
(D0) Maximize f(y) +Pm
i=1uigi(y)−12D z,h
∇2f(y) +Pm
i=1ui∇2gi(y)i zE subject to
∇f(y) +
m
X
i=1
ui∇gi(y) + h
∇2f(y) +
m
X
i=1
ui∇2gi(y) i
z= 0, y∈Rn, u∈Rm, u≥0, z∈Rn,
where ∇F(y) and ∇2F(y) are, respectively, the gradient and Hessian of the functionF :Rn→Revaluated atyand ha, bidenotes the inner product of the
vectors a and b. Imposing somewhat complicated conditions onf, gi, i ∈m, andz, he proved weak, strong, and converse duality theorems for(P0)and(D0). Reconsidering Mangasarian's second-order problem, Mond [5] established some duality results under relatively simpler conditions involving a certain second- order generalization of the concept of convexity, pointed out some possible computational advantages of second-order duality results, and also studied a pair of second-order symmetric dual problems. Subsequently, Mond's original notion of second-order convexity was generalized by other authors in dierent ways and utilized for establishing various second-order duality results for several classes of nonlinear programming problems.For brief accounts of the evolution of these generalized second-order convexity concepts, the reader is referred to [1, 2, 7, 12, 13], and for a fairly extensive list of references pertaining to second- and higher-order duality results the reader may consult [8].
Presently, various second- and higher-order duality results exist in the related literature for several classes of mathematical programming problems with a nite number of constraints.However, so far no such results are available for any kind of semiinnite fractional programming problems. To the best of our knowledge, all the second-order duality results established in this paper are new in the area of semiinnite programming.
The rest of this paper is organized as follows. In Section 2, we present a few denitions and auxiliary results which will be needed in the sequel. In Section 3, we consider two second-order parametric duality models with simple con- straint structures and establish duality under an appropriate (α, β, γ, η, ρ, θ)- pseudosonvexity hypothesis imposed on a Lagrangian-type function.In Section 4, we present two duality models with relatively more exible constraint struc- tures for which duality can be proved under a greater variety of generalized (α, β, γ, η, ρ, θ)-sonvexity conditions.We continue our discussion of second-order duality in Section 5 where we use a partitioning scheme and construct two generalized second-order parametric duality models and obtain several duality results making various generalized (α, β, γ, η, ρ, θ)-sonvexity assumptions. In Section 6, we formulate two second-order nonparametric duality models and prove appropriate duality theorems. Finally, in Section 7 we summarize our main results and also point out some further research opportunities arising from certain modications of the principal problem model considered in this paper.
2. PRELIMINARIES
In this section we recall, for convenience of reference, the denitions of certain classes of generalized convex functions which will be needed in the
sequel. For a brief discussion of the origins and predecessors of these functions as well as numerous relevant references, the reader is referred to [9].
The concept of second-order convexity generalizing that of convexity, as mentioned earlier, was originally proposed by Mond [5]. Subsequently, this concept was extended to the class of invex functions by Hanson [3] who demon- strated its use in establishing some duality relations in nonlinear programming.
Following are slightly modied versions of the classes of second-order invex functions introduced in [3].
Let f be a twice dierentiable real-valued function dened onX.
Denition 2.1. The function f is said to be second-order η-invex (invex with respect to η) at x∗ if there exists a function η :X×X → Rn such that for each x∈X and z∈Rn,
f(x)−f(x∗)≥ h∇f(x∗) +∇2f(x∗)z, η(x, x∗)i −1
2hz,∇2f(x∗)zi.
Denition 2.2. The functionf is said to be second-orderη-pseudoinvex at x∗ if there exists a functionη:X×X→Rnsuch that for each x∈X(x6=x∗) and z∈Rn,
h∇f(x∗) +∇2f(x∗)z, η(x, x∗)i ≥0 ⇒f(x)=f(x∗)−1
2hz,∇2f(x∗)zi.
Denition 2.3. The function f is said to be second-orderη-quasiinvex at x∗ if there exists a function η : X×X → Rn such that for each x ∈ X and z∈Rn,
f(x)5f(x∗)−1
2hz,∇2f(x∗)zi ⇒ h∇f(x∗) +∇2f(x∗)z, η(x, x∗)i ≤0.
As pointed out in Section 1, dierent generalizations of invex functions are currently available in the literature of mathematical programming. For more information about these and other related generalized convex functions, the reader is referred to [9].
Here we consider the second-order extensions of the classes of generalized (α, β, γ, η, ρ, θ)-invex functions dened in [9]. More specically, we dene the second-order versions of the notions of(α, β, γ, η, ρ, θ)-invexity,(α, β, γ, η, ρ, θ)- pseudoinvexity, and(α, β, γ, η, ρ, θ)-quasiinvexity which will be used frequently in our duality theorems. We shall use the word sonvex for second-order invex.
