IN SEMIINFINITE FRACTIONAL PROGRAMMING.
PART I: SUFFICIENT OPTIMALITY CONDITIONS
G.J. ZALMAI
Communicated by the former editorial board
In this paper, we formulate and discuss a fairly large number of sets of global sufficient optimality criteria using various generalized (α, β, γ, η, ρ, θ)-invexity assumptions for a semiinfinite fractional programming problem.
AMS 2010 Subject Classification: 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, sufficient optimality conditions.
1. INTRODUCTION
In this paper, we first propose further generalizations of the new classes of invex functions recently defined in [11], and then, using the new functions, we state and prove a fairly large number of sufficient optimality results 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
REV. ROUMAINE MATH. PURES APPL.,58(2013),1,9–34
Tj and Sk for all x ∈ X, and g(x) >0 for all x satisfying the constraints of (P).
Nonlinear programming problems like (P) but with a finite number of constraints, that is, when the functionsGj are independent oft, and the func- tions Hk are independent of s, are known in the literature of mathematical programming as fractional programming problems. These problems have been the subject of vigorous investigations in the past four decades which have suc- ceeded in producing a phenomenal profusion of publications dealing with var- ious aspects of numerous classes of static and dynamic fractional optimization problems. One of the principal reasons for such a continual immense interest in this particular optimization paradigm appears to be its capability in providing realistically representative models for a number of significant classes of prob- lems in the fields of operations research, management science, and economics.
This feature is primarily due to the fact that in many areas, including re- source allocation, transportation, production planning, sequencing, inventory control, financial planning, maintenance and replacement scheduling, and re- liability assessment, ratios such as profit/capital, profit/revenue, return/cost, return/risk, cost/time, profit/time, etc., can serve as useful measures of system performance. Proper characterization and utilization of these measures often requires optimization of certain ratios which, in turn, gives rise to the formula- tion of fractional programming problems. In noneconomic situations, fractional programming problems have arisen in information theory, stochastic program- ming, numerical analysis, approximation theory, cluster analysis, graph the- ory, multifacility location theory, decomposition of large-scale mathematical programming problems, and goal programming, among others. For compre- hensive surveys and extensive lists of references dealing with several aspects of fractional programming, including modeling properties, actual and potential areas of applications, optimality conditions, duality formulations, sensitivity and stability analysis, and computational algorithms, the reader is referred to [12, 18, 19, 52, 54, 55].
A mathematical programming problem with a finite number of variables and infinitely many constraints is called a semiinfinite programming problem.
Problems of this type have been utilized for the modeling and analysis of a wide range of theoretical as well as concrete, real-world, practical problems. More specifically, semiinfinite programming concepts and techniques have found rel- evance and applications in approximation theory, statistics, game theory, en- gineering design (design of control systems, digital filters, electronic circuits, etc.), boundary value problems, defect minimization for operator equations, ge- ometry, random graphs, graphs related to Newton flows, reliability testing, en- vironmental protection planning, decision making under uncertainty, semidefi-
nite programming, geometric programming, disjunctive programming, optimal control problems, robotics, and continuum mechanics, among others. For a wealth of information pertaining to various aspects of semiinfinite program- ming, including areas of applications, optimality conditions, duality relations, and numerical algorithms, the reader is referred to [15, 23, 24, 27–31, 34–37, 41, 50, 56, 57]. Some relatively more recent applications of semiinfinite pro- gramming to data envelopment are discussed in [38], to anticipatory systems and gene-environment networks in [22, 58–61], to the analysis and implemen- tation of Vasicek-type interest rate models in [16], to an interesting gemstone cutting problem in [62], to infinite kernel learning in [46, 47], and to basket option pricing in [20].
In stark contrast to the case of conventional fractional programming, semiinfinite programming problems with fractional objective functions have not received much attention in the related literature. Some small steps toward ameliorating this situation were taken in the recently published papers [66]
and [67]. In [66], we establish a set of necessary optimality conditions and numerous sets of global sufficient optimality criteria under various generalized (F, β, φ, ρ, θ)-univexity assumptions. In [67], we use these optimality conditions to formulate and discuss several parametric and parameter-free duality models for (P).
In the present study, we shall adopt a Dinkelbach-type approach [21] to formulate and discuss a fairly large number of sets of global sufficient optimality results for (P) using some new classes of generalized invex funstions, namely, (strictly) (α, β, γ, η, ρ, θ)-invex functions, (strictly) (α, β, γ, η, ρ, θ)-pseudoinvex functions, and (prestrictly) (α, β, γ, η, ρ, θ)-quasiinvex functions. The use of these optimality results in the construction and analysis of various parametric and parameter-free duality models for (P) is discussed in [63–65].
The rest of this paper is organized as follows. In Section 2, we present a number of definitions and auxiliary results which will be needed in the se- quel. In Section 3, we begin our discussion of sufficient optimality conditions where we clearly indicate how one can formulate and prove numerous sets of sufficiency criteria under a variety of generalized (α, β, γ, η, ρ, θ)-invexity as- sumptions that are placed on the individual as well as certain combinations of the problem functions. Utilizing a partitioning scheme, in Section 4 we estab- lish several sets of generalized sufficient optimality results each of which is in fact a family of such results whose members can easily be identified by appro- priate choices of certain sets and functions. Finally, in Section 5 we summarize our main results and also point out some further research opportunities arising from certain modifications of the principal problem model considered in this paper.
