Generalized (𝑑-𝜌-𝜂-𝜃)-Type I Univex Functions in Multiobjective Optimization

Abstract

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 𝑅^𝑛 will be followed throughout this paper: 𝑥≧𝑦⇔𝑥_𝑖≧𝑦_𝑖, 𝑖=1,2,…,𝑛; 𝑥≥𝑦⇔𝑥≧𝑦, 𝑥≠𝑦; 𝑥>𝑦⇔𝑥_𝑖>𝑦_𝑖, 𝑖=1,2,…,𝑛. Let 𝐷 be a nonempty subset of 𝑅^𝑛, 𝜂∶𝐷×𝐷→𝑅^𝑛, 𝑥_𝑜 be an arbitrary point of 𝐷 and ℎ∶𝐷→𝑅,  𝜙∶𝑅→𝑅,  𝑏∶𝐷×𝐷→𝑅₊. Also, we denote 𝑅^𝑝_≥={𝑦∶𝑦∈𝑅^𝑝 and 𝑦≥0} and 𝑅^𝑘_≧={𝑦∶𝑦∈𝑅^𝑘 and 𝑦≧0}.

Definition 2.1 (Ben-Israel and Mond [11]). Let 𝐷⊆𝑅^𝑛 be an invex set. A function ℎ is called preinvex on 𝐷 with respect to 𝜂, if for all 𝑥,𝑥_𝑜∈𝐷, 𝜆ℎ(𝑥)+(1−𝜆)ℎ𝑥_𝑜≧ℎ𝑥_𝑜+𝜆𝜂𝑥,𝑥_𝑜,∀𝜆∈[0,1]. (2.1)

Definition 2.2 (Clarke [12]). The function ℎ is said to be locally Lipschitz at 𝑥_𝑜∈𝐷, if there exists a neighbourhood 𝑣(𝑥_𝑜) of 𝑥_𝑜 and a constant 𝑘>0 such that ℎ𝑦−ℎ(𝑥)≦𝑘𝑦−𝑥∀𝑥,𝑦∈𝑣𝑥_𝑜, (2.2) 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. [5]). A differentiable function ℎ is said to be univex at 𝑥_𝑜 if for all 𝑥∈𝐷, we have 𝑏𝑥,𝑥_𝑜𝜙ℎ(𝑥)−ℎ𝑥_𝑜≧▿ℎ𝑥_𝑜^𝑇𝜂𝑥,𝑥_𝑜. (2.3)
We consider the following nonlinear multiobjective programming problem: Minimize𝑓(𝑥)=𝑓₁(𝑥),𝑓₂(𝑥),…,𝑓_𝑝(𝑥)subjectto𝑔(𝑥)≦0, (MP) where 𝑥∈𝐷 and the functions 𝑓∶𝐷→𝑅^𝑝, 𝑔∶𝐷→𝑅^𝑘. Let 𝑋={𝑥∈𝐷∶𝑔(𝑥)≦0} be a set of feasible solutions of (MP). For 𝑥_𝑜∈𝐷, if we denote by 𝐽𝑥_𝑜=𝑗∈{1,2,…,𝑘}∶𝑔_𝑗𝑥_𝑜,𝐽̃𝑥=0_𝑜=𝑗∈{1,2,…,𝑘}∶𝑔_𝑗𝑥_𝑜<0,𝐽𝑥_𝑜=𝑗∈{1,2,…,𝑘}∶𝑔_𝑗𝑥_𝑜>0, (2.4)then𝐽𝑥_𝑜∪𝐽̃𝑥_𝑜∪𝐽𝑥_𝑜={1,2,…,𝑘}. (2.5)
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 𝑥_𝑜∈𝑋 is said to be an efficient solution of (MP), if there exists no 𝑥∈𝑋 such that 𝑓(𝑥)≤𝑓𝑥_𝑜. (2.6)

Definition 2.5. A point 𝑥_𝑜∈𝑋 is said to be a weak efficient solution of (MP), if there exists no 𝑥∈𝑋 such that 𝑓(𝑥)<𝑓𝑥_𝑜. (2.7)

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,…,𝑓_𝑝)∶𝐷→𝑅^𝑝 and 𝑔=(𝑔₁,…,𝑔_𝑘)∶𝐷→𝑅^𝑘 are directionally differentiable at 𝑥_𝑜∈𝑋,𝜂∶𝑋×𝐷→𝑅^𝑛, 𝑏_𝑜 and 𝑏₁ are nonnegative functions defined on 𝑋×𝐷, 𝜙_𝑜∶𝑅^𝑝→𝑅^𝑝 and 𝜙₁∶𝑅^𝑘→𝑅^𝑘, while 𝜌∈𝑅^𝑝+𝑘 and 𝜃(⋅,⋅)∶𝑋×𝐷→𝑅^𝑛 be vector-valued functions.

