A new class of generalized functions (d-ρ-η-θ)-type I univex is introduced for a nonsmooth multiobjective programming problem. Based upon these generalized functions, sufficient optimality conditions are established. Weak, strong, converse, and strict converse duality theorems are also derived for Mond-Weir-type multiobjective dual program.

1. Introduction

Generalizations of convexity related to optimality conditions and duality for nonlinear single objective or multiobjective optimization problems have been of much interest in the recent past and thus explored the extent of optimality conditions and duality applicability in mathematical programming problems. Invexity theory was originated by Hanson [1]. Many authors have then contributed in this direction.

For a nondifferentiable multiobjective programming problem, there exists a generalisation of invexity to locally Lipschitz functions with gradients replaced by the Clarke generalized gradient. Zhao [2] extended optimality conditions and duality in nonsmooth scalar programming assuming Clarke generalized subgradients under type I functions. However, Antczak [3] used directional derivative in association with a hypothesis of an invex kind following Ye [4]. On the other hand, Bector et al. [5] generalized the notion of convexity to univex functions. Rueda et al. [6] obtained optimality and duality results for several mathematical programs by combining the concepts of type I and univex functions. Mishra [7] obtained optimality results and saddle point results for multiobjective programs under generalized type I univex functions which were further generalized to univex type I-vector-valued functions by Mishra et al. [8]. Jayswal [9] introduced new classes of generalized α-univex type I vector valued functions and established sufficient optimality conditions and various duality results for Mond-Weir type dual program. Generalizing the work of Antczak [3], recently Nahak and Mohapatra [10] obtained duality results for multiobjective programming problem under (d-ρ-η-θ) invexity assumptions.

In this paper, by combining the concepts of Mishra et al. [8] and Nahak and Mohapatra [10], we introduce a new generalized class of (d-ρ-η-θ)-type I univex functions and establish weak, strong, converse, and strict converse duality results for Mond-Weir type dual.

2. Preliminaries and Definitions

The following convention of vectors in Rn will be followed throughout this paper: x≧y⇔xi≧yi, i=1,2,…,n; x≥y⇔x≧y, x≠y; x>y⇔xi>yi, i=1,2,…,n. Let D be a nonempty subset of Rn, η:D×D→Rn, xo be an arbitrary point of D and h:D→R, ϕ:R→R, b:D×D→R+. Also, we denote R≥p={y:y∈Rp and y≥0} and R≧k={y:y∈Rk and y≧0}.

Definition 2.1 (Ben-Israel and Mond [<xref ref-type="bibr" rid="B11">11</xref>]).

Let D⊆Rn be an invex set. A function h is called preinvex on D with respect to η, if for all x,xo∈D,
(2.1)λh(x)+(1-λ)h(xo)≧h(xo+λη(x,xo)),∀λ∈[0,1].

The function h is said to be locally Lipschitz at xo∈D, if there exists a neighbourhood v(xo) of xo and a constant k>0 such that
(2.2)|h(y)-h(x)|≦k‖y-x‖∀x,y∈v(xo),
where ∥·∥ denotes the euclidean norm. Also, we say that h is locally Lipschitz on D if it is locally Lipschitz at every point of D.

Definition 2.3 (Bector et al. [<xref ref-type="bibr" rid="B5">5</xref>]).

A differentiable function h is said to be univex at xo if for all x∈D, we have
(2.3)b(x,xo)ϕ(h(x)-h(xo))≧[▿h(xo)]Tη(x,xo).

We consider the following nonlinear multiobjective programming problem:
(MP)Minimizef(x)=(f1(x),f2(x),…,fp(x))subjecttog(x)≦0,
where x∈D and the functions f:D→Rp, g:D→Rk. Let X={x∈D:g(x)≦0} be a set of feasible solutions of (MP). For xo∈D, if we denote by
(2.4)J(xo)={j∈{1,2,…,k}:gj(xo)=0},J~(xo)={j∈{1,2,…,k}:gj(xo)<0},J¯(xo)={j∈{1,2,…,k}:gj(xo)>0},then(2.5)J(xo)∪J~(xo)∪J¯(xo)={1,2,…,k}.

Since the objectives in multiobjective programming problems generally conflict with one another, an optimal solution is chosen from the set of efficient or weak efficient solution in the following sense by Miettinen [13].