Denition 2.4. The function f is said to be (strictly) (α, β, γ, η, ρ, θ)- sonvex at x∗ ∈ X if there exist functions α : X ×X → R, β : X ×X → R, γ : X ×X → R+ ≡ [0,∞), η : X ×X → Rn, ρ : X ×X → R, and θ:X×X→Rn, such that for each x∈X(x6=x∗) andz∈Rn,
1
α(x, x∗)γ(x, x∗)
eα(x,x∗)[f(x)−f(x∗)+12hz,∇2f(x∗)zi]−1 (>)≥ 1
β(x, x∗)
∇f(x∗) +∇2f(x∗)z, eβ(x,x∗)η(x,x∗)−1
+ρ(x, x∗)kθ(x, x∗)k2 if α(x, x∗)6= 0and β(x, x∗)6= 0, for allx∈X, 1
α(x, x∗)γ(x, x∗)
eα(x,x∗)[f(x)−f(x∗)+12hz,∇2f(x∗)zi]−1
(>)≥
∇f(x∗) +∇2f(x∗)z, η(x, x∗)
+ρ(x, x∗)kθ(x, x∗)k2
if α(x, x∗)6= 0and β(x, x∗) = 0, for allx∈X, γ(x, x∗)[f(x)−f(x∗) +1
2hz,∇2f(x∗)zi]
(>)≥ 1 β(x, x∗)
∇f(x∗) +∇2f(x∗)z, eβ(x,x∗)η(x,x∗)−1
+ρ(x, x∗)kθ(x, x∗)k2 if α(x, x∗) = 0and β(x, x∗)6= 0, for allx∈X, γ(x, x∗)[f(x)−f(x∗) +1
2hz,∇2f(x∗)zi](>)≥D
∇f(x∗) +∇2f(x∗)z, η(x, x∗) E
+ρ(x, x∗)kθ(x, x∗)k2 if α(x, x∗) = 0and β(x, x∗) = 0, for allx∈X, where
eβ(x,x∗)η(x,x∗)−1
≡
eβ(x,x∗)η1(x,x∗)−1, . . . , eβ(x,x∗)ηn(x,x∗)−1 and k · kis a norm onRn.
The functionf is said to be (strictly) (α, β, γ, η, ρ, θ)-sonvex onX if it is (strictly)(α, β, γ, η, ρ, θ)-sonvex at eachx∗∈X.
Denition 2.5. The function f is said to be (strictly) (α, β, γ, η, ρ, θ)- pseudosonvex atx∗ ∈X if there exist functions α:X×X→R, β:X×X → R, γ :X×X →R+, η:X×X→Rn, ρ:X×X→R, and θ:X×X→Rn, such that for all x∈X(x6=x∗) andz∈Rn,
1 β(x, x∗)
∇f(x∗) +∇2f(x∗)z, eβ(x,x∗)η(x,x∗)−1
≥ −ρ(x, x∗)kθ(x, x∗)k2
⇒ 1
α(x, x∗)γ(x, x∗)
eα(x,x∗)[f(x)−f(x∗)+12hz,∇2f(x∗)zi]−1
(>)≥0 if α(x, x∗)6= 0and β(x, x∗)6= 0, for allx∈X, ∇f(x∗) +∇2f(x∗)z, η(x, x∗)
≥ −ρ(x, x∗)kθ(x, x∗)k2
⇒ 1
α(x, x∗)γ(x, x∗)
eα(x,x∗)[f(x)−f(x∗)+12hz,∇2f(x∗)zi]−1
(>)≥0 if α(x, x∗)6= 0and β(x, x∗) = 0, for allx∈X,
1 β(x, x∗)
∇f(x∗) +∇2f(x∗)z, eβ(x,x∗)η(x,x∗)−1
≥ −ρ(x, x∗)kθ(x, x∗)k2
⇒ γ(x, x∗)[f(x)−f(x∗) +1
2hz,∇2f(x∗)zi](>)≥0
if α(x, x∗) = 0and β(x, x∗)6= 0, for allx∈X, ∇f(x∗) +∇2f(x∗)z, η(x, x∗)
≥ −ρ(x, x∗)kθ(x, x∗)k2
⇒ γ(x, x∗)[f(x)−f(x∗) +1
2hz,∇2f(x∗)zi](>)≥0
if α(x, x∗) = 0and β(x, x∗) = 0, for allx∈X.
The function f is said to be (strictly)(α, β, γ, η, ρ, θ)-pseudosonvex onX if it is (strictly) (α, β, γ, η, ρ, θ)-pseudosonvex at each x∗ ∈X.
Denition 2.6. The function f is said to be (prestrictly)(α, β, γ, η, ρ, θ)- quasisonvex at x∗ ∈X if there exist functions α:X×X → R, β :X×X → R, γ :X×X →R+, η:X×X→Rn, ρ:X×X→R, and θ:X×X→Rn, such that for each x∈X and z∈Rn,
1
α(x, x∗)γ(x, x∗)
eα(x,x∗)[f(x)−f(x∗)+12hz,∇2f(x∗)zi]−1
(<)≤0
⇒ 1
β(x, x∗)
∇f(x∗) +∇2f(x∗)z, eβ(x,x∗)η(x,x∗)−1
≤ −ρ(x, x∗)kθ(x, x∗)k2, if α(x, x∗)6= 0and β(x, x∗)6= 0, for allx∈X, 1
α(x, x∗)γ(x, x∗)
eα(x,x∗)[f(x)−f(x∗)+12hz,∇2f(x∗)zi]−1
(<)≤0
⇒
∇f(x∗) +∇2f(x∗)z, η(x, x∗)
≤ −ρ(x, x∗)kθ(x, x∗)k2,
if α(x, x∗)6= 0and β(x, x∗) = 0, for allx∈X, γ(x, x∗)[f(x)−f(x∗) +1
2hz,∇2f(x∗)zi](<)≤0
⇒ 1
β(x, x∗)
∇f(x∗) +∇2f(x∗)z, eβ(x,x∗)η(x,x∗)−1
≤ −ρ(x, x∗)kθ(x, x∗)k2, if α(x, x∗) = 0and β(x, x∗)6= 0, for allx∈X, γ(x, x∗)[f(x)−f(x∗) +1
2hz,∇2f(x∗)zi](<)≤0 ⇒ ∇f(x∗) +∇2f(x∗)z, η(x, x∗)
≤ −ρ(x, x∗)kθ(x, x∗)k2
if α(x, x∗) = 0and β(x, x∗) = 0, for allx∈X.