Evidently, all the sufficient optimality results established in this paper can easily be modified and restated for the following classes of nonlinear pro- gramming problems, which are special cases of (P):
(P1) Minimize
x∈F
f(x),
whereF (assumed to be nonempty) is the feasible set of (P), that is,
F={x∈X:Gj(x, t)≤0 for allt∈Tj, j ∈q, Hk(x, s) = 0 for alls∈Sk, k∈r};
(P2) Minimize
x∈G
f(x) g(x); (P3) Minimize
x∈G
f(x), where
G={x∈X: ˜Gj(x)≤0, j∈q, H˜k(x) = 0, k∈r}.
Here X, f, andg are as defined in the description of (P), and ˜Gj, j ∈q, and H˜k, k ∈r, are real-valued functions defined onX.
2. PRELIMINARIES
In this section we recall, for convenience of reference, the definitions of certain new classes of generalized invex functions which will be needed in the sequel. These functions were recently introduced by Antczak [11] which are certain extensions and variants of the invex functions originally introduced by Hanson [32]. For brief accounts of the evolution of these new classes of generalized convex functions as well as their relevance and applicability to optimality conditions and duality relations for various classes of mathematical programming problems, the reader is referred to [1–11, 40, 43, 53]. We begin by defining an invex function which has been instrumental in creating a vast array of interesting and important classes of generalized convex functions.
Definition 2.1. Let f be a differentiable real-valued function defined on Rn. Then f is said to be η-invex (invex with respect to η) at y if there exists a function η:Rn×Rn→Rn such that for eachx∈Rn,
f(x)−f(y)≥ h∇f(y), η(x, y)i,
where ∇f(y) = (∂f(y)/∂y1, ∂f(y)/∂y2, . . . , ∂f(y)/∂yn) is the gradient off at y, andha, bi denotes the inner product of the vectors aand b; f is said to be η-invex on Rn if the above inequality holds for allx, y∈Rn.
From this definition it is clear that every differentiable real-valued convex function is invex with η(x, y) = x−y. This generalization of the concept of convexity was originally proposed by Hanson [32] who showed that for a nonlinear programming problem of the form
Minimizef(x) subject to gi(x)≤0, i∈m, x∈Rn,
where the differentiable functionsf, gi:Rn→R, i∈m, are invex with respect to the same function η :Rn×Rn → Rn, the Karush-Kuhn-Tucker necessary optimality conditions are also sufficient. The terminvex (forinvariant convex) was coined by Craven [17] to signify the fact that the invexity property, unlike convexity, remains invariant under bijective coordinate transformations.
In a similar manner, one can readily defineη-pseudoinvex andη-quasiinvex functions as generalizations of differentiable pseudoconvex and quasiconvex functions.
The notion of invexity has been generalized in several directions. For our present purposes, we shall need the recent extension of invexity, namely, B −(˜p,r)-invexity defined in [11].˜ Below, we reproduce the definitions of B−(˜p,r)-invex,˜ B−(˜p,r)-pseudoinvex, and˜ B−(˜p,r)-quasiinvex functions.˜
Let f be a differentiable real-valued function defined on X.
Definition 2.2 [11]. The function f is said to be B −(˜p,r)-invex with˜ respect to η and b at x∗ ∈ X if there exist a function η : X×X → Rn, a functionb:X×X→R+≡[0,∞), and real numbers ˜r and ˜psuch that for all x∈X,
1
˜
rb(x, x∗)
er[f(x)−f(x˜ ∗)]−1
≥ 1
˜ p
∇f(x∗),epη(x,x˜ ∗)−1
for ˜p6= 0 and ˜r6= 0, 1
˜
rb(x, x∗)
er[f(x)−f(x˜ ∗)]−1
≥
∇f(x∗), η(x, x∗)
for ˜p= 0 and ˜r 6= 0, b(x, x∗)[f(x)−f(x∗)]≥ 1
˜ p
∇f(x∗),epη(x,x˜ ∗)−1
for ˜p6= 0 and ˜r= 0, b(x, x∗)[f(x)−f(x∗)]≥
∇f(x∗), η(x, x∗)
for ˜p= 0 and ˜r = 0, where
epη(x,x˜ ∗)−1
≡ epη˜ 1(x,x∗)−1, . . . ,epη˜ n(x,x∗)−1 .
Definition 2.3 [11]. The function f is said to be B−(˜p,r)-pseudoinvex˜ with respect toη and b atx∗ ∈X if there exist a functionη :X×X →Rn, a functionb:X×X→R+, and real numbers ˜r and ˜p such that for allx∈X,
1
˜ p
∇f(x∗),epη(x,x˜ ∗)−1
≥0 ⇒ 1
˜
rb(x, x∗)
e˜r[f(x)−f(x∗)]−1
≥0 for ˜p6= 0 and ˜r6= 0,
∇f(x∗), η(x, x∗)
≥0⇒ 1
˜
rb(x, x∗)
er[f(x)−f(x˜ ∗)]−1
≥0 for ˜p= 0 and ˜r6= 0, 1
˜ p
∇f(x∗),epη(x,x˜ ∗)−1
≥0 ⇒ b(x, x∗)[f(x)−f(x∗)]≥0 for ˜p6= 0 and ˜r= 0, ∇f(x∗), η(x, x∗)
≥0 ⇒ b(x, x∗)[f(x)−f(x∗)]≥0 for ˜p= 0 and ˜r= 0.