Definition 2.4.

A point xo∈X is said to be an efficient solution of (MP), if there exists no x∈X such that
(2.6)f(x)≤f(xo).

Definition 2.5.

A point xo∈X is said to be a weak efficient solution of (MP), if there exists no x∈X such that
(2.7)f(x)<f(xo).

Now we define a new class of (d-ρ-η-θ)-type I univex functions which generalize the work of Mishra et al. [8] and Nahak and Mohapatra [10]. Let functions f=(f1,…,fp):D→Rp and g=(g1,…,gk):D→Rk are directionally differentiable at xo∈X,η:X×D→Rn, bo and b1 are nonnegative functions defined on X×D, ϕo:Rp→Rp and ϕ1:Rk→Rk, while ρ∈Rp+k and θ(·,·):X×D→Rn be vector-valued functions.

Definition 2.6.

(f,g) is said to be (d-ρ-η-θ)-type I univex at xo∈D if for all x∈X(2.8)bo(x,xo)ϕo(f(x)-f(xo))≧f′(xo;η(x,xo))+ρ1‖θ(x,xo)‖2,-b1(x,xo)ϕ1(g(xo))≧g′(xo;η(x,xo))+ρ2‖θ(x,xo)‖2.

If the inequalities in f are strict (whenever x≠xo), then (f,g) is said to be semistrictly (d-ρ-η-θ)-type I univex at xo.

Remark 2.7.

(i) If ρ1,ρ2=0, bo(x,xo)=b1(x,xo)=1, ϕo(t)=t,ϕ1(t)=t, then above definition becomes that of d-type I function [14].

(ii) If in the above definition, the functions f and g are differentiable functions such that f′(xo;η(x,xo))=[▿f(xo)]Tη(x,xo), g′(xo;η(x,xo))=[▿g(xo)]Tη(x,xo); ρ1, ρ2=0; bo(x,xo)=b1(x,xo)=1, ϕo(t)=t, ϕ1(t)=t, then we obtain the definition of type I function [15].

Definition 2.8.

(f,g) is said to be aweak strictly pseudo-quasi (d-ρ-η-θ)-type I univex at xo∈D if for all x∈X(2.9)bo(x,xo)ϕo(f(x)-f(xo))≤0⟹f′(xo;η(x,xo))<-ρ1‖θ(x,xo)‖2,b1(x,xo)ϕ1(g(xo))≧0⟹g′(xo;η(x,xo))≦-ρ2‖θ(x,xo)‖2.

Definition 2.9.

(f,g) is said to be strong pseudo-quasi (d-ρ-η-θ)-type I univex at xo∈D if for all x∈X(2.10)bo(x,xo)ϕo(f(x)-f(xo))≤0⟹f′(xo;η(x,xo))≤-ρ1‖θ(x,xo)‖2,b1(x,xo)ϕ1(g(xo))≧0⟹g′(xo;η(x,xo))≦-ρ2‖θ(x,xo)‖2.

Definition 2.10.

(f,g) is said to be weak strictly-pseudo (d-ρ-η-θ)-type I univex at xo∈D if for all x∈X(2.11)bo(x,xo)ϕo(f(x)-f(xo))≤0⟹f′(xo;η(x,xo))<-ρ1‖θ(x,xo)‖2,b1(x,xo)ϕ1(g(xo))≧0⟹g′(xo;η(x,xo))<-ρ2‖θ(x,xo)‖2.

Definition 2.11.

(f,g) is said to be aweak quasistrictly-pseudo (d-ρ-η-θ)-type I univex at xo∈D if for all x∈X(2.12)bo(x,xo)ϕo(f(x)-f(xo))≤0⟹f′(xo;η(x,xo))≦-ρ1‖θ(x,xo)‖2,b1(x,xo)ϕ1(g(xo))≧0⟹g′(xo;η(x,xo))≤-ρ2‖θ(x,xo)‖2.

Remark 2.12.

In the above definitions, if f and g are differentiable functions such that f′(xo;η(x,xo))=[▿f(xo)]Tη(x,xo); g′(xo;η(x,xo))=[▿g(xo)]Tη(x,xo); ρ1=ρ2=0, then we obtain the functions given in Mishra et al. [8].