The function f is said to be (prestrictly) (α, β, γ, η, ρ, θ)-quasisonvex on X if it is (prestrictly) (α, β, γ, η, ρ, θ)-quasisonvex at each x∗∈X.
From the above denitions it is clear that iff is(α, β, γ, η, ρ, θ)-sonvex at x∗, then it is both(α, β, γ, η, ρ, θ)-pseudosonvex and(α, β, γ, η, ρ, θ)-quasisonvex at x∗, if f is (α, β, γ, η, ρ, θ)-quasisonvex at x∗, then it is prestrictly (α, β, γ, η, ρ, θ)-quasisonvex at x∗, and if f is strictly(α, β, γ, η, ρ, θ)-pseudosonvex at x∗, then it is (α, β, γ, η, ρ, θ)-quasisonvex at x∗.
In the proofs of the duality theorems, sometimes it may be more conve- nient to use certain alternative but equivalent forms of the above denitions.
These are obtained by considering the contrapositive statements. For example, (α, β, γ, η, ρ, θ)-pseudosonvexity (when α(x, x∗) 6= 0 and β(x, x∗) 6= 0, for all x∈X) can be dened in the following equivalent way:
The function f is said to be (α, β, γ, η, ρ, θ)-pseudosonvex at x∗ ∈ X if there exist functions α : X×X → R, β : X×X → R, γ : X ×X → R+, η : X×X →Rn, ρ:X×X →R, and θ:X×X →Rn, such that for all x∈X and z∈Rn,
1
α(x, x∗)γ(x, x∗)
eα(x,x∗)[f(x)−f(x∗)+12hz,∇2f(x∗)zi]−1
<0
⇒ 1
β(x, x∗) D
∇f(x∗)+∇2f(x∗)z, eβ(x,x∗)η(x,x∗)−1E
<−ρ(x, x∗)kθ(x, x∗)k2. In the remainder of this paper, we assume that the functions f, g, z → Gj(z, t), and z → Hk(z, s), are twice continuously dierentiable on X, for all t∈Tj, j∈q, ands∈Sk, k∈r.
With regard to the choice of the type of generalized invex functions, spec- ied in Denitions 2.4 - 2.6, to be used in the statements and proofs of our duality theorems, we shall consistently use the cases in which the functions α and β are nonzero for all (x, y) ∈ X×X. All the duality results established in this paper can be modied, restated, and proved for the other cases in a straightforward manner.
3. DUALITY MODEL I
In this section, we consider two duality models with relatively simple constraint structures and prove weak, strong, and strict converse duality theo- rems. The main assumption used in these duality theorems is an appropriate (α, β, γ, η, ρ, θ)-pseudosonvexity condition imposed on a Lagrangian-type func- tion. More general duality models and results will be discussed in the subse- quent sections.
Let H=
n
(y, z, v, λ, ν, ν0, Jν0, Kν\ν0,¯t,s) :¯ y∈X; z∈Rn; λ∈R;
0≤ν0 ≤ν ≤n+ 1; v∈Rν, vi >0, 1≤i≤ν0;
Jν0 = (j1, j2, . . . , jν0),1≤ji ≤q; Kν\ν0 = (kν0+1, . . . , kν),1≤ki ≤r;
¯t= (t1, t2, . . . , tν0), ti ∈Tji; ¯s= (sν0+1, . . . , sν), si∈Skio . Consider the following two problems:
(DI) sup
(y,z,v,λ,ν,ν0,Jν0,Kν\ν0,¯t,¯s)∈H
λ subject to
(3.1) ∇f(y)−λ∇g(y) +
ν0
X
m=1
vm∇Gjm(y, tm) +
ν
X
m=ν0+1
vm∇Hkm(y, sm)+
h
∇2f(y)−λ∇2g(y) +
ν0
X
m=1
vm∇2Gjm(y, tm)+
ν
X
m=ν0+1
vm∇2Hkm(y, sm)i z= 0, (3.2) f(y)−λg(y) +
ν0
X
m=1
vmGjm(y, tm) +
ν
X
m=ν0+1
vmHkm(y, sm)−
1 2 D
z,h
∇2f(y)−λ∇2g(y) +
ν0
X
m=1
vm∇2Gjm(y, tm)+
ν
X
m=ν0+1
vm∇2Hkm(y, sm) i
z E
≥0;
( ˜DI) sup
(y,z,v,λ,ν,ν0,Jν0,Kν\ν
0,¯t,¯s)∈H
λ subject to (3.2) and
(3.3) 1 β(x, y)
D∇f(y)−λ∇g(y) +
ν0
X
m=1
vm∇Gjm(y, tm)+
ν
X
m=ν0+1
vm∇Hkm(y, sm) + h
∇2f(y)−λ∇2g(y) +
ν0
X
m=1
vm∇2Gjm(y, tm)+
ν
X
m=ν0+1
vm∇2Hkm(y, sm) i
z, eβ(x,y)η(x,y)−1 E
≥0, for allx∈F,
whereβ is a function fromX×X toRandη is a function fromX×X toRn. Comparing (DI) and ( ˜DI), we see that ( ˜DI) is relatively more general than (DI) in the sense that any feasible solution of (DI) is also feasible for ( ˜DI), but the converse is not necessarily true. Furthermore, we observe that (3.1) is a system of n equations, whereas (3.3) is a single inequality. Clearly, from a computational point of view, (DI)is preferable to ( ˜DI) because of the dependence of (3.3) on the feasible set of (P).
Despite these apparent dierences, it turns out that the statements and proofs of all the duality theorems for (P)−(DI) and (P)−( ˜DI) are almost identical and, therefore, we shall consider only the pair(P)−(DI).
