IN SEMIINFINITE FRACTIONAL PROGRAMMING.
PART II: FIRST-ORDER DUALITY MODELS
G.J. ZALMAI
Communicated by the former editorial board
In this paper, we discuss a fairly large number of first-order duality results un- der various generalized (α, β, γ, η, ρ, θ)-invexity assumptions for a semiinfinite fractional programming problem.
AMS 2010 Subject Classification: 49N15, 90C30, 90C32, 90C34, 90C46.
Key words: Semiinfinite programming, fractional programming, generalized (α, β, γ, η, ρ, θ)-invex functions, infinitely many equality and inequality constraints, first-order dual problems, duality theorems.
1. INTRODUCTION
In this paper, we state and prove a fairly large number of parametric and parameter-free duality results under various generalized (α, β, γ, η, ρ, θ)- invexity assumptions for the following semiinfinite 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 all s∈Sk, k∈r,
x∈X,
whereqandr are 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 andSk are compact subsets of complete metric spaces,f andg are real-valued functions defined on X, for each j ∈q, z →Gj(z, t) is a real-valued function defined on X for all t ∈ Tj, for each k ∈ r, z → Hk(z, s) is a real-valued function defined on X for all s ∈Sk, for each j ∈ q and k ∈r, t → Gj(x, t) ands→Hk(x, s) are continuous real-valued functions defined, respectively, on Tj andSkfor allx∈X, andg(x)>0 for allxsatisfying the constraints of (P).
REV. ROUMAINE MATH. PURES APPL.,58(2013),1,35–62
The present study is essentially a continuation of the investigation ini- tiated in the companion paper [2] where some information about fractional programming is presented, the current status of semiinfinite programming is briefly discussed and numerous key references are cited, an overview of the con- cept of (α, β, γ, η, ρ, θ)-invexity and some of its extensions is given, and a fairly large number of sets of global sufficient optimality results under various gener- alized (α, β, γ, η, ρ, θ)-invexity assumptions are established. For the necessary background material and preliminaries, the reader is referred to [2]. Here we shall make use of the sufficient optimality results developed in [2] and a set of necessary optimality conditions, which will be recalled in the next section, and construct several first-order duality models for (P) and prove appropriate duality theorems. Second- and higher-order counterparts of these results are discussed in [3, 4].
Although presently various duality results exist in the related literature for several classes of static and dynamic fractional optimization problems with a finite number of constraints, so far no such results are available for any kind of semiinfinitefractional programming problems involving generalized (α, β, γ, η, ρ, θ)-invex functions. To the best of our knowledge, all the duality results established in this paper are new in the area of semiinfinite programming.
The rest of the paper is organized as follows. In Section 2, we recall an auxiliary result which will be needed in the sequel. We begin our discussion of duality for (P) in Section 3 where we formulate two parametric duality models with somewhat restricted constraint structures, and prove weak, strong, and strict converse duality theorems under appropriate generalized (α, β, γ, η, ρ, θ)- invexity assumptions. In Section 4, we present another pair of parametric duality models with relatively more general constraint structures that allow for a greater variety of generalized (α, β, γ, η, ρ, θ)-invexity hypotheses under which duality can be established. We continue our investigation of parametric duality for (P) in Section 5 where we consider two generalized duality models and prove several sets of duality results. These duality models are, in fact, two families of dual problems for (P) whose members can easily be identified by appropriate choices of certain sets and functions. The nonparametric counterparts of these generalized duality models are taken up in Section 6, and a number of duality results are established. Finally, in Section 7, we summarize our main results and also point out some additional research opportunities arising from certain modifications of the principal problem model (P).
2. PRELIMINARIES
For proving our duality theorems, we shall need a few basic definitions and auxiliary results. For the definitions of the new classes of generalized
(α, β, γ, η, ρ, θ)-invex functions that will be utilized in the statements and proofs of our duality theorems, the reader is referred to [2], and the two aux- iliary results that will be needed in the sequel for proving strong and strict converse duality theorems are stated below.
Theorem 2.1. [5]. Let x∗ ∈ F and λ∗ = f(x∗)/g(x∗), let f and g be continuously differentiable at x∗, for each j ∈q, let the function z →Gj(z, t) be continuously differentiable at x∗ for all t ∈ Tj, and for each k ∈ r, let the function z → Hk(z, s) be continuously differentiable at x∗ for all s∈ Sk. If x∗ is an optimal solution of (P), if the generalized Guignard constraint qualification holds at x∗, and if the set cone{∇Gj(x∗, t) : t ∈ Tˆj(x∗), j ∈ q}+span{∇Hk(x∗, s) : s ∈ Sk, k ∈ r} is closed, then there exist integers ν0∗ and ν∗, with 0 ≤ ν0∗ ≤ ν∗ ≤ n+ 1, such that there exist ν0∗ indices jm, with 1≤jm≤q, together withν0∗ points tm∈Tˆjm(x∗), m∈ν0∗, ν∗−ν0∗ indiceskm, with 1 ≤km ≤r, together with ν∗−ν0∗ points sm ∈Skm for m ∈ ν∗\ν0∗, and ν∗ real numbers vm∗, with v∗m>0 for m∈ν0∗, with the property that
∇f(x∗)−λ∗∇g(x∗) +
ν0∗
X
m=1
v∗m∇Gjm(x∗, tm) +
ν∗
X
m=ν0∗+1
vm∗∇Hkm(x∗, sm) = 0, where Tˆj(x∗) ={t∈Tj :Gj(x∗, t) = 0}andν∗\ν0∗ is the complement of the set ν0∗ relative to ν∗.