3. Sufficient Optimality Conditions

In this section, we discuss sufficient optimality conditions for a point to be an efficient solution of (MP) under generalized (d-ρ-η-θ)-type I univex assumptions. In the following theorems, μ=(μ1,…,μp)∈Rp and λ=(λ1,…,λJ(xo))∈J(xo).

Theorem 3.1.

Suppose there exists a feasible solution xo for (MP), vector functions η:X×D→Rn and vectors μ>0 and λ≧0, such that

(f,gJ(xo)) is a strong pseudo-quasi (d-ρ-η-θ)-type I univex at xo,

for any u∈Rp, u≤0⇒ϕo(u)≤0 and v∈RJ(xo),v≧0⇒ϕ1(v)≧0;bo(x,xo)>0, b1(x,xo)≧0,

∑i=1pμiρi1+∑j∈J(xo)λjρj2≧0,

then xo is an efficient solution of (MP).
Proof.

Suppose xo is not an efficient solution of (MP), then there exists x∈X such that f(x)≤f(xo).

Since gj(xo)=0,j∈J(xo), therefore by hypothesis (iii), we get
(3.1)bo(x,xo)ϕo(f(x)-f(xo))≤0,b1(x,xo)ϕ1(gJ(xo)(xo))≧0.
which using hypothesis (ii) yields
(3.2)f′(xo;η(x,xo))≤-ρ1‖θ(x,xo)‖2,gJ(xo)′(xo;η(x,xo))≦-ρJ(xo)2‖θ(x,xo)‖2.
Also μ>0 and λ≧0, so, we get
(3.3)∑i=1pμifi′(xo;η(x,xo))<-∑i=1pμiρi1‖θ(x,xo)‖2,∑j∈J(xo)λjgj′(xo;η(x,xo))≦-∑j∈J(xo)λjρj2‖θ(x,xo)‖2.
Adding the above inequalities, we obtain
(3.4)∑i=1pμifi′(xo;η(x,xo))+∑j∈J(xo)λjgj′(xo;η(x,xo))<-(∑i=1pμiρi1+∑j∈J(xo)λjρj2)‖θ(x,xo)‖2≦0(Byhypothesis(iv)),
which contradicts hypothesis (i). Hence the proof.

Theorem 3.2.

Suppose there exists a feasible solution xo for (MP), vector functions η:X×D→Rn and vectors μ≥0 and λ≧0, such that

(f,gJ(xo)) is a weak strictly-pseudo-quasi (d-ρ-η-θ)-type I univex at xo,

for any u∈Rp,u≤0⇒ϕo(u)≤0 and v∈RJ(xo), v≧0⇒ϕ1(v)≧0; bo(x,xo)>0, b1(x,xo)≧0,

∑i=1pμiρi1+∑j∈J(xo)λjρj2≧0,

then xo is an efficient solution of (MP).
Proof.

Suppose xo is not an efficient solution of (MP), then there exists x∈X such that f(x)≤f(xo).

As gj(xo)=0, j∈J(xo), so, hypothesis (iii) yields
(3.5)bo(x,xo)ϕo(f(x)-f(xo))≤0,b1(x,xo)ϕ1(gJ(xo)(xo))≧0.
By hypothesis (ii), the above inequalities imply
(3.6)f′(xo;η(x,xo))<-ρ1‖θ(x,xo)‖2,gJ(xo)′(xo;η(x,xo))≦-ρJ(xo)2∥θ(x,xo)∥2.
Since μ≥0 and λ≧0, we get
(3.7)∑i=1pμifi′(xo;η(x,xo))<-∑i=1pμiρi1‖θ(x,xo)‖2,∑j∈J(xo)λjgj′(xo;η(x,xo))≦-∑j∈J(xo)λjρj2‖θ(x,xo)‖2.
Adding the above inequalities, we obtain
(3.8)∑i=1pμifi′(xo;η(x,xo))+∑j∈J(xo)λjgj′(xo;η(x,xo))<-(∑i=1pμiρi1+∑j∈J(xo)λjρj2)‖θ(x,xo)‖2≦0(usinghypothesis(iv)),
which contradicts hypothesis (i). Hence the proof.

Theorem 3.3.