The next two theorems show that(DI) is a dual problem for(P). Theorem 3.1 (Weak Duality). Let x andw≡(y, z, v, λ, ν, ν0, Jν0, Kν\ν0,
¯t,s)¯ be arbitrary feasible solutions of (P) and (DI), respectively, and assume that the Lagrangian-type function
ξ→L(ξ, v, λ,¯t,¯s) =f(ξ)−λg(ξ) +
ν0
X
m=1
vmGjm(ξ, tm) +
ν
X
m=ν0+1
vmHkm(ξ, sm) is (α, β, γ, η,0, θ)-pseudosonvex at y and γ(x, y) >0, where ¯t≡(t1, t2, . . . , tν0) and
¯
s≡(sν0+1, sν0+2, . . . , sν). Thenϕ(x)≥λ.
Proof. From (3.1) we observe that 1
β(x, y) D
∇f(y)−λ∇g(y) +
ν0
X
m=1
vm∇Gjm(y, tm) +
ν
X
m=ν0+1
vm∇Hkm(y, sm) +h
∇2f(y)−λ∇2g(y) +
ν0
X
m=1
vm∇2Gjm(y, tm) +
ν
X
m=ν0+1
vm∇2Hkm(y, sm)i
z, eβ(x,y)η(x,y)−1E
= 0, which in view of our (α, β, γ, η,0, θ)-pseudosonvexity assumption implies that
1
α(x, y)γ(x, y)
eα(x,y)[L(x,v,λ,¯t,¯s)−L(y,v,λ,¯t,¯s)+12hz,∇2L(y,v,λ,¯t,¯s)zi]−1
≥0.
We need to consider two cases: α(x, y)>0 andα(x, y)<0. If we assume that α(x, y)>0 and recall thatγ(x, y)>0, then the above inequality becomes
eα(x,y)[L(x,v,λ,¯t,¯s)−L(y,v,λ,¯t,¯s)+12hz,∇2L(y,v,λ,¯t,¯s)zi]≥1, which implies that
L(x, v, λ,t,¯¯s)≥L(y, v, λ,t,¯¯s)−1
2hz,∇2L(y, v, λ,¯t,s)zi.¯
In view of (3.2), the right-hand side of the above inequality is greater than or equal to zero, and hence, we have L(x, v, λ,¯t,¯s) ≥0. Inasmuch as x∈ F,and vm >0, m∈ν0, this inequality simplies tof(x)−λg(x) ≥0, and hence, we conclude thatϕ(x)≥λ.
If we assume that α(x, y)<0, we arrive at the same conclusion.
Theorem 3.2 (Strong Duality). Let x∗ be a normal optimal solution of (P) and assume that for each feasible solution (y, z, v, λ, ν, ν0, Jν0, Kν\ν0, ¯t,s)¯ of (DI), the conditions specied in Theorem 3.1 are satised. Then, there exist v∗, λ∗, ν∗, ν0∗, Jν∗
0, Kν∗\ν∗
0,¯t∗, and s¯∗ such that (x∗, z∗ = 0, v∗, λ∗, ν∗, ν0∗, Jν∗
0, Kν∗\ν∗
0,t¯∗,¯s∗) is an optimal solution of (DI) andϕ(x∗) =λ∗.
Proof. Since x∗ is a normal optimal solution of (P), by Theorem 2.1 of [10], there exist v∗, λ∗, ν∗, ν0∗, Jν∗
0, Kν∗\ν∗
0,¯t∗, and s¯∗ such that (x∗, z∗ = 0, v∗, λ∗, ν∗, ν0∗, Jν∗
0, Kν∗\ν∗
0,¯t∗,¯s∗)is a feasible solution of(DI)andϕ(x∗) =λ∗. If it were not optimal, then there would exist a feasible solution(˜x,z,˜ v,˜ λ,˜ ν,˜ ν˜0, Jν˜0, Kν\˜˜ ν0,t,¯˜¯˜s) of(DI) such thatλ > λ˜ ∗ =ϕ(x∗), contradicting Theorem 3.1.
Therefore,(x∗, z∗ = 0, v∗, λ∗, ν∗, ν0∗, Jν0∗, Kν∗\ν∗
0,¯t∗,¯s∗)is an optimal solution of (DI).
We also have the following converse duality result for(P) and (DI). Theorem 3.3 (Strict Converse Duality). Let x∗ be a normal optimal so- lution of (P), letw˜≡(˜x,z,˜ v,˜ ˜λ,ν,˜ ν˜0, Jν˜0, Kν\˜˜ ν0,t,¯˜¯˜s) be an optimal solution of (DI), and assume that the functionξ →L(ξ,v,˜ λ,˜ ¯˜t,s)¯˜ is strictly(α, β, γ, η,0, θ)- pseudosonvex at x˜. Then x˜=x∗, that is, x˜ is an optimal solution of (P), and ϕ(x∗) = ˜λ.
Proof. (a): Suppose to the contrary that x˜ 6= x∗. By Theorem 2.1 of [10], there exist v∗, λ∗, ν∗, ν0∗, Jν0∗, Kν∗\ν∗
0,¯t∗, and s¯∗ such that (x∗, z∗ = 0, v∗, λ∗, ν∗, ν0∗, Jν∗
0, Kν∗\ν∗
0,¯t∗,¯s∗)is a feasible solution of(DI)andϕ(x∗) =λ∗. Now, proceeding as in the proof of Theorem 3.1 (with x replaced by x∗ and w by w˜), and using the conditions set forth above, we arrive at the strict inequality f(x∗)−λg(x˜ ∗) > 0, and so ϕ(x∗) > λ˜, which contradicts the fact thatϕ(x∗) =λ∗≤λ˜. Therefore, we conclude that x˜=x∗ andϕ(x∗) = ˜λ.