Definition 2.4 [11]. The function f is said to be B −(˜p,r)-quasiinvex˜ with respect toη and b atx∗ ∈X if there exist a functionη :X×X →Rn, a functionb:X×X→R+, and real numbers ˜r and ˜p such that for allx∈X,
1
˜
rb(x, x∗)
e˜r[f(x)−f(x∗)]−1
≤0 ⇒ 1
˜ p
∇f(x∗),epη(x,x˜ ∗)−1
≤0 for ˜p6= 0 and ˜r6= 0,
1
˜
rb(x, x∗)
e˜r[f(x)−f(x∗)]−1
≤0⇒
∇f(x∗), η(x, x∗)
≤0 for ˜p= 0 and ˜r6= 0, b(x, x∗)[f(x)−f(x∗)]≤0 ⇒ 1
˜ p
∇f(x∗),epη(x,x˜ ∗)−1
≤0 for ˜p6= 0 and ˜r= 0, b(x, x∗)[f(x)−f(x∗)]≤0 ⇒
∇f(x∗), η(x, x∗)
≤0 for ˜p= 0 and ˜r= 0.
In this paper, we shall utilize the following slightly modified and more general versions of the above classes of generalized invex functions.
Definition2.5. The functionf is said to be (strictly) (α, β, γ, η, ρ, θ)-invex atx∗ ∈Xif there exist functionsα:X×X →R, β :X×X →R, γ :X×X → R+, η:X×X→Rn, ρ:X×X →R, andθ:X×X →Rn such that for all x∈X(x6=x∗),
1
α(x, x∗)γ(x, x∗)
eα(x,x∗)[f(x)−f(x∗)]−1
(>)≥ 1 β(x, x∗)
∇f(x∗),eβ(x,x∗)η(x,x∗)−1 +ρ(x, x∗)kθ(x, x∗)k2ifα(x, x∗)6= 0 andβ(x, x∗)6= 0 for allx∈X, 1
α(x, x∗)γ(x, x∗)
eα(x,x∗)[f(x)−f(x∗)]−1
(>)≥
∇f(x∗), η(x, x∗)
+ρ(x, x∗)kθ(x, x∗)k2 ifα(x, x∗)6= 0 andβ(x, x∗) = 0 for allx∈X,
γ(x, x∗)[f(x)−f(x∗)](>)≥ 1 β(x, x∗)
∇f(x∗),eβ(x,x∗)η(x,x∗)−1
+ρ(x, x∗)kθ(x, x∗)k2 ifα(x, x∗) = 0 and β(x, x∗)6= 0 for all x∈X, γ(x, x∗)[f(x)−f(x∗)](>)≥
∇f(x∗), η(x, x∗)
+ρ(x, x∗)kθ(x, x∗)k2 ifα(x, x∗) = 0 andβ(x, x∗) = 0 for allx∈X, wherek · k is a norm on Rn.
The function f is said to be (strictly) (α, β, γ, η, ρ, θ)-invex on X if it is (strictly) (α, β, γ, η, ρ, θ)-invex at eachx∗∈X.
Definition 2.6. The function f is said to be (strictly) (α, β, γ, η, ρ, θ)- pseudoinvex 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 allx∈X(x6=x∗),
1 β(x, x∗)
∇f(x∗),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∗)]−1
(>)≥0 ifα(x, x∗)6= 0 and β(x, x∗)6= 0 for all x∈X, ∇f(x∗), η(x, x∗)
≥ −ρ(x, x∗)kθ(x, x∗)k2
⇒ 1
α(x, x∗)γ(x, x∗)
eα(x,x∗)[f(x)−f(x∗)]−1
(>)≥0 ifα(x, x∗)6= 0 and β(x, x∗) = 0 for allx∈X, 1
β(x, x∗)
∇f(x∗),eβ(x,x∗)η(x,x∗)−1
≥ −ρ(x, x∗)kθ(x, x∗)k2
⇒γ(x, x∗)[f(x)−f(x∗)](>)≥0 ifα(x, x∗) = 0 and β(x, x∗)6= 0 for all x∈X, ∇f(x∗), η(x, x∗)
≥ −ρ(x, x∗)kθ(x, x∗)k2
⇒ γ(x, x∗)[f(x)−f(x∗)](>)≥
∇f(x∗), η(x, x∗) ifα(x, x∗) = 0 and β(x, x∗) = 0 for all x∈X.
The function f is said to be (strictly) (α, β, γ, η, ρ, θ)-pseudoinvex onX if it is (strictly) (α, β, γ, η, ρ, θ)-pseudoinvex at eachx∗∈X.