Suppose there exists a feasible solution xo for (MP), vector functions η:X×D→Rn and vectors μ≥0 and λ≧0, such that

(f,gJ(xo)) is a weak strictly-pseudo (d-ρ-η-θ)-type I univex at xo,

for any u∈Rp,u≤0⇒ϕo(u)≤0 and v∈RJ(xo), v≧0⇒ϕ1(v)≧0; bo(x,xo)>0, b1(x,xo)≧0,

∑i=1pμiρi1+∑j∈J(xo)λjρj2≧0,

then xo is an efficient solution of (MP).
Proof.

Suppose xo is not an efficient solution of (MP), then there exists x∈X such that f(x)≤f(xo).

As gj(xo)=0, j∈J(xo), so hypothesis (iii) implies
(3.9)bo(x,xo)ϕo(f(x)-f(xo))≤0,b1(x,xo)ϕ1(gJ(xo)(xo))≧0.
Since hypothesis (ii) holds, above inequalities imply
(3.10)f′(xo;η(x,xo))<-ρ1‖θ(x,xo)2‖,gJ(xo)′(xo;η(x,xo))<-ρJ(xo)2‖θ(x,xo)‖2.
Also μ≥0 and λ≧0, so we obtain(3.11)∑i=1pμifi′(xo;η(x,xo))<-∑i=1pμiρi1‖θ(x,xo)‖2,∑j∈J(xo)λjgj′(xo;η(x,xo))≦-∑j∈J(xo)λjρj2‖θ(x,xo)‖2.
On adding and using hypothesis (iv), above inequalities yield
(3.12)∑i=1pμifi′(xo;η(x,xo))+∑j∈J(xo)λjgj′(xo;η(x,xo))<-(∑i=1pμiρi1+∑j∈J(xo)λjρj2)‖θ(x,xo)‖2≦0
which contradicts hypothesis (i). Hence the proof.

Theorem 3.4.

Suppose there exists a feasible solution xo for (MP), vector functions η:X×D→Rn and vectors μ≧0 and λ>0, such that

(f,gJ(xo)) is weak quasi-strictly-pseudo (d-ρ-η-θ)-type I univex at xo,

for any u∈Rp,u≤0⇒ϕo(u)≤0 and v∈RJ(xo),v≧0⇒ϕ1(v)≧0;bo(x,xo)>0,b1(x,xo)≧0,

∑i=1pμiρi1+∑j∈J(xo)λjρj2≧0,

then xo is an efficient solution of (MP).
Proof.

Suppose xo is not an efficient solution of (MP), then there exists x∈X such that f(x)≤f(xo).

Since gj(xo)=0, j∈J(xo), therefore hypothesis (iii) yields
(3.13)bo(x,xo)ϕo(f(x)-f(xo))≤0,b1(x,xo)ϕ1(gJ(xo)(xo))≧0.
By hypothesis (ii), we get
(3.14)f′(xo;η(x,xo))≦-ρ1‖θ(x,xo)‖2,gJ(xo)′(xo;η(x,xo))≤-ρJ(xo)2‖θ(x,xo)‖2.
Also μ≧0 and λ>0, so, we obtain
(3.15)∑i=1pμifi′(xo;η(x,xo))≦-∑i=1pμiρi1‖θ(x,xo)‖2,(3.16)∑j∈J(xo)λjgj′(xo;η(x,xo))<-∑j∈J(xo)λjρj2‖θ(x,xo)‖2.

On adding and using hypothesis (iv), above inequalities yield
(3.17)∑i=1pμifi′(xo;η(x,xo))+∑j∈J(xo)λjgj′(xo;η(x,xo))<-(∑i=1pμiρi1+∑j∈J(xo)λjρj2)‖θ(x,xo)‖2≦0,
which contradicts hypothesis (i). Hence the proof.

Now, following Antczak [3], we state following necessary optimality conditions.

Theorem 3.5 (Karush-Kuhn-Tucker type necessary optimality conditions).