Eliminating λ∗ and redefining the Lagrange multipliers vm∗, m ∈ ν∗, we obtain the following parameter-free version of Theorem 2.1.
Theorem 2.2. Let x∗ ∈ F, let f and g be continuously differentiable at x∗, for each j∈q, let the function z →Gj(z, t) be continuously differentiable at x∗ for all t ∈ Tj, and for each k ∈ r, let the function z → Hk(z, s) be continuously differentiable at x∗ for all s∈Sk. Ifx∗ is an optimal solution of (P), if the generalized Guignard constraint qualification holds atx∗, and if the set cone{∇Gj(x∗, t) : t ∈ Tˆj(x∗), j ∈ q}+span{∇Hk(x∗, s) : s ∈ Sk, k ∈ r}
is closed, then there exist integers ν0∗ and ν∗, with 0 ≤ ν0∗ ≤ ν∗ ≤ n+ 1, such that there exist ν0∗ indices jm, with 1 ≤ jm ≤q, together with ν0∗ points tm ∈ Tˆjm(x∗), m ∈ ν0∗, ν∗−ν0∗ indices km, with 1 ≤ km ≤ r, together with ν∗−ν0∗ points sm∈Skm for m∈ν∗\ν0∗, and ν∗ real numbers vm∗, withvm∗ >0 for m∈ν0∗, with the property that
g(x∗)∇f(x∗)−f(x∗)∇g(x∗)+
ν0∗
X
m=1
vm∗∇Gjm(x∗, tm)+
ν∗
X
m=ν0∗+1
vm∗∇Hkm(x∗, sm) = 0.
We shall call x a normal feasible solution of (P) if x satisfies all the constraints of (P), if the generalized Guignard constraint qualification holds
at x, and if the setcone{∇Gj(x, t) :t∈Tˆj(x), j ∈ q}+span{∇Hk(x, s) :s∈ Sk, k∈r} is closed.
The forms and features of these results will serve as our guide in this paper for constructing several parametric and nonparametric duality models for (P) and proving appropriate duality theorems.
In the remainder of this paper, we assume that the functionsf, g, Gj(·, t), t ∈ Tj, j ∈ q, and Hk(·, s), s ∈ Sk, k ∈ r, are continuously differentiable on the open setX.
With regard to the choice of the type of generalized invex functions, specified in Definitions 2.5–2.7 in [2], 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 modified, restated, and proved for the other cases in a straightforward manner.
3. DUALITY MODEL I
In this section, we consider two dual problems with relatively simple constraint structures and prove weak, strong, and strict converse duality the- orems under appropriate (α, β, γ, η,0, θ)-invexity conditions. More general du- ality models and results for (P) will be discussed in the subsequent sections.
Let H=
n
(y, v, λ, ν, ν0, Jν0, Kν\ν0,¯t,s) :¯ y∈X; 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∈Ski
o . Consider the following two problems:
(DI) sup
(y,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) = 0, (3.2) f(y)−λg(y) +
ν0
X
m=1
vmGjm(y, tm) +
ν
X
m=ν0+1
vmHkm(y, sm)≥0;
( ˜DI) sup
(y,v,λ,ν,ν0,Jν0,Kν\ν0,¯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),eβ(x,y)η(x,y)−1E
≥0 for allx∈F, whereβ is a function from X×X toRand ηis a function from X×X toRn.
The structures of the two problems designated above as (DI) and ( ˜DI) are based directly on the form and contents of the necessary optimality condi- tions of Theorem 2.1. This is, of course, the standard method for constructing Wolfe-type dual problems. Comparing (DI) and ( ˜DI), we see that ( ˜DI) is rel- atively 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 in- equality. 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 differences, 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 and(y, 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
z→L(z, v, λ,¯t,s) =¯ f(z)−λg(z) +
ν0
X
m=1
vmGjm(z, tm) +
ν
X
m=ν0+1
vmHkm(z, sm) is (α, β, γ, η,0, θ)-pseudoinvex at y and γ(x, y) > 0, where ¯t ≡ (t1, t2, . . . , tν0) and
¯
s≡(sν0+1, sν0+2, . . . , sν). Then ϕ(x)≥λ.
Proof. From (3.1) we observe that 1
β(x, y) D
∇f(y)−λ∇g(y) +
ν0
X
m=1
∇vmGjm(y, tm) +
ν
X
m=ν0+1
∇vmHkm(y, sm),eβ(x,y)η(x,y)−1E
= 0, which in view of our (α, β, γ, η,0, θ)-pseudoinvexity 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)]−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 that γ(x, y)>0, then the above inequality becomes
eα(x,y)[L(x,v,λ,¯t,¯s)−L(y,v,λ,t,¯¯s)] ≥1, which implies that
L(x, v, λ,¯t,¯s)≥L(y, v, λ,¯t,¯s).