Definition 2.7. The function f is said to be (prestrictly) (α, β, γ, η, ρ, θ)- quasiinvex at x∗ ∈ X if there exist functionsα :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 allx∈X,
1
α(x, x∗)γ(x, x∗)
eα(x,x∗)[f(x)−f(x∗)]−1
(<)≤0
⇒ 1
β(x, x∗)
∇f(x∗),eβ(x,x∗)η(x,x∗)−1
≤ −ρ(x, x∗)kθ(x, x∗)k2 ifα(x, x∗)6= 0 and β(x, x∗)6= 0 for all x∈X,
1
α(x, x∗)γ(x, x∗)
eα(x,x∗)[f(x)−f(x∗)]−1
(<)≤0
⇒
∇f(x∗), η(x, x∗)
≤ −ρ(x, x∗)kθ(x, x∗)k2 ifα(x, x∗)6= 0 and β(x, x∗) = 0 for allx∈X, γ(x, x∗)[f(x)−f(x∗)](<)≤0
⇒ 1
β(x, x∗)
∇f(x∗),eβ(x,x∗)η(x,x∗)−1
≤ −ρ(x, x∗)kθ(x, x∗)k2 ifα(x, x∗) = 0 and β(x, x∗)6= 0 for all x∈X,
γ(x, x∗)[f(x)−f(x∗)](<)≤0 ⇒
∇f(x∗), η(x, x∗)
≤ −ρ(x, x∗)kθ(x, x∗)k2 ifα(x, x∗) = 0 and β(x, x∗) = 0 for all x∈X.
The functionf is said to be (prestrictly) (α, β, γ, η, ρ, θ)-quasiinvex onX if it is (prestrictly) (α, β, γ, η, ρ, θ)-quasiinvex at eachx∗ ∈X.
From the above definitions it is clear that if f is (α, β, γ, η, ρ, θ)-invex at x∗, then it is both (α, β, γ, η, ρ, θ)-pseudoinvex and (α, β, γ, η, ρ, θ)-quasiinvex atx∗, iffis (α, β, γ, η, ρ, θ)-quasiinvex atx∗, then it is prestrictly (α, β, γ, η, ρ, θ)- quasiinvex at x∗, and if f is strictly (α, β, γ, η, ρ, θ)-pseudoinvex at x∗, then it is (α, β, γ, η, ρ, θ)-quasiinvex atx∗.
In the proofs of the sufficient optimality 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, (α, β, γ, η, ρ, θ)-pseudoinvexity (whenα(x, x∗)6= 0 andβ(x, x∗)6=
0 for all x ∈ X) can be defined in the following equivalent way: the function f is said to be (α, β, γ, η, ρ, θ)-pseudoinvex 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 allx∈X,
1
α(x, x∗)γ(x, x∗)
eα(x,x∗)[f(x)−f(x∗)]−1
<0⇒
⇒ 1 β(x, x∗)
∇f(x∗),eβ(x,x∗)η(x,x∗)−1
<−ρ(x, x∗)kθ(x, x∗)k2. The concept of η-invexity has been extended in many ways, and various types of generalized η-invex functions have been utilized for establishing a variety of sufficient optimality criteria and duality relations for several classes of nonlinear programming problems. For more information about invex functions, the reader may consult [13, 14, 25, 26, 33, 42, 44, 51], and for recent surveys of these and other related functions, the reader is referred to [39, 49].
In formulating our sufficient optimality criteria, we have followed as our guide the form and features of the necessary optimality conditions for (P). For a precise statement of this result, the reader is referred to [63].
3. SUFFICIENT OPTIMALITY CONDITIONS
In this section, we present a multitude of sufficiency results in which various generalized (α, β, γ, η, ρ, θ)-invexity assumptions are imposed on the individual as well as certain combinations of the problem functions.
With regard to the choice of the type of generalized invex functions, specified in Definitions 2.5–2.7, to be used in the statements and proofs of the our sufficient optimality results, we shall consistently use the cases in which the functions α and β are nonzero for all (x, y) ∈ X×X. All the sufficiency results established in this paper can be modified, restated, and proved for the other cases in a straightforward manner.
We begin our discussion of sufficiency criteria for (P) with two simple sets of conditions. In the first set only (α, β, γ, η, ρ, θ)-invexity assumptions are imposed on the problem functions, whereas in the second set a Lagrangian- type function is assumed to satisfy a certain (α, β, γ, η, ρ, θ)-pseudoinvexity requirement.
Theorem 3.1. Let x∗ ∈ F, let λ∗ = ϕ(x∗), let the functions f, g, z → Gj(z, t), and z→Hk(z, s) be differentiable at x∗ for allt∈Tj ands∈Sk, j∈ q, k∈r, and assume that there exist integersν0andν, with0≤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 indiceskm, with1≤km ≤r, together withν−ν0 points sm ∈Skm, m∈ν\ν0, and ν real numbers v∗m, with v∗m>0 for m∈ν0, with the property that
(3.1) ∇f(x∗)−λ∗∇g(x∗) +
ν0
X
m=1
vm∗∇Gjm(x∗, tm) +
ν
X
m=ν0+1
vm∗∇Hkm(x∗, sm) = 0.