If

xo is a weakly efficient solution of (MP),

gj is continuous at xo for j∈J~(xo),

there exists a vector functions η:X×D→Rn,

for all i=1,p¯ and j∈J(xo), fi and gj are directionally differentiable at xo and the functions fi′(xo;η(x,xo)),i=1,p¯ and gj′(xo;η(x,xo)),j∈J(xo) are preinvex functions of x on X,

the function g satisfies the generalized Slater's constraint qualification at xo,

then there exists μ∈R⩾p and λ∈R≧k such that
(3.18)∑i=1pμifi′(xo;η(x,xo))+∑j=1kλjgj′(xo;η(x,xo))≧0∀x∈X,λjgj(xo)=0,j=1,k¯.4. Mond-Weir Type Duality

In this section, we consider Mond-Weir type dual of (MP) and establish weak, strong, converse, and strict converse duality theorems. In this section, we denote gλ=(λ1g1,…,λkgk). (MWD)Maxf(y)subjectto∑i=1pμifi′(y,η(x,y))+∑j=1kλjgj′(y,η(x,y))≧0∀x∈X,λjgj(y)≧0,j=1,k¯,
where y∈D, μ∈R⩾p, λ∈R≧k, η:X×D→Rn. Let W be the set of feasible points of (MWD).

Theorem 4.1 (Weak Duality).

Let x and (y,μ,λ,η) be the feasible solutions for (MP) and (MWD) respectively. If

(f,gλ) is a weak strictly-pseudo-quasi (d-ρ-η-θ)-type I univex at y,

for any u∈Rp,u≤0⇒ϕo(u)≤0 and v∈Rk,v≧0⇒ϕ1(v)≧0;bo(x,y)>0,b1(x,y)≧0,

∑i=1pμiρi1+∑j=1kρj2≧0,

then
(4.1)f(x)≰f(y).Proof.

Suppose to the contrary that
(4.2)f(x)≤f(y).
Since λjgj(y)≧0, j=1,k¯, hypothesis (ii) yields
(4.3)bo(x,y)ϕo(f(x)-f(y))≤0,b1(x,y)ϕ1(gλ(y))≧0.
As hypothesis (i) holds, therefore the above inequalities imply
(4.4)f′(y;η(x,y))<-ρ1‖θ(x,y)‖2,gλ′(y;η(x,y))≦-ρ2‖θ(x,y)‖2.
Also μ∈R⩾p, so, we obtain
(4.5)∑i=1pμifi′(y;η(x,y))<-∑i=1pμiρi1‖θ(x,y)‖2,∑j=1kλjgj′(y;η(x,y))≦-∑j=1kρj2‖θ(x,y)‖2.
On adding above inequalities and using hypothesis (iii), we get
(4.6)∑i=1pμifi′(y;η(x,y))+∑j=1kλjgj′(y;η(x,y))<-(∑i=1pμiρi1+∑j=1kρj2)‖θ(x,y)‖2≦0,
which is a contradiction to the dual constraint. Hence the proof.

The proofs of the following weak duality theorems are similar to Theorem 4.1 and hence are omitted.

Theorem 4.2 (Weak Duality).

Let x and (y,μ,λ,η) be the feasible solutions for (MP) and (MWD), respectively, with μi>0, i=1,p¯. If

(f,gλ) is a strong pseudo-quasi (d-ρ-η-θ)-type I univex at y,

for any u∈Rp,u≤0⇒ϕo(u)≤0 and v∈Rk,v≧0⇒ϕ1(v)≧0;bo(x,y)>0,b1(x,y)≧0,

∑i=1pμiρi1+∑j=1kρj2≧0,

then f(x)≰f(y).
Theorem 4.3 (Weak Duality).

Let x and (y,μ,λ,η) be the feasible solutions for (MP) and (MWD), respectively. If

(f,gλ) is weak strictly-pseudo (d-ρ-η-θ)-type I univex at y,

for any u∈Rp,u≤0⇒ϕo(u)≤0 and v∈Rk,v≧0⇒ϕ1(v)≧0;bo(x,y)>0,b1(x,y)≧0,

∑i=1pμiρi1+∑j=1kρj2≧0,

then f(x)≰f(y).
Corollary 4.4.

Let xo and (yo,μ,λ,η) be the feasible solutions for (MP) and (MWD), respectively, such that f(xo)=f(yo). If the weak duality holds between (MP) and (MWD) for all feasible solutions of two problems, then xo is efficient for (MP) and (yo,μ,λ,η) is efficient for (MWD).

Proof.