4. DUALITY MODEL II
In this section, we consider two duality models with special constraint structures that allow for a greater variety of generalized(α, β, γ, η, ρ, θ)-sonvexity conditions under which duality can be established. These duality models have the following forms:
(DII) sup
(y,z,v,λ,ν,ν0,Jν0,Kν\ν0,¯t,¯s)∈H
λ subject to
(4.1) ∇f(y)−λ∇g(y) +
ν0
X
m=1
vm∇Gjm(y, tm) +
ν
X
m=ν0+1
vm∇Hkm(y, sm)+
h∇2f(y)−λ∇2g(y) +
ν0
X
m=1
vm∇2Gjm(y, tm) +
ν
X
m=ν0+1
vm∇2Hkm(y, sm)i z= 0,
(4.2) f(y)−λg(y)−1
2hz,[∇2f(y)−λ∇2g(y)]zi ≥0, (4.3) Gjm(y, tm)−1
2hz,∇2Gjm(y, tm)zi ≥0, m∈ν0, (4.4) vmHkm(y, sm)−1
2hz, vm∇2Hkm(y, sm)zi ≥0, m∈ν\ν0;
( ˜DII) sup
(y,z,v,λ,ν,ν0,Jν0,Kν\ν
0,¯t,¯s)∈H
λ subject to (3.3) and (4.2)(4.4).
The remarks and observations made earlier about the relationships be- tween(DI) and ( ˜DI) are, of course, also valid for(DII) and( ˜DII).
The next two theorems show that(DII) is a dual problem for(P). Theorem 4.1 (Weak Duality). Let x andw≡(y, z, v, λ, ν, ν0, Jν0, Kν\ν0,
¯t,s)¯ be arbitrary feasible solutions of (P) and (DII), respectively, and assume that any one of the following ve sets of hypotheses is satised:
(a) (i) ξ→f(ξ)−λg(ξ)is(α, β,γ, η,¯ ρ, θ)¯ -pseudosonvex atyandγ(x, y)>0; (ii) for each m ∈ ν0, ξ → Gjm(ξ, tm) is (α, β,ˆγm, η,ρˆm, θ)-quasisonvex
at y;
(iii) for each m ∈ ν\ν0, ξ → vmHkm(ξ, sm) is (α, β,γ˘m, η,ρ˘m, θ)- quasisonvex at y;
(iv) ρ(x, y) +¯ Pν0
m=1vmρˆm(x, y) +Pν
m=ν0+1ρ˘m(x, y)≥0;
(b) (i) ξ→f(ξ)−λg(ξ)is(α, β,γ, η,¯ ρ, θ)¯ -pseudosonvex atyandγ(x, y)>0; (ii) ξ →Pν0
m=1vmGjm(ξ, tm) is (α, β,γ, η,ˆ ρ, θ)ˆ -quasisonvex at y; (iii) for each m ∈ ν\ν0, ξ → vmHkm(ξ, sm) is (α, β,γ˘m, η,ρ˘m, θ)-
quasisonvex at y; (iv) ρ(x, y) + ˆ¯ ρ(x, y) +Pν
m=ν0+1ρ˘m(x, y)≥0;
(c) (i) ξ→f(ξ)−λg(ξ)is(α, β,γ, η,¯ ρ, θ)¯ -pseudosonvex atyandγ(x, y)>0; (ii) for each m ∈ ν0, ξ → Gjm(ξ, tm) is (α, β,ˆγm, η,ρˆm, θ)-quasisonvex
at y; (iii) ξ →Pν
m=ν0+1vmHkm(ξ, sm) is(α, β,˘γ, η,ρ, θ)˘ -quasisonvex aty; (iv) ρ(x, y) +¯ Pν0
m=1vmρˆm(x, y) + ˘ρ(x, y)≥0;
(d) (i) ξ→f(ξ)−λg(ξ)is(α, β,γ, η,¯ ρ, θ)¯ -pseudosonvex atyandγ(x, y)>0; (ii) ξ →Pν0
m=1vmGjm(ξ, tm) is (α, β,γ, η,ˆ ρ, θ)ˆ -quasisonvex at y; (iii) ξ →Pν
m=ν0+1vmHkm(ξ, sm) is(α, β,˘γ, η,ρ, θ)˘ -quasisonvex aty; (iv) ρ(x, y) + ˆ¯ ρ(x, y) + ˘ρ(x, y)≥0;
(e) (i) ξ→f(ξ)−λg(ξ)is(α, β,γ, η,¯ ρ, θ)¯ -pseudosonvex atyandγ(x, y)>0; (ii) ξ → Pν0
m=1vmGjm(ξ, tm) +Pν
m=ν0+1vmHkm(ξ, sm) is (α, β,γ, η,ˆ ˆ
ρ, θ)-quasisonvex at y; (iii) ρ(x, y) + ˆ¯ ρ(x, y)≥0.
Then, ϕ(x)≥λ.
Proof. (a): From the primal feasibility ofx and (4.3) we see that for each m∈ν0,
Gjm(x, tm)≤0≤Gjm(y, tm)−1
2hz,∇2Gjm(y, tm)zi, and hence,
1
α(x, y)ˆγm(x, y)
eα(x,y){Gjm(x,tm)−[Gjm(y,tm)−12hz,∇2Gjm(y,tm)zi]}−1
≤0, which in view of (ii) implies that
1 β(x, y)
∇Gjm(y, tm) +∇2Gjm(y, tm)z, eβ(x,y)η(x,y)−1
≤ −ˆρm(x, y)kθ(x, y)k2. Asvm >0 for eachm∈ν0, the above inequalities yield
(4.5) 1 β(x, y)
DXν0
m=1
vm[∇Gjm(y, tm) +∇2Gjm(y, tm)z], eβ(x,y)η(x,y)−1E
≤ −
ν0
X
m=1
vmρˆm(x, y)kθ(x, y)k2. Similarly, from the primal feasibility of x, (4.4), and (iii) we deduce that (4.6) 1
β(x, y)
D Xν
m=ν0+1
vm[∇Hkm(y, sm) +∇2Hkm(y, sm)z], eβ(x,y)η(x,y)−1 E
≤ −
ν
X
m=ν0+1
˘
ρm(x, y)kθ(x, y)k2.