Assume, furthermore, that either one of the following two sets of conditions holds:
(a) (i) the function z → f(z)−λ∗g(z) is (α, β,γ, η,¯ ρ, θ)-invex at¯ x∗ and
¯
γ(x, x∗)>0 for all x∈F;
(ii) the function z → Gjm(z, tm) is (α, β,γˆm, η,ρˆm, θ)-invex at x∗ for each m∈ν0;
(iii) the functionz→v∗mHkm(z, sm)is(α, β,γ˘m, η,ρ˘m, θ)-invex atx∗ for each m∈ν\ν0;
(iv) ¯ρ(x, x∗)+Pν0
m=1v∗mρˆm(x, x∗)+Pν
m=ν0+1ρ˘m(x, x∗)≥0for allx∈F; (b) the Lagrangian-type function
z→L(z, v∗, λ∗,¯t,s) =¯ f(z)−λ∗g(z)+
ν0
X
m=1
vm∗Gjm(z, tm)+
ν
X
m=ν0+1
v∗mHkm(z, sm) is(α, β, γ, η,0, θ)-pseudoinvex atx∗ andγ(x, x∗)>0for allx∈F, where
¯t≡(t1, t2, . . . , tν0) and s¯≡(sν0+1, sν0+2, . . . , sν).
Then x∗ is an optimal solution of (P).
Proof. (a): Suppose to the contrary thatx∗ is not an optimal solution of (P), and hence for some x∈F, ϕ(x)< ϕ(x∗) =λ∗. This implies that
(3.2) f(x)−λ∗g(x)<0.
Since the exponential functionz→ezis monotonically increasing,α(x, x∗)6= 0, and ¯γ(x, x∗) > 0 for all x ∈ F, it follows from (3.2) (in both cases when α(x, x∗)>0 and α(x, x∗)<0) that the following strict inequality holds:
(3.3) 1
α(x, x∗)γ¯(x, x∗)
eα(x,x∗)[f(x)−λ∗g(x)]−1
<0.
In view of our (α, β, γ, η, ρ, θ)-invexity assumptions in (i)–(iii), we have
(3.4) 1
α(x, x∗)¯γ(x, x∗)
eα(x,x∗){f(x)−λ∗g(x)−[f(x∗)−λ∗g(x∗)]}−1
≥ 1 β(x, x∗)
D∇f(x∗)−λ∗∇g(x∗),eβ(x,x∗)η(x,x∗)−1E
+ ¯ρ(x, x∗)kθ(x, x∗)k2,
(3.5) 1
α(x, x∗)ˆγm(x, x∗)
eα(x,x∗)[Gjm(x,tm)−Gjm(x∗,tm)]−1
≥ 1 β(x, x∗)
D∇Gjm(x∗, tm),eβ(x,x∗)η(x,x∗)−1E
+ ˆρm(x, x∗)kθ(x, x∗)k2, m∈ν0,
(3.6) 1
α(x, x∗)˘γm(x, x∗)
eα(x,x∗)[v∗mHkm(x,sm)−vm∗Hkm(x∗,sm)]−1
≥ 1 β(x, x∗)
D
vm∗∇Hkm(x∗, sm),eβ(x,x∗)η(x,x∗)−1E
+ ˘ρm(x, x∗)kθ(x, x∗)k2, m∈ν\ν0. Multiplying (3.5) by v∗m and then summing over m ∈ν0, summing (3.6) over m∈ν\ν0, and finally adding (3.4) and the resulting inequalities, we get
1 α(x, x∗)
n
¯ γ(x, x∗)
eα(x,x∗){f(x)−λ∗g(x)−[f(x∗)−λ∗g(x∗)]}−1
+
ν0
X
m=1
vm∗γˆm(x, x∗)
eα(x,x∗)[Gjm(x,tm)−Gjm(x∗,tm)]−1
+
ν
X
m=ν0+1
˘
γm(x, x∗)
eα(x,x∗)[v∗mHkm(x,sm)−v∗mHkm(x∗,sm)]−1 o
≥ 1 β(x, x∗)
D∇f(x∗)−λ∗∇g(x∗) +
ν0
X
m=1
vm∗∇Gjm(x∗, tm)
+
ν
X
m=ν0+1
vm∗∇Hkm(x∗, sm), eβ(x,x∗)η(x,x∗)−1E + h
¯
ρ(x, x∗) +
ν0
X
m=1
v∗mρˆm(x, x∗) +
ν
X
m=ν0+1
˘
ρm(x, x∗) i
kθ(x, x∗)k2. Now using (3.1) and (iv), and noticing thatϕ(x∗) =λ∗; x, x∗∈F, Gjm(x∗, tm)
= 0 for all m ∈ν0, and ˆγm(x, x∗)≥0 for each m∈ν0, we see that the above inequality reduces to
1
α(x, x∗)γ¯(x, x∗)
eα(x,x∗)[f(x)−λ∗g(x)]−1
≥0,
which contradicts (3.3). Therefore, we conclude that x∗ is an optimal solution of (P).
(b) : Letxbe an arbitrary feasible solution of (P). From (3.1) we observe that
1 β(x, x∗)
D
∇f(x∗)−λ∗∇g(x∗) +
ν0
X
m=1
vm∗∇Gjm(x∗, tm)
+
ν
X
m=ν0+1
v∗m∇Hkm(x∗, sm),eβ(x,x∗)η(x,x∗)−1 E
= 0, which in view of our (α, β, γ, η,0, θ)-pseudoinvexity assumption implies that
1
α(x, x∗)γ(x, x∗)
eα(x,x∗)[L(x,v∗,λ∗,¯t,¯s)−L(x∗,v∗,λ∗,¯t,¯s)]−1
≥0.