Suppose that xo is not efficient for (MP), then for some x∈X(4.7)f(x)≤f(xo)=f(yo).
which contradicts weak duality theorems as (yo,μ,λ,η) is feasible for (MWD) and x is feasible for (MP). So, xo is efficient for (MP). Similarly (yo,μ,λ,η) is efficient for (MWD).

Theorem 4.5 (Strong Duality).

Let xo be a weakly efficient solution of (MP), gj is continuous at xo for j∈J~(xo), f,g are directionally differentiable at xo with fi′(xo,η(x,xo)), and gj′(xo,η(x,xo)) as preinvex functions on X. Also if g satisfies the generalized Slater's constraint qualification at xo, then ∃μ∈R⩾p,λ∈R≧k such that (xo,μ,λ,η) is feasible for (MWD) and the objective function values of (MP) and (MWD) are equal. Moreover, if any of weak duality theorem holds, then (xo,μ,λ,η) is an efficient solution of (MWD).

Proof.

Since xo is a weakly efficient solution of (MP), therefore by Theorem 3.5, there exists μ∈R⩾p,λ∈R≧k such that
(4.8)∑i=1pμifi′(xo,η(x,xo))+∑j=1kλjgj′(xo,η(x,xo))≧0∀x∈X,λjgj(xo)=0,j=1,k¯.

It follows that (xo,μ,λ,η)∈W and therefore feasible for (MWD). Clearly objective function values of (MP) and (MWD) are equal at optimal points.

Suppose (xo,μ,λ,η) is not an efficient solution for (MWD). Then ∃(y~,μ~,λ~,η~)∈W such that f(xo)≤f(y~), which contradicts weak duality theorems. Therefore (xo,μ,λ,η) is an efficient solution of (MWD). Hence the proof.

Theorem 4.6 (Converse Duality).

Let (yo,μ,λ,η) be a feasible solution of (MWD). If

(f,gλ) is a weak strictly-pseudo-quasi (d-ρ-η-θ)-type I univex at yo,

for any u∈Rp,u≤0⇒ϕo(u)≤0 and v∈Rk,v≧0⇒ϕ1(v)≧0; bo(xo,yo)>0,b1(xo,yo)≧0,

∑i=1pμiρi1+∑j=1kρj2≧0,

then yo is an efficient solution of (MP).
Proof.

Suppose that yo is not an efficient solution of (MP). Then ∃xo∈X such that
(4.9)f(xo)≤f(y0).
Now proceeding as in Theorem 4.1 (Weak Duality), we obtain a contradiction. Hence yo is an efficient solution of (MP).

Theorem 4.7 (Strict Converse Duality).

Let xo and (yo,μ,λ,η) be the feasible solutions of (MP) and (MWD), respectively. If

f(xo)≦f(yo),

(f,gλ) is a weak quasi-strictly-pseudo (d-ρ-η-θ)-type I univex at yo,

for any u∈Rp,u≦0⇒ϕo(u)≤0 and v∈Rk,v≧0⇒ϕ1(v)≧0;bo(xo,yo)>0,b1(xo,yo)≧0,

∑i=1pμiρi1+∑j=1kρj2≧0,

then xo=yo.
Proof.

Suppose xo≠yo.

Since yo is a feasible solution of (MWD), therefore by hypothesis (i) and hypothesis (iii), we get
(4.10)bo(xo,yo)ϕo(f(xo)-f(yo))≤0,b1(xo,yo)ϕ1(gλ(yo))≧0.
By hypothesis (ii), we obtain
(4.11)f′(yo;η(xo,yo))≦-ρ1‖θ(xo,yo)‖2,gλ′(yo;η(xo,yo))≤-ρ2‖θ(xo,yo)‖2.
Since μ∈R⩾p, therefore the above inequalities yield
(4.12)∑i=1pμifi′(yo;η(xo,yo))≦-∑i=1pμiρi1‖θ(xo,yo)‖2,(4.13)∑j=1kλjgj′(yo;η(xo,yo))<-∑j=1kρj2‖θ(xo,yo)‖2,
which on adding gives
(4.14)∑i=1pμifi′(yo;η(xo,yo))+∑j=1kλjgj′(yo;η(xo,yo))<-(∑i=1pμiρi1+∑j=1kρj2)‖θ(xo,yo)‖2≦0(usinghypothesis(iv)),
which is a contradiction to feasibility of yo. Hence xo=yo.

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