Combining (4.1) with (4.5) and (4.6), and using (iv), we obtain 1
β(x, y)
∇f(y)−λ∇g(y) + [∇2f(y)−λ∇2g(y)]z, eβ(x,y)η(x,y)−1
≥hXν0
m=1
vmρˆm(x, y) +
ν
X
m=ν0+1
˘
ρm(x, y)i
kθ(x, y)k2 ≥ −¯ρ(x, y)kθ(x, y)k2, which in view of (i) implies that
1
α(x, y)γ¯(x, y)
eα(x,y){f(x)−λg(x)−{f(y)−λg(y)−1
2hz,[∇2f(y)−λ∇2g(y)]zi}}−1
≥0.
Since ¯γ(x, y)>0, this inequality implies that f(x)−λg(x)≥f(y)−λg(y)− 1
2hz,[∇2f(y)−λ∇2g(y)]zi ≥0,
where the second inequality follows from (4.2). Hence, we have the desired weak duality inequalityϕ(x) = f(x)g(x) ≥λ.
(b)(e): The proofs are similar to that of part (a).
Theorem 4.2 (Strong Duality). Let x∗ be a normal optimal solution of (P) and assume that any one of the ve sets of conditions set forth in Theorem 4.1 is satised for all feasible solutions of (DII). Then there exist v∗, λ∗, ν∗, ν0∗, Jν0∗, Kν∗\ν∗
0,¯t∗, and s¯∗ such that (x∗, z∗ = 0, v∗, λ∗, ν∗, ν0∗, Jν∗0, Kν∗\ν∗
0,t¯∗,¯s∗) is an optimal solution of (DII) and ϕ(x∗) =λ∗. Proof. The proof is similar to that of Theorem 3.2.
Theorem 4.3 (Strict Converse Duality). Let x∗ be a normal optimal so- lution of (P), let w˜ ≡ (˜x,z,˜ ˜v,˜λ,ν,˜ ν˜0, Jν˜0, K˜ν\˜ν0, t,¯˜¯˜s) be an optimal solution of (DII), and assume that any one of the ve sets of conditions specied in Theorem 4.1 is satised and that the function ξ → f(ξ)−λg(ξ)˜ is strictly (α, β,¯γ, η,ρ, θ)-pseudosonvex at¯ x. Then˜ x˜=x∗ and ϕ(x∗) = ˜λ.
Proof. (a): Suppose to the contrary that x˜ 6= x∗. By Theorem 2.1 of [10], there exist v∗, λ∗, ν∗, ν0∗, Jν∗
0, Kν∗\ν∗
0,¯t∗, and s¯∗ such that(x∗, z∗ = 0, v∗, λ∗, ν∗, ν0∗, Jν∗
0, Kν∗\ν∗
0,¯t∗,s¯∗) is a feasible solution of (DII) and ϕ(x∗) = λ∗. Now, proceeding as in the proof of Theorem 4.1 (with x replaced by x∗ and w by w˜) and using our strict (α, β,γ, η,¯ ρ, θ)¯ -pseudosonvexity assumption, we arrive at the strict inequality f(x∗)−λg(x˜ ∗) > 0, yielding ϕ(x∗) >λ˜, which contradicts the fact that ϕ(x∗) =λ∗ ≤˜λ. Therefore, we conclude that x˜=x∗ and ϕ(x∗) = ˜λ.
(b)(e): The proofs are similar to that of part (a).