We need to consider two cases: α(x, x∗) >0 and α(x, x∗) < 0. If we assume that α(x, x∗) > 0 and recall that γ(x, x∗) > 0, then the above inequality becomes
eα(x,x∗)[L(x,v∗,λ∗,¯t,¯s)−L(x∗,v∗,λ∗,¯t,¯s)] ≥1, which implies that
L(x, v∗, λ∗,¯t,s)¯ ≥L(x∗, v∗, λ∗,¯t,s).¯
Because x∗ ∈ F, tm ∈ Tˆjm(x∗), m ∈ ν0, and λ∗ = ϕ(x∗), the right-hand side of the above inequality is 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)−λ∗g(x) ≥0, and hence ϕ(x∗) = λ∗ ≤ϕ(x). Sincex ∈F was arbitrary, we conclude from this inequality thatx∗ is an optimal solution of (P).
If we assume that α(x, x∗)<0, we arrive at the same conclusion.
Theorem 3.2. Let x∗ ∈ F, let λ∗ =ϕ(x∗), let the functions f, g, z → Gj(z, t), and z→Hk(z, s) be differentiable at x∗ for allt∈Tj ands∈Sk, j∈ q, k∈r, and assume that there exist integersν0andν, with0≤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 indiceskm, with1≤km ≤r, together withν−ν0 points sm ∈Skm, m∈ν\ν0, and ν real numbers v∗m with v∗m >0 for m ∈ν0, such that (3.1) holds. Assume, furthermore, that any one of the following five sets of hypotheses is satisfied:
(a) (i) z→f(z)−λ∗g(z)is(α, β,γ, η,¯ ρ, θ)-pseudoinvex at¯ x∗ and¯γ(x, x∗)>
0 for all x∈F;
(ii) z→Gjm(z, tm) is (α, β,γˆm, η,ρˆm, θ)-quasiinvex at x∗ for each m∈ν0;
(iii) z → vm∗Hkm(z, sm) is (α, β,γ˘m, η,ρ˘m, θ)-quasiinvex at x∗ for each m∈ν\ν0;
(iv) ¯ρ(x, x∗) +Pν0
m=1vm∗ρˆm(x, x∗) +Pν
m=ν0+1ρ˘m(x, x∗)≥0 for all x∈F;
(b) (i) z→f(z)−λ∗g(z)is(α, β,γ, η,¯ ρ, θ)-pseudoinvex at¯ x∗ and¯γ(x, x∗)>
0 for all x∈F; (ii) z→Pν0
m=1vm∗Gjm(z, tm) is (α, β,γ, η,ˆ ρ, θ)-quasiinvex atˆ x∗; (iii) z → vm∗Hkm(z, sm) is (α, β,γ˘m, η,ρ˘m, θ)-quasiinvex at x∗ for each
m∈ν\ν0;
(iv) ¯ρ(x, x∗) + ˆρ(x, x∗) +Pν
m=ν0+1ρ˘m(x, x∗)≥0 for allx∈F;
(c) (i) z→f(z)−λ∗g(z)is(α, β,γ, η,¯ ρ, θ)-pseudoinvex at¯ x∗ and¯γ(x, x∗)>
0 for all x∈F;
(ii) z→Gjm(z, tm) is (α, β,γˆm, η,ρˆm, θ)-quasiinvex at x∗ for each m∈ ν0;
(iii) z→Pν
m=ν0+1vm∗Hkm(z, sm) is(α, β,˘γ, η,ρ, θ)-quasiinvex at˘ x∗; (iv) ¯ρ(x, x∗) +Pν0
m=1vm∗ρˆm(x, x∗) + ˘ρ(x, x∗)≥0 for allx∈F;
(d) (i) z→f(z)−λ∗g(z)is(α, β,γ, η,¯ ρ, θ)-pseudoinvex at¯ x∗ and¯γ(x, x∗)>
0 for all x∈F; (ii) z→Pν0
m=1vm∗Gjm(z, tm) is (α, β,γ, η,ˆ ρ, θ)-quasiinvex atˆ x∗; (iii) z→Pν
m=ν0+1vm∗Hkm(z, sm) is(α, β,˘γ, η,ρ, θ)-quasiinvex at˘ x∗; (iv) ¯ρ(x, x∗) + ˆρ(x, x∗) + ˘ρ(x, x∗)≥0 for allx∈F;
(e) (i) z→f(z)−λ∗g(z)is(α, β,γ, η,¯ ρ, θ)-pseudoinvex at¯ x∗ and¯γ(x, x∗)>
0 for all x∈F; (ii) z→Pν0
m=1v∗mGjm(z, tm)+Pν
m=ν0+1vm∗Hkm(z, sm)is(α, β,γ, η,ˆ ρ, θ)-ˆ quasiinvex at x∗;
(iii) ¯ρ(x, x∗) + ˆρ(x, x∗)≥0 for all x∈F. Then x∗ is an optimal solution of (P).
Proof. Letx be an arbitrary feasible solution of (P).
(a): Suppose to the contrary thatx∗is not an optimal solution of (P). As shown in the proof of part (a) of Theorem 3.1, this supposition leads to (3.3).