In view of the dual feasibility of (y, v, λ, ν, ν0, Jν0, Kν\ν0,t,¯¯s) and (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 simplifies to f(x)−λg(x)≥0. Hence,ϕ(x) =f(x)/g(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, v, λ, ν, ν0, Jν0, Kν\ν0, ¯t,s)¯ of (DI), the conditions specified in Theorem 3.1 are satisfied. Then there exist v∗, λ∗, ν∗, ν0∗, Jν∗
0, Kν∗\ν∗
0,t¯∗, and s¯∗ such that (x∗, v∗, λ∗, ν∗, ν0∗, Jν∗
0, Kν∗\ν0∗,t¯∗,¯s∗) is an optimal solution of (DI) andϕ(x∗) =λ∗.
Proof. Sincex∗is a normal optimal solution of (P), by Theorem 2.1, there existv∗, λ∗, ν∗, ν0∗, Jν∗
0, Kν∗\ν∗
0,¯t∗, and ¯s∗such thatw∗ ≡(x∗, v∗, λ∗, ν∗, ν0∗, Jν∗
0, Kν∗\ν∗
0,t¯∗,¯s∗) is a feasible solution of (DI) and ϕ(x∗) = λ∗. If w∗ were not optimal, then there would exist a feasible solution (˜x,v,˜ ˜λ,˜ν,ν˜0, Jν˜0, Kν\˜ ν˜0,¯˜t,s)¯˜ of (DI) such thatϕ(x∗) =λ∗<˜λ, which contradicts Theorem 3.1. Therefore, we conclude that w∗ 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), let (˜x,˜v,˜λ,ν,˜ ν˜0, Jν˜0, Kν\˜˜ ν0,¯˜t,s)¯˜ be an optimal solution of (DI), and assume that the function z → L(z,˜v,˜λ,t,¯˜¯˜s) is strictly (α, β, γ, η,0, θ)- pseudoinvex at x˜ and γ(x∗,x)˜ > 0. Then x˜ = x∗, that is, x˜ is an optimal solution of (P) and ϕ(x∗) = ˜λ.
Proof. (a): Suppose to the contrary that ˜x6=x∗. By Theorem 2.1, there exist v∗, λ∗, ν∗, ν0∗, Jν∗
0, Kν∗\ν∗
0,t¯∗, and s¯∗ such that (x∗, 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 (withxreplaced byx∗and (y, v, λ, ν, ν0, Jν0, Kν\ν0,
¯t,s) by (˜¯ x,v,˜ λ,˜ ν,˜ ν˜0, Jν˜0, Kν\˜˜ ν0,t,¯˜¯˜s)) and using the condition set forth above, we arrive at the strict inequalityf(x∗)−˜λg(x∗)>0, and so ϕ(x∗)>λ, which˜ contradicts the fact that ϕ(x∗) =λ∗ ≤˜λ. Therefore, we conclude that ˜x=x∗ and ϕ(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)-invexity conditions under which duality can be established.
Consider the following two problems:
(DII) sup
(y,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) = 0,
(4.2) f(y)−λg(y)≥0,
(4.3) Gjm(y, tm)≥0, m∈ν0, (4.4) vmHkm(y, sm)≥0, m∈ν\ν0;
( ˜DII) sup
(y,v,λ,ν,ν0,Jν0,Kν\ν0,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).
Theorem4.1 (Weak Duality). Letxandw≡(y, 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 five sets of hypotheses is satisfied:
(a) (i) z→f(z)−λg(z)is(α, β,¯γ, η,ρ, θ)-pseudoinvex at¯ yand¯γ(x, y)>0;
(ii) for eachm∈ν0, z→Gjm(z, tm) is(α, β,γˆm, η,ρˆm, θ)-quasiinvex at y;
(iii) for each m ∈ν\ν0, z → vmHkm(z, sm) is (α, β,γ˘m, η,ρ˘m, θ)-quasi- invex aty;
(iv) ¯ρ(x, y) +Pν0
m=1vmρˆm(x, y) +Pν
m=ν0+1ρ˘m(x, y)≥0;
(b) (i) z→f(z)−λg(z)is(α, β,¯γ, η,ρ, θ)-pseudoinvex at¯ yand¯γ(x, y)>0;
(ii) z→Pν0
m=1vmGjm(z, tm) is (α, β,γ, η,ˆ ρ, θ)-quasiinvex atˆ y;
(iii) for each m ∈ν\ν0, z → vmHkm(z, sm) is (α, β,γ˘m, η,ρ˘m, θ)-quasi- invex aty;
(iv) ¯ρ(x, y) + ˆρ(x, y) +Pν
m=ν0+1ρ˘m(x, y)≥0;
(c) (i) z→f(z)−λg(z)is(α, β,¯γ, η,ρ, θ)-pseudoinvex at¯ yand¯γ(x, y)>0;
(ii) for each m∈ν0, z→Gjm(z, tm) is (α, β,γ, η,ˆ ρˆm, θ)-quasiinvex at y;
(iii) z→Pν
m=ν0+1vmHkm(z, sm) is(α, β,˘γ, η,ρ, θ)-quasiinvex at˘ y;
(iv) ¯ρ(x, y) +Pν0
m=1vmρˆm(x, y) + ˘ρ(x, y)≥0;
(d) (i) z→f(z)−λg(z)is(α, β,¯γ, η,ρ, θ)-pseudoinvex at¯ yand¯γ(x, y)>0;
(ii) z→Pν0
m=1vmGjm(z, tm) is (α, β,γ, η,ˆ ρ, θ)-quasiinvex atˆ y;
(iii) z→Pν
m=ν0+1vmHkm(z, sm) is(α, β,˘γ, η,ρ, θ)-quasiinvex at˘ y;
(iv) ¯ρ(x, y) + ˆρ(x, y) + ˘ρ(x, y)≥0;
(e) (i) z→f(z)−λg(z)is(α, β,¯γ, η,ρ, θ)-pseudoinvex at¯ yand¯γ(x, y)>0;
(ii) z → Pν0
m=1vmGjm(z, tm) +Pν
m=ν0+1vmHkm(z, sm) is (α, β,γ, η,ˆ ˆ
ρ, θ)-quasiinvex at y;
(iii) ¯ρ(x, y) + ˆρ(x, y)≥0.