Theorem 4.4 (Weak Duality). Let x and (y, z, v, λ, ν, ν0, Jν0, Kν\ν0,¯t,s)¯ be arbitrary feasible solutions of (P) and (DII), respectively, and assume that any one of the following ve sets of hypotheses is satised:
(a) (i) ξ →f(ξ)−λg(ξ) is prestrictly (α, β,γ, η,¯ ρ, θ)¯ -quasisonvex at y and
¯
γ(x, y)>0;
(ii) for each m ∈ ν0, ξ → Gjm(ξ, tm) is (α, β,ˆγm, η,ρˆm, θ)-quasisonvex at y;
(iii) for each m ∈ ν\ν0, ξ → vmHkm(ξ, sm) is (α, β,γ˘m, η,ρ˘m, θ)- quasisonvex at y;
(iv) ρ(x, y) +¯ Pν0
m=1vmρˆm(x, y) +Pν
m=ν0+1ρ˘m(x, y)>0;
(b) (i) ξ →f(ξ)−λg(ξ) is prestrictly (α, β,γ, η,¯ ρ, θ)¯ -quasisonvex at y and
¯
γ(x, y)>0; (ii) ξ →Pν0
m=1vmGjm(ξ, tm) is (α, β,γ, η,ˆ ρ, θ)ˆ -quasisonvex at y; (iii) for each m ∈ ν\ν0, ξ → vmHkm(ξ, sm) is (α, β,γ˘m, η,ρ˘m, θ)-
quasisonvex at y; (iv) ρ(x, y) + ˆ¯ ρ(x, y) +Pν
m=ν0+1ρ˘m(x, y)>0;
(c) (i) ξ →f(ξ)−λg(ξ) is prestrictly (α, β,γ, η,¯ ρ, θ)¯ -quasisonvex at y and
¯
γ(x, y)>0;
(ii) for each m ∈ ν0, ξ → Gjm(ξ, tm) is (α, β,ˆγm, η,ρˆm, θ)-quasisonvex at y;
(iii) ξ →Pν
m=ν0+1vmHkm(ξ, sm) is(α, β,˘γ, η,ρ, θ)˘ -quasisonvex aty; (iv) ρ(x, y) +¯ Pν0
m=1vmρˆm(x, y) + ˘ρ(x, y)>0;
(d) (i) ξ →f(ξ)−λg(ξ) is prestrictly (α, β,γ, η,¯ ρ, θ)-quasisonvex at¯ y and
¯
γ(x, y)>0; (ii) ξ →Pν0
m=1vmGjm(ξ, tm) is (α, β,γ, η,ˆ ρ, θ)ˆ -quasisonvex at y; (iii) ξ →Pν
m=ν0+1vmHkm(ξ, sm) is(α, β,˘γ, η,ρ, θ)˘ -quasisonvex aty; (iv) ρ(x, y) + ˆ¯ ρ(x, y) + ˘ρ(x, y)>0;
(e) (i) ξ →f(ξ)−λg(ξ) is prestrictly (α, β,γ, η,¯ ρ, θ)¯ -quasisonvex at y and
¯
γ(x, y)>0; (ii) ξ →Pν0
m=1vmGjm(ξ, tm)+Pν
m=ν0+1vmHkm(ξ, sm)is(α, β,γ, η,ˆ ρ, θ)ˆ - quasisonvex at y;
(iii) ρ(x, y) + ˆ¯ ρ(x, y)>0. Then, ϕ(x)≥λ.
Proof. (a): Because of our assumptions specied in (ii) and (iii), (4.5) and (4.6) remain valid for the present case. From (4.1), (4.5), (4.6), and (iv)
we deduce that 1
β(x, y)
∇f(y)−λ∇g(y) + [∇2f(y)−λ∇2g(y)]z, eβ(x,y)η(x,y)−1
≥hXν0
m=1
vmρˆm(x, y) +
ν
X
m=ν0+1
˘ ρm(x, y)
i
kθ(x, y)k2 >−¯ρ(x, y)kθ(x, y)k2, which in view of (i) implies that
1
α(x, y)γ¯(x, y)
eα(x,y){f(x)−λg(x)−[f(y)−λg(y)]+12hz,[∇2f(y)−λ∇2g(y)]zi}−1
≥0.
Since ¯γ(x, y)>0, it follows from this inequality that f(x)−λg(x)−[f(y)−λg(y)] + 1
2hz,[∇2f(y)−λ∇2g(y)]zi ≥0.
Because of (4.2), this inequality reduces tof(x)−λg(x)≥0, which leads to the conclusion that ϕ(x)≥λ.
(b)(e): The proofs are similar to that of part (a).
Theorem 4.5 (Strong Duality). Let x∗ be a normal optimal solution of (P) and assume that any one of the ve sets of conditions set forth in Theorem 4.4 is satised for all feasible solutions of (DII). Then, there exist v∗, λ∗, ν∗, ν0∗, Jν0∗, Kν∗\ν∗0,¯t∗, and ¯s∗ such that (x∗, z∗ = 0, v∗, λ∗, ν∗, ν0∗, Jν∗0, Kν∗\ν∗
0,t¯∗,¯s∗) is an optimal solution of (DII) and ϕ(x∗) =λ∗. Proof. The proof is similar to that of Theorem 3.2.
Theorem 4.6 (Strict Converse Duality). Let x∗ be a normal optimal so- lution of (P), let w˜ ≡ (˜x,z,˜ ˜v,˜λ,ν,˜ ν˜0, Jν˜0, K˜ν\˜ν0, t,¯˜¯˜s) be an optimal solution of (DII), and assume that any one of the ve sets of conditions set forth in Theorem 4.4 is satised for the feasible solution w˜ of (DII), and that the func- tion ξ → f(ξ)−λg(ξ)˜ is (α, β,γ, η,¯ ρ, θ)¯ -quasisonvex at x˜. Then x˜ = x∗ and ϕ(x∗) = ˜λ.
Proof. The proof is similar to that of Theorem 3.3.