Since x ∈F and tm ∈Tˆjm(x∗) for each m ∈ ν0, it is clear that Gjm(x, tm) ≤ 0 =Gjm(x∗, tm), which implies that
1
α(x, x∗)γˆm(x, x∗)
eα(x,x∗)[Gjm(x,tm)−Gjm(x∗,tm)]−1
≤0
because α(x, x∗) 6= 0 and ˆγm(x, x∗)≥ 0 for all x∈ F and m ∈ν0. In view of (ii), this inequality implies that
1 β(x, x∗)
∇Gjm(x∗, tm),eβ(x,x∗)η(x,x∗)−1
≤ −ρˆm(x, x∗)kθ(x, x∗)k2, m∈ν0. Multiplying this inequality by v∗m and then summing overm∈ν0, we get
(3.7) 1
β(x, x∗) DXν0
m=1
vm∗∇Gjm(x∗, tm),eβ(x,x∗)η(x,x∗)−1E
≤ −
ν0
X
m=1
vm∗ρˆm(x, x∗)kθ(x, x∗)k2.
In a similar manner, our assumptions in (iii) lead to the following inequality:
(3.8) 1
β(x, x∗)
D Xν
m=ν0+1
vm∗∇Hkm(x∗, sm),eβ(x,x∗)η(x,x∗)−1 E
≤ −
ν
X
m=ν0+1
˘
ρm(x, x∗)kθ(x, x∗)k2. Now combining (3.1), (3.7), and (3.8), and using (iv), we obtain
1 β(x, x∗)
∇f(x∗)−λ∗∇g(x∗),eβ(x,x∗)η(x,x∗)−1
≥ −ρ(x, x¯ ∗)kθ(x, x∗)k2, which in view of (i) implies that
1
α(x, x∗)¯γ(x, x∗)
eα(x,x∗){f(x)−λ∗g(x)−[f(x∗)−λ∗g(x∗)]}−1
≥0.
Sinceλ∗=ϕ(x∗), this inequality contradicts (3.3), and hence we conclude that x∗ is an optimal solution of (P).
(b)–(e) : The proofs are similar to that of part (a).
Theorem 3.3. Let x∗ ∈ F, let λ∗ =ϕ(x∗), let the functions f g, z → Gj(z, t), and z→Hk(z, s) be differentiable at x∗ for allt∈Tj ands∈Sk, j∈ q, k∈r, and assume that there exist integersν0andν, with0≤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 indiceskm, with1≤km ≤r, together withν−ν0 points sm ∈Skm, m∈ν\ν0, and ν real numbers v∗m with v∗m >0 for m ∈ν0, such that (3.1) holds. Assume, furthermore, that any one of the following five sets of hypotheses is satisfied:
(a) (i) z→f(z)−λ∗g(z)is prestrictly(α, β,¯γ, η,ρ, θ)-quasiinvex at¯ x∗ and
¯
γ(x, x∗)>0 for all x∈F;
(ii) z→Gjm(z, tm) is (α, β,γˆm, η,ρˆm, θ)-quasiinvex at x∗ for each m∈ ν0;
(iii) z → vm∗Hkm(z, sm) is (α, β,γ˘m, η,ρ˘m, θ)-quasiinvex at x∗ for each m∈ν\ν0;
(iv) ¯ρ(x, x∗)+Pν0
m=1v∗mρˆm(x, x∗)+Pν
m=ν0+1ρ˘m(x, x∗)>0for allx∈F; (b) (i) z→f(z)−λ∗g(z)is prestrictly(α, β,¯γ, η,ρ, θ)-quasiinvex at¯ x∗ and
¯
γ(x, x∗)>0 for all x∈F; (ii) z→Pν0
m=1vm∗Gjm(z, tm) is (α, β,γ, η,ˆ ρ, θ)-quasiinvex atˆ x∗; (iii) z → vm∗Hkm(z, sm) is (α, β,γ˘m, η,ρ˘m, θ)-quasiinvex at x∗ for each
m∈ν\ν0;
(iv) ¯ρ(x, x∗) + ˆρ(x, x∗) +Pν
m=ν0+1ρ˘m(x, x∗)>0 for all x∈F;
(c) (i) z→f(z)−λ∗g(z)is prestrictly(α, β,¯γ, η,ρ, θ)-quasiinvex at¯ x∗ and
¯
γ(x, x∗)>0 for all x∈F;
(ii) z→Gjm(z, tm) is (α, β,γˆm, η,ρˆm, θ)-quasiinvex at x∗ for each m∈ ν0;
(iii) z→Pν
m=ν0+1vm∗Hkm(z, sm) is(α, β,˘γ, η,ρ, θ)-quasiinvex at˘ x∗; (iv) ¯ρ(x, x∗) +Pν0
m=1vm∗ρˆm(x, x∗) + ˘ρ(x, x∗)>0 for allx∈F;
(d) (i) z→f(z)−λ∗g(z)is prestrictly(α, β,¯γ, η,ρ, θ)-quasiinvex at¯ x∗ and
¯
γ(x, x∗)>0 for all x∈F; (ii) z→Pν0
m=1vm∗Gjm(z, tm) is (α, β,γ, η,ˆ ρ, θ)-quasiinvex atˆ x∗; (iii) z→Pν
m=ν0+1vm∗Hkm(z, sm) is(α, β,˘γ, η,ρ, θ)-quasiinvex at˘ x∗; (iv) ¯ρ(x, x∗) + ˆρ(x, x∗) + ˘ρ(x, x∗)>0 for allx∈F;
(e) (i) z→f(z)−λ∗g(z)is prestrictly(α, β,¯γ, η,ρ, θ)-quasiinvex at¯ x∗ and
¯
γ(x, x∗)>0 for all x∈F; (ii) z→Pν0
m=1v∗mGjm(z, tm)+Pν
m=ν0+1v∗mHkm(z, sm)is(α, β,ˆγ, η,ρ, θ)-ˆ quasiinvex at x∗;
(iii) ¯ρ(x, x∗) + ˆρ(x, x∗)>0 for all x∈F. Then x∗ is an optimal solution of (P).