Then ϕ(x)≥λ.
Proof. (a): From the primal feasibility ofxand (4.3) we see thatGjm(x, tm)
≤0≤Gjm(y, tm) for each m∈ν0, and hence 1
α(x, y)ˆγm(x, y)
eα(x,y)[Gjm(x,tm)−Gjm(y,tm)]−1
≤0, which in view of (ii) implies that
1 β(x, y)
∇Gjm(y, tm),eβ(x,y)η(x,y)−1
≤ −ˆρm(x, y)kθ(x, y)k2. Asvm >0 for each m∈ν0, the above inequalities yield
(4.5) 1 β(x, y)
DXν0
m=1
vm∇Gjm(y, tm),eβ(x,y)η(x,y)−1E
≤ −
ν0
X
m=1
vmρˆm(x, y)kθ(x, y)k2. Similarly, from the primal feasibility ofx, (4.4), and (iii) we deduce that (4.6)
1 β(x, y)
D Xν
m=ν0+1
vm∇Hkm(y, sm),eβ(x,y)η(x,y)−1E
≤ −
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),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
≥0.
Since ¯γ(x, y)>0, this inequality implies that
f(x)−λg(x)≥f(y)−λg(y),
and therefore using (4.2) we have the desired inequality ϕ(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 five sets of conditions set forth in Theorem 4.1 is satisfied for all feasible solutions of (DII). Then there exist v∗, λ∗, ν∗, ν0∗, Jν0∗, Kν∗\ν∗
0,¯t∗, and s¯∗ such that (x∗, v∗, λ∗, ν∗, ν0∗, Jν0∗, 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, v,˜ λ,˜ ν,˜ ν˜0, J˜ν0, K˜ν\˜ν0, ¯˜t, s)¯˜ be an optimal solution of (DII), and assume that any one of the five sets of conditions specified in Theorem 4.1 is satisfied and that the function z → f(z)−˜λg(z) is strictly (α, β,¯γ, η,ρ, θ)-pseudoinvex at¯ x. Then˜ x˜ =x∗, that is, x˜ is an optimal solu- tion of (P), and ϕ(x∗) = ˜λ.
Proof. (a): Suppose to the contrary that ˜x 6= x∗. By Theorem 2.1, there existv∗, λ∗, ν∗, ν0∗, Jν∗
0, Kν∗\ν∗
0,¯t∗, and ¯s∗such that (x∗, v∗, λ∗, ν∗, ν0∗, Jν∗
0, Kν∗\ν0∗,t¯∗,¯s∗) is a feasible solution of (DII) andϕ(x∗) =λ∗. Now proceeding as in the proof of Theorem 4.1 (with x replaced byx∗ and wby ˜w) and using our strict (α, β,¯γ, η,ρ, θ)-pseudoinvexity assumption, we arrive at the strict¯ inequality f(x∗)−λg(x˜ ∗) > 0 and so 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, 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 five sets of hypotheses is satisfied:
(a) (i) z → f(z)−λg(z) is prestrictly (α, β,γ, η,¯ ρ, θ)-quasiinvex at¯ y and
¯
γ(x, y)>0;
(ii) for eachm∈ν0, z→Gjm(z, tm)is(α, β,γˆm, η,ρˆm, θ)-quasiinvex at y;
(iii) for each m ∈ν\ν0, z → vmHkm(z, sm) is (α, β,γ˘m, η,ρ˘m, θ)-quasi- invex aty;
(iv) ¯ρ(x, y) +Pν0
m=1vmρˆm(x, y) +Pν
m=ν0+1ρ˘m(x, y)>0;
(b) (i) z → f(z)−λg(z) is prestrictly (α, β,γ, η,¯ ρ, θ)-quasiinvex at¯ y and
¯
γ(x, y)>0;
(ii) z→Pν0
m=1vmGjm(z, tm) is (α, β,γ, η,ˆ ρ, θ)-quasiinvex atˆ y;
(iii) for each m ∈ν\ν0, z → vmHkm(z, sm) is (α, β,γ˘m, η,ρ˘m, θ)-quasi- invex aty;
(iv) ¯ρ(x, y) + ˆρ(x, y) +Pν
m=ν0+1ρ˘m(x, y)>0;
(c) (i) z → f(z)−λg(z) is prestrictly (α, β,γ, η,¯ ρ, θ)-quasiinvex at¯ y and
¯
γ(x, y)>0;
(ii) for eachm∈ν0, z→Gjm(z, tm)is(α, β,γˆm, η,ρˆm, θ)-quasiinvex at y;
(iii) z→Pν
m=ν0+1vmHkm(z, sm) is(α, β,˘γ, η,ρ, θ)-quasiinvex at˘ y;
(iv) ¯ρ(x, y) +Pν0
m=1vmρˆm(x, y) + ˘ρ(x, y)>0;
(d) (i) z → f(z)−λg(z) is prestrictly (α, β,γ, η,¯ ρ, θ)-quasiinvex at¯ y and
¯
γ(x, y)>0;
(ii) z→Pν0
m=1vmGjm(z, tm) is (α, β,γ, η,ˆ ρ, θ)-quasiinvex atˆ y;
(iii) z→Pν
m=ν0+1vmHkm(z, sm) is(α, β,˘γ, η,ρ, θ)-quasiinvex at˘ y;
(iv) ¯ρ(x, y) + ˆρ(x, y) + ˘ρ(x, y)>0;
(e) (i) z → f(z)−λg(z) is prestrictly (α, β,γ, η,¯ ρ, θ)-quasiinvex at¯ y and
¯
γ(x, y)>0;
(ii) z → Pν0
m=1vmGjm(z, tm) +Pν
m=ν0+1vmHkm(z, sm) is (α, β,γ, η,ˆ ˆ
ρ, θ)-quasiinvex at y;
(iii) ¯ρ(x, y) + ˆρ(x, y)>0;
Then ϕ(x)≥λ.