Theorem 4.7 (Weak Duality). Let x and (y, z, v, λ, ν, ν0, Jν0, Kν\ν0,¯t,s)¯ be arbitrary feasible solutions of (P) and (DII), respectively, and assume that any one of the following seven sets of hypotheses is satised:
(a) (i) ξ →f(ξ)−λg(ξ) is prestrictly (α, β,γ, η,¯ ρ, θ)-quasisonvex at¯ y and
¯
γ(x, y)>0;
(ii) for each m ∈ ν0, ξ → Gjm(ξ, tm) is strictly (α, β,γˆm, η,ρˆm, θ)- pseudosonvex at y;
(iii) for each m ∈ ν\ν0, ξ → vmHkm(ξ, sm) is (α, β,γ˘m, η,ρ˘m, θ)- quasisonvex at y;
(iv) ρ(x, y) +¯ Pν0
m=1vmρˆm(x, y) +Pν
m=ν0+1ρ˘m(x, y)≥0;
(b) (i) ξ →f(ξ)−λg(ξ) is prestrictly (α, β,γ, η,¯ ρ, θ)¯ -quasisonvex at y and
¯
γ(x, y)>0; (ii) ξ→Pν0
m=1vmGjm(ξ, tm)is strictly(α, β,γ, η,ˆ ρ, θ)ˆ -pseudosonvex aty; (iii) for each m ∈ ν\ν0, ξ → vmHkm(ξ, sm) is (α, β,γ˘m, η,ρ˘m, θ)-
quasisonvex at y;
(iv) ρ(x, y) + ˆ¯ ρ(x, y) +Pν
m=ν0+1ρ˘m(x, y)≥0;
(c) (i) ξ →f(ξ)−λg(ξ) is prestrictly (α, β,γ, η,¯ ρ, θ)¯ -quasisonvex at y and
¯
γ(x, y)>0;
(ii) for each m ∈ ν0, ξ → Gjm(ξ, tm) is (α, β,ˆγm, η,ρˆm, θ)-quasisonvex at y;
(iii) for eachm∈ν\ν0, ξ→vmHkm(ξ, sm) is strictly(α, β,γ˘m, η,ρ˘m, θ)- pseudosonvex at y;
(iv) ρ(x, y) +¯ Pν0
m=1vmρˆm(x, y) +Pν
m=ν0+1ρ˘m(x, y)≥0;
(d) (i) ξ →f(ξ)−λg(ξ) is prestrictly (α, β,γ, η,¯ ρ, θ)¯ -quasisonvex at y and
¯
γ(x, y)>0;
(ii) for each m ∈ ν0, ξ → Gjm(ξ, tm) is (α, β,ˆγm, η,ρˆm, θ)-quasisonvex at y;
(iii) ξ →Pν
m=ν0+1vmHkm(ξ, sm)is strictly(α, β,γ, η,˘ ρ, θ)-pseudosonvex˘ at y;
(iv) ρ(x, y) +¯ Pν0
m=1vmρˆm(x, y) + ˘ρ(x, y)≥0;
(e) (i) ξ →f(ξ)−λg(ξ) is prestrictly (α, β,γ, η,¯ ρ, θ)¯ -quasisonvex at y and
¯
γ(x, y)>0; (ii) ξ→Pν0
m=1vmGjm(ξ, tm)is strictly(α, β,γ, η,ˆ ρ, θ)ˆ -pseudosonvex aty; (iii) ξ →Pν
m=ν0+1vmHkm(ξ, sm) is(α, β,˘γ, η,ρ, θ)˘ -quasisonvex aty; (iv) ρ(x, y) + ˆ¯ ρ(x, y) + ˘ρ(x, y)≥0;
(f) (i) ξ →f(ξ)−λg(ξ) is prestrictly (α, β,γ, η,¯ ρ, θ)¯ -quasisonvex at y and
¯
γ(x, y)>0; (ii) ξ →Pν0
m=1vmGjm(ξ, tm) is (α, β,γ, η,ˆ ρ, θ)ˆ -quasisonvex at y; (iii) ξ →Pν
m=ν0+1vmHkm(ξ, sm)is strictly(α, β,γ, η,˘ ρ, θ)˘ -pseudosonvex at y;
(iv) ρ(x, y) + ˆ¯ ρ(x, y) + ˘ρ(x, y)≥0;
(g) (i) ξ →f(ξ)−λg(ξ) is prestrictly (α, β,γ, η,¯ ρ, θ)¯ -quasisonvex at y and
¯
γ(x, y)>0;
(ii) ξ →Pν0
m=1vmGjm(ξ, tm) +Pν
m=ν0+1vmHkm(ξ, sm) is strictly(α, β,
ˆ
γ, η,ρ, θ)ˆ -pseudosonvex at y; (iii) ρ(x, y) + ˆ¯ ρ(x, y)≥0.
Then, ϕ(x)≥λ.
Proof. (a): Suppose to the contrary thatϕ(x)< λ. Using this inequality and (4.2), we get
f(x)−λg(x)<0≤f(y)−λg(y)−1
2hz,[∇2f(y)−λ∇2g(y)]zi and hence,
1
α(x, y)γ¯(x, y)
eα(x,y){f(x)−λg(x)−f(y)+λg(y)+1
2hz,[∇2f(y)−λ∇2g(y)]z}−1
<0, which in view of (i) implies
(4.7) 1 β(x, y)
∇f(y)−λ∇g(y) + [∇2f(y)−λ∇2g(y)]z, eβ(x,y)η(x,y)−1
≤ −¯ρ(x, y)kθ(x, y)k2. From the primal feasibility of x and (4.3) we see that for eachm∈ν0,
Gjm(x, tm)≤0≤Gjm(y, tm)−1
2hz,∇2Gjm(y, tm)zi and hence,
1
α(x, y)γˆm(x, y)
eα(x,y)[Gjm(x,tm)−Gjm(y,tm)+12hz,∇2Gjm(y,tm)zi]−1
≤0, which in view of (ii) implies that
1 β(x, y)
∇Gjm(y, tm) +∇2Gjm(y, tm)z, eβ(x,y)η(x,y)−1
<−ˆρm(x, y)kθ(x, y)k2. Asvm >0 for eachm∈ν0, the above inequalities yield
(4.8) 1 β(x, y)
DXν0
m=1
vm[∇Gjm(y, tm) +∇2Gjm(y, tm)z], eβ(x,y)η(x,y)−1 E
<−
ν0
X
m=1
vmρˆm(x, y)kθ(x, y)k2. Now, combining (4.1), (4.6) (which is valid for the present case because of our assumptions set forth in (iii)), (4.8), and (iv), we get
1 β(x, y)
∇f(y)−λ∇g(y) + [∇2f(y)−λ∇2g(y)]z}, eβ(x,y)η(x,y)−1
>
−¯ρ(x, y)kθ(x, y)k2.
which contradicts (4.7). Therefore, we conclude that ϕ(x)≥λ. (b)(g): The proofs are similar to that of part (a).