Proof. Letx be an arbitrary feasible solution of (P).
(a): Suppose to the contrary that x∗ is not an optimal solution of (P).
As observed in the proof of part (a) of Theorem 3.1, this supposition implies that (3.3) holds. Because of our assumptions specified in (ii) and (iii), (3.7) and (3.8) remain valid for the present case. From (3.1), (3.7), (3.8), and (iv) we deduce that
1 β(x, x∗)
∇f(x∗)−λ∗∇g(x∗),eβ(x,x∗)η(x,x∗)−1
≥
hXν0
m=1
vm∗ρˆm(x, x∗) +
ν
X
m=ν0+1
ˆ
ρm(x, x∗)i
kθ(x, x∗)k2 >−¯ρ(x, x∗)kθ(x, x∗)k2, which in view of (i) implies that
1
α(x, x∗)¯γ(x, x∗)
eα(x,x∗){f(x)−λ∗g(x)−[f(x∗)−λ∗g(x∗)]}−1
≥0.
Since f(x∗)−λ∗g(x∗) = 0, this inequality contradicts (3.3), and hence we conclude that x∗ is an optimal solution of (P).
(b)–(e): The proofs are similar to that of part (a).
Theorem 3.4. Let x∗ ∈ F, let λ∗ = ϕ(x∗), let the functions f g, z → Gj(z, t), and z→Hk(z, s) be differentiable at x∗ for allt∈Tj ands∈Sk, j∈ q, k∈r, and assume that there exist integersν0andν, with0≤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 indiceskm, with1≤km ≤r, together withν−ν0 points sm ∈Skm, m∈ν\ν0, and ν real numbers v∗m with v∗m >0 for m ∈ν0, such that (3.1) holds. Assume, furthermore, that any one of the following seven sets of hypotheses is satisfied:
(a) (i) z→f(z)−λ∗g(z)is prestrictly(α, β,¯γ, η,ρ, θ)-quasiinvex at¯ x∗ and
¯
γ(x, x∗)>0 for all x∈F;
(ii) z → Gjm(z, tm) is strictly (α, β,γˆm, η,ρˆm, θ)-pseudoinvex at x∗ for each m∈ν0;
(iii) z → vm∗Hkm(z, sm) is (α, β,γ˘m, η,ρ˘m, θ)-quasiinvex at x∗ for each m∈ν\ν0;
(iv) ¯ρ(x, x∗)+Pν0
m=1v∗mρˆm(x, x∗)+Pν
m=ν0+1ρ˘m(x, x∗)≥0for allx∈F; (b) (i) z→f(z)−λ∗g(z)is prestrictly(α, β,¯γ, η,ρ, θ)-quasiinvex at¯ x∗ and
¯
γ(x, x∗)>0 for all x∈F; (ii) z → Pν0
m=1vm∗Gjm(z, tm) is strictly (α, β,ˆγ, η,ρ, θ)-pseudoinvex atˆ x∗;
(iii) z → vm∗Hkm(z, sm) is (α, β,γ˘m, η,ρ˘m, θ)-quasiinvex at x∗ for each m∈ν\ν0;
(iv) ¯ρ(x, x∗) + ˆρ(x, x∗) +Pν
m=ν0+1ρ˘m(x, x∗)≥0 for all x∈F;
(c) (i) z→f(z)−λ∗g(z)is prestrictly(α, β,¯γ, η,ρ, θ)-quasiinvex at¯ x∗ and
¯
γ(x, x∗)>0 for all x∈F;
(ii) z→Gjm(z, tm) is (α, β,γˆm, η,ρˆm, θ)-quasiinvex at x∗ for each m∈ ν0;
(iii) z → vm∗Hkm(z, sm) is strictly (α, β,˘γm, η,ρ˘m, θ)-pseudoinvex at x∗ for each m∈ν\ν0;
(iv) ¯ρ(x, x∗)+Pν0
m=1v∗mρˆm(x, x∗)+Pν
m=ν0+1ρ˘m(x, x∗)≥0for allx∈F; (d) (i) z→f(z)−λ∗g(z)is prestrictly(α, β,¯γ, η,ρ, θ)-quasiinvex at¯ x∗ and
¯
γ(x, x∗)>0 for all x∈F;
(ii) z→Gjm(z, tm) is (α, β,γˆm, η,ρˆm, θ)-quasiinvex at x∗ for each m∈ ν0;
(iii) z →Pν
m=ν0+1v∗mHkm(z, sm) is strictly (α, β,γ, η,˘ ρ, θ)-pseudoinvex˘ at x∗;
(iv) ¯ρ(x, x∗) +Pν0
m=1vm∗ρˆm(x, x∗) + ˘ρ(x, x∗)≥0 for allx∈F;