Proof. (a) : Because of our assumptions specified 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),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
≥0.
Since ¯γ(x, y) > 0, it follows from this inequality that f(x)−λg(x) ≥f(y)− λg(y)≥0, where the second inequality follows from (4.2). Therefore, we have the desired inequality 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 five sets of conditions set forth in
Theorem 4.4 is satisfied for all feasible solutions of (DII). Then there exist v∗, λ∗, ν∗, ν0∗, Jν∗
0, Kν∗\ν∗
0,¯t∗, and s¯∗ such that (x∗, 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,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 five sets of conditions set forth in Theorem 4.4 is satisfied for the feasible solutionw˜ of (DII) and that the func- tion z → f(z)−λg(z)˜ is (α, β,¯γ, η,ρ, θ)-quasiinvex at¯ x. Then˜ x˜ = x∗ and ϕ(x∗) = ˜λ.
Proof. The proof is similar to that of Theorem 4.3.
Theorem 4.7 (Weak Duality). Let x and(y, 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 satisfied:
(a) (i) z → f(z)−λg(z) is prestrictly (α, β,γ, η,¯ ρ, θ)-quasiinvex at¯ y and
¯
γ(x, y)>0;
(ii) for each m ∈ ν0, z → Gjm(z, tm) is strictly (α, β,γˆm, η,ρˆm, θ)- pseudoinvex at y;
(iii) for each m ∈ν\ν0, z → vmHkm(z, sm) is (α, β,γ˘m, η,ρ˘m, θ)-quasi- invex aty;
(iv) ¯ρ(x, y) +Pν0
m=1vmρˆm(x, y) +Pν
m=ν0+1ρ˘m(x, y)≥0;
(b) (i) z → f(z)−λg(z) is prestrictly (α, β,γ, η,¯ ρ, θ)-quasiinvex at¯ y and
¯
γ(x, y)>0;
(ii) z→Pν0
m=1vmGjm(z, tm)is strictly(α, β,γ, η,ˆ ρ, θ)-pseudoinvex atˆ y;
(iii) for each m ∈ν\ν0, z → vmHkm(z, sm) is (α, β,γ˘m, η,ρ˘m, θ)-quasi- invex aty;
(iv) ¯ρ(x, y) + ˆρ(x, y) +Pν
m=ν0+1ρ˘m(x, y)≥0;
(c) (i) z → f(z)−λg(z) is prestrictly (α, β,γ, η,¯ ρ, θ)-quasiinvex at¯ y and
¯
γ(x, y)>0;
(ii) for eachm∈ν0, z→Gjm(z, tm) is(α, β,ˆγm, η,ρˆm, θ)-quasiinvex at y;
(iii) for eachm∈ν\ν0, z →vmHkm(z, sm)is strictly(α, β,γ˘m, η,ρ˘m, θ)- pseudoinvex at y;
(iv) ¯ρ(x, y) +Pν0
m=1vmρˆm(x, y) +Pν
m=ν0+1ρ˘m(x, y)≥0;
(d) (i) z → f(z)−λg(z) is prestrictly (α, β,γ, η,¯ ρ, θ)-quasiinvex at¯ y and
¯
γ(x, y)>0;
(ii) for eachm∈ν0, z→Gjm(z, tm) is(α, β,ˆγm, η,ρˆm, θ)-quasiinvex at y;
(iii) z →Pν
m=ν0+1vmHkm(z, sm) is strictly (α, β,γ, η,˘ ρ, θ)-pseudoinvex˘ at y;
(iv) ¯ρ(x, y) +Pν0
m=1vmρˆm(x, y) + ˘ρ(x, y)≥0;
(e) (i) z → f(z)−λg(z) is prestrictly (α, β,γ, η,¯ ρ, θ)-quasiinvex at¯ y and
¯
γ(x, y)>0;
(ii) z→Pν0
m=1vmGjm(z, tm) is strictly (α, β,γ, η,ˆ ρ, θ)-pseudoinvex atˆ y;
(iii) z→Pν
m=ν0+1vmHkm(z, sm) is(α, β,˘γ, η,ρ, θ)-quasiinvex at˘ y;
(iv) ¯ρ(x, y) + ˆρ(x, y) + ˘ρ(x, y)≥0;
(f) (i) z → f(z)−λg(z) is prestrictly (α, β,γ, η,¯ ρ, θ)-quasiinvex at¯ y and
¯
γ(x, y)>0;
(ii) z→Pν0
m=1vmGjm(z, tm) is (α, β,γ, η,ˆ ρ, θ)-quasiinvex atˆ y;
(iii) z →Pν
m=ν0+1vmHkm(z, sm) is strictly (α, β,γ, η,˘ ρ, θ)-pseudoinvex˘ at y;
(iv) ¯ρ(x, y) + ˆρ(x, y) + ˘ρ(x, y)≥0;
(g) (i) z → f(z)−λg(z) is prestrictly (α, β,γ, η,¯ ρ, θ)-quasiinvex at¯ y and
¯
γ(x, y)>0;
(ii) z→Pν0
m=1vmGjm(z, tm) +Pν
m=ν0+1vmHkm(z, sm) is strictly(α, β, ˆ
γ, η,ρ, θ)-pseudoinvex atˆ y;
(iii) ¯ρ(x, y) + ˆρ(x, y)≥0.
Then ϕ(x)≥λ.
Proof. (a): Suppose to the contrary that ϕ(x) < λ. This means that f(x)−λg(x)<0, and hence
1
α(x, y)¯γ(x, y)
eα(x,y){f(x)−λg(x)−[f(y)−λg(y)]}−1
<0, which in view of (i) implies
(4.7) 1
β(x, y)
∇f(y)−λ∇g(y),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 Gjm(x, tm) ≤ 0 ≤ Gjm(y, tm) for eachm∈ν0, and hence
1
α(x, y)ˆγm(x, y)
eα(x,y)[Gjm(x,tm)−Gjm(y,tm)]−1
≤0, which in view of (ii) implies that
1 β(x, y)
∇Gjm(y, tm),eβ(x,y)η(x,y)−1
<−ˆρm(x, y)kθ(x, y)k2.
Asvm >0 for each m∈ν0, the above inequalities yield (4.8)
1 β(x, y)
DXν0
m=1
vm∇Gjm(y, tm),eβ(x,y)η(x,y)−1E
<−
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),eβ(x,y)η(x,y)−1
>−¯ρ(x, y)kθ(x, y)k2, which contradicts (4.7). Therefore, we conclude thatϕ(x)≥λ.
(b)–(g): The proofs are similar to that of part (a).
Theorem 4.8 (Strong Duality). Let x∗ be a normal optimal solution of (P) and assume that any one of the seven sets of conditions set forth in Theorem 4.7 is satisfied for all feasible solutions of (DII). Then there exist v∗, λ∗, ν∗, ν0∗, Jν0∗, Kν∗\ν∗0,¯t∗, and s¯∗ such that (x∗, v∗, λ∗, ν∗, ν0∗, Jν0∗, 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 4.9 (Strict Converse Duality). Let x∗ be a normal optimal so- lution of (P), let w˜ ≡(˜x,˜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 seven sets of conditions set forth in Theorem 4.7 is satisfied for the feasible solution w˜ of (DII), and the func- tion z → f(z)−λg(z)˜ is (α, β,¯γ, η,ρ, θ)-quasiinvex at¯ x. Then˜ x˜ = x∗ and ϕ(x∗) = ˜λ.
Proof. (a): By Theorem 2.1, there exist v∗, λ∗, ν∗, ν0∗, Jν∗
0, Kν∗\ν∗
0,¯t∗, and s¯∗ such that (x∗, v∗, λ∗, ν∗, ν0∗, Jν0∗, Kν∗\ν∗
0,¯t∗,s¯∗) is a feasible solution of (DII) andϕ(x∗) =λ∗. Suppose to the contrary that ˜x6=x∗. From the primal feasibility of x∗ and (4.3) we see thatGjm(x∗, t∗m)≤0≤Gjm(˜x,t˜m) for each m∈ν0, and hence
1
α(x∗,x)˜ γˆm(x∗,x)˜
eα(x∗,˜x)[Gjm(x∗,t∗m)−Gjm(˜x,˜tm)]−1
≤0, which in view of (ii) implies that
1 β(x∗,x)˜
∇Gjm(˜x,˜tm),eβ(x∗,˜x)η(x∗,˜x)−1
<−ρˆm(x∗,x)kθ(x˜ ∗,x)k˜ 2. As ˜vm >0 for each m∈ν˜0, the above inequalities yield
(4.9) 1 β(x∗,x)˜
DXν˜0
m=1
˜
vm∇Gjm(˜x,˜tm),eβ(x∗,˜x)η(x∗,˜x)−1E
<−
˜ ν0
X
m=1
˜
vmρˆm(x∗,x)kθ(x˜ ∗,x)k˜ 2.
Now, combining (4.1), (4.6) (which is valid for the present case because of our assumptions set forth in (iii)), (4.9), and (iv), we get
1 β(x∗,x)˜
∇f(˜x)−λ∇g(˜˜ x),eβ(x∗,˜x)η(x∗,˜x)−1
>−¯ρ(x∗,x)kθ(x˜ ∗,x)k˜ 2, which by virtue of the (α, β,γ, η,¯ ρ, θ)-quasiinvexity of the function¯ z→f(z)− λg(z) at ˜˜ ximplies that
1
α(x∗,x)˜ γ¯(x∗,x)˜
eα(x∗,˜x){f(x∗)−˜λg(x∗)−[f(˜x)−λg(˜˜ x)]}−1
>0.
Since ¯γ(x∗,x)˜ >0 and f(˜x)−˜λg(˜x) ≥0, this inequality implies that f(x∗)− λg(x˜ ∗)>0, and soϕ(x∗)>λ, which contradicts the fact that˜ ϕ(x∗) =λ∗ ≤λ.˜ Therefore, we conclude that ˜x=x∗ and ϕ(x∗) = ˜λ.
(b)–(g): The proofs are similar to that of part (a).
5. DUALITY MODEL III
In this section, we discuss several families of duality results under various generalized (α, β, γ, η, ρ, θ)-invexity hypotheses imposed on certain combina- tions of the problem functions. This is accomplished by employing a certain partitioning scheme which was originally proposed in [1] for the purpose of constructing generalized dual problems for nonlinear programming problems.
For this we need some additional notation.
Letν0 andνbe integers, with 1≤ν0 ≤ν≤n+1, and let{J0, J1, . . . , JM} and{K0, K1, . . . , KM}be partitions of the setsν0andν\ν0, respectively; thus, Ji ⊆ν0 for each i∈M ∪ {0}, Ji∩Jj =∅ for each i, j ∈M∪ {0} with i6=j, and ∪Mi=0Ji = ν0. Obviously, similar properties hold for {K0, K1, . . . , KM}.
Moreover, if m1 and m2 are the numbers of the partitioning sets of ν0 and ν\ν0, respectively, then M = max{m1, m2} and Ji = ∅ or Ki = ∅ for i >
min{m1, m2}.
In addition, we use the real-valued functions z → Φ(z, v, λ, ¯t, s), and¯ z → Λτ(z, v,¯t,s) defined, for fixed¯ v, λ, ν, ν0, Jν0, Kν\ν0,t, and ¯¯ s, on X as follows:
Φ(z, v, λ,¯t,s) =¯ f(z)−λg(z) + X
m∈J0
vmGjm(z, tm) + X
m∈K0
vmHkm(z, sm), Λτ(z, v,¯t,s) =¯ X
m∈Jτ
vmGjm(z, tm) + X
m∈Kτ
vmHkm(z, sm), τ ∈M∪ {0}.
Making use of the sets and functions defined above, we can state our general duality models as follows:
(DIII) sup
(y,v,λ,ν,ν0,Jν0,Kν\ν0,¯t,¯s)∈H
λ subject to
(5.1) ∇f(y)−λ∇g(y) +
ν0
X
m=1
vm∇Gjm(y, tm) +
ν
X
m=ν0+1
vm∇Hkm(y, sm) = 0, (5.2) f(y)−λg(y) + X
m∈J0
vmGjm(y, tm) + X
m∈K0
vmHkm(y, sm)≥0,
(5.3) X
m∈Jτ
vmGjm(y, tm) + X
m∈Kτ
vmHkm(y, sm)≥0, τ ∈M;
( ˜DIII) sup
(y,v,λ,ν,ν0,Jν0,Kν\ν0,¯t,¯s)∈H
λ subject to (3.3), (5.2), and (5.3).
The remarks and observations made earlier about the relationships be- tween (DI) and ( ˜DI) are, of course, also valid for (DIII) and ( ˜DIII).
The next two theorems show that (DIII) is a dual problem for (P).
Theorem 5.1 (Weak Duality). Let x and w ≡ (y, v, λ, ν, ν0, Jν0, Kν\ν0,
¯t,s)¯ be arbitrary feasible solutions of (P) and (DIII), respectively, and assume that any one of the following four sets of hypotheses is satisfied:
(a) (i) z→Φ(z, v, λ,¯t,s)¯ is(α, β,γ, η,¯ ρ, θ)-pseudoinvex at¯ yandγ¯(x, y)>0;
(ii) for each τ∈M , z→Λτ(z, v,¯t,s)¯ is(α, β,γ˜τ, η,ρ˜τ, θ)-quasiinvex at y;
(iii) ¯ρ(x, y) +PM
τ=1ρ˜τ(x, y)≥0;
(b) (i) z→Φ(z, v, λ,¯t,s)¯ is prestrictly (α, β,γ, η,¯ ρ, θ)-quasiinvex at¯ y and
¯
γ(x, y)>0;
(ii) for each τ∈M , z→Λτ(z, v,¯t,s)¯ is(α, β,γ˜τ, η,ρ˜τ, θ)-quasiinvex at y;
(iii) ¯ρ(x, y) +PM
τ=1ρ˜τ(x, y)>0;
(c) (i) z→Φ(z, v, λ,¯t,s)¯ is prestrictly (α, β,γ, η,¯ ρ, θ)-quasiinvex at¯ y and
¯
γ(x, y)>0;
(ii) for each τ ∈ M , z → Λτ(z, v,¯t,¯s) is strictly (α, β,˜γτ, η,ρ˜τ, θ)- pseudoinvex at y;
(iii) ¯ρ(x, y) +PM
τ=1ρ˜τ(x, y)≥0;
(d) (i) z→Φ(z, v, λ,¯t,s)¯ is(α, β,¯γ, η,ρ, θ)-quasiinvex at¯ yand¯γ(x, y)>0;
(ii) for each τ ∈ M1, z → Λτ(z, v,¯t,s)¯ is (α, β,˜γτ, η,ρ˜τ, θ)-quasiinvex at y, and for each τ ∈ M2 6= ∅, z → Λτ(z, v,¯t,s)¯ is strictly (α, β,˜γτ, η,ρ˜τ, θ)-pseudoinvex at y, where {M1, M2} is a partition of M;
(iii) ¯ρ(x, y) +PM
τ=1ρ˜τ(x, y)≥0.
Then ϕ(x)≥λ.
Proof. (a): It is clear that (5.1) can be expressed as follows:
(5.4) ∇f(y)−λ∇g(y) + X
m∈J0
vm∇Gjm(y, tm) + X
m∈K0
vm∇Hkm(y, sm)
+
M
X
τ=1
h X
m∈Jτ
vm∇Gjm(y, tm) + X
m∈Kτ
vm∇Hkm(y, sm)i
= 0.
Since for each τ ∈M, Λτ(x, v,t,¯¯s) = X
m∈Jτ
vmGjm(x, tm) + X
m∈Kτ
vmHkm(x, sm)
≤0 (by the primal feasibility of x and positivity ofvm, m∈ν0)
≤ X
m∈Jτ
vmGjm(y, tm) + X
m∈Kτ
vmHkm(y, sm) (by (5.3))
= Λτ(y, v,¯t,¯s), it follows that
1
α(x, y)γ˜τ(x, y)
eα(x,y)[Λτ(x,v,¯t,¯s)−Λτ(y,v,¯t,¯s)]−1
≤0, which in view of (ii) implies that
1 β(x, y)
D X
m∈Jτ
vm∇Gjm(y, tm) + X
m∈Kτ
vm∇Hkm(y, sm),eβ(x,y)η(x,y)−1 E
≤ −˜ρτ(x, y)kθ(x, y)k2. Summing over τ, we obtain
(5.5) 1 β(x, y)
DXM
τ=1
h X
m∈Jτ
vm∇Gjm(y, tm)+ X
m∈Kτ
vm∇Hkm(y, sm) i
,eβ(x,y)η(x,y)−1E
≤ −
M
X
τ=1
˜
ρτ(x, y)kθ(x, y)k2. Combining (5.4) and (5.5), and using (iii) we get
1 β(x, y)
D∇f(y)−λ∇g(y) + X
m∈J0
vm∇Gjm(y, tm)
+ X
m∈K0
vm∇Hkm(y, sm),eβ(x,y)η(x,y)−1 E
≥
M
X
τ=1
˜
ρτ(x, y)kθ(x, y)k2
≥ −¯ρ(x, y)kθ(x, y)k2,
which by virtue of (i) implies that 1
α(x, y)γ(x, y)¯
eα(x,y)[Φ(x,v,λ,¯t,¯s)−Φ(y,v,λ,¯t,¯s)]−1
≥0,
Since ¯γ(x, y) > 0, we can deduce from this inequality that Φ(x, v, λ,¯t,s)¯ ≥ Φ(y, v, λ,¯t,s)¯ ≥0, where the second inequality follows from (5.2). Therefore, we have
0≤Φ(x, v, λ,¯t,s)¯
=f(x)−λg(x) + X
m∈J0
vmGjm(x, tm) + X
m∈K0
vmHkm(x, sm)
≤f(x)−λg(x) (by the primal feasibility of x).
Hence,ϕ(x)≥λ.
(b): The proof is similar to that of part (a).
(c) : Suppose to the contrary thatϕ(x)< λ. This givesf(x)−λg(x)<0.
Using this inequality and keeping in mind that vm > 0 for each m ∈ ν0, we have
Φ(x, v, λ,¯t,s)¯
≤f(x)−λg(x) (by the primal feasibility ofx)
<0
≤Φ(y, v, λ,¯t,s) (by (5.2)),¯ and hence
1
α(x, y)γ(x, y)¯
eα(x,y)[Φ(x,v,λ,¯t,¯s)−Φ(y,v,λ,¯t,¯s)]−1
<0, which in view of (i) implies that
(5.6) 1 β(x, y)
D
∇f(y)−λ∇g(y) + X
m∈J0
vm∇Gjm(y, tm)
+ X
m∈K0
vm∇Hkm(y, sm),eβ(x,y)η(x,y)−1 E
≤ −¯ρ(x, y)kθ(x, y)k2. As shown in the proof of part (a), for each τ ∈ M, we have Λτ(x, v,t,¯¯s) ≤ Λτ(y, v,t,¯s), and hence¯
1
α(x, y)γ˜τ(x, y)
eα(x,y)[Λτ(x,v,¯t,¯s)−Λτ(y,v,¯t,¯s)]−1
≤0, which in view of (ii) implies that
1 β(x, y)
D X
m∈Jτ
vm∇Gjm(y, tm) + X
m∈Kτ
vm∇Hkm(y, sm),eβ(x,y)η(x,y)−1 E
<−˜ρτ(x, y)kθ(x, y)k2.