Definition 2.6. (𝑓,𝑔) is said to be (𝑑-𝜌-𝜂-𝜃)-type I univex at 𝑥_𝑜∈𝐷 if for all 𝑥∈𝑋𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜𝑓(𝑥)−𝑓𝑥_𝑜≧𝑓^′𝑥_𝑜;𝜂𝑥,𝑥_𝑜+𝜌¹𝜃𝑥,𝑥_𝑜²,−𝑏₁𝑥,𝑥_𝑜𝜙₁𝑔𝑥_𝑜≧𝑔^′𝑥_𝑜;𝜂𝑥,𝑥_𝑜+𝜌²𝜃(𝑥,𝑥_𝑜)². (2.8)
If the inequalities in 𝑓 are strict (whenever 𝑥≠𝑥_𝑜), then (𝑓,𝑔) is said to be semistrictly (𝑑-𝜌-𝜂-𝜃)-type I univex at 𝑥_𝑜.

Remark 2.7. (i) If 𝜌¹,𝜌²=0,  𝑏_𝑜(𝑥,𝑥_𝑜)=𝑏₁(𝑥,𝑥_𝑜)=1,  𝜙_𝑜(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^′(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜))=[▿𝑓(𝑥_𝑜)]^𝑇𝜂(𝑥,𝑥_𝑜),  𝑔^′(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜))=[▿𝑔(𝑥_𝑜)]^𝑇𝜂(𝑥,𝑥_𝑜);  𝜌¹, 𝜌²=0; 𝑏_𝑜(𝑥,𝑥_𝑜)=𝑏₁(𝑥,𝑥_𝑜)=1, 𝜙_𝑜(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 𝑥_𝑜∈𝐷 if for all 𝑥∈𝑋𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜𝑓(𝑥)−𝑓𝑥_𝑜≤0⟹𝑓^′𝑥_𝑜;𝜂𝑥,𝑥_𝑜<−𝜌¹𝜃𝑥,𝑥_𝑜²,𝑏₁𝑥,𝑥_𝑜𝜙₁𝑔𝑥_𝑜≧0⟹𝑔^′𝑥_𝑜;𝜂𝑥,𝑥_𝑜≦−𝜌²𝜃𝑥,𝑥_𝑜². (2.9)

Definition 2.9. (𝑓,𝑔) is said to be strong pseudo-quasi (𝑑-𝜌-𝜂-𝜃)-type I univex at 𝑥_𝑜∈𝐷 if for all 𝑥∈𝑋𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜𝑓(𝑥)−𝑓𝑥_𝑜≤0⟹𝑓^′𝑥_𝑜;𝜂𝑥,𝑥_𝑜≤−𝜌¹𝜃𝑥,𝑥_𝑜²,𝑏₁𝑥,𝑥_𝑜𝜙₁𝑔𝑥_𝑜≧0⟹𝑔^′𝑥_𝑜;𝜂𝑥,𝑥_𝑜≦−𝜌²𝜃𝑥,𝑥_𝑜². (2.10)

Definition 2.10. (𝑓,𝑔) is said to be weak strictly-pseudo (𝑑-𝜌-𝜂-𝜃)-type I univex at 𝑥_𝑜∈𝐷 if for all 𝑥∈𝑋𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜𝑓(𝑥)−𝑓𝑥_𝑜≤0⟹𝑓^′𝑥_𝑜;𝜂𝑥,𝑥_𝑜<−𝜌¹𝜃𝑥,𝑥_𝑜²,𝑏₁𝑥,𝑥_𝑜𝜙₁𝑔𝑥_𝑜≧0⟹𝑔^′𝑥_𝑜;𝜂𝑥,𝑥_𝑜<−𝜌²𝜃𝑥,𝑥_𝑜². (2.11)

Definition 2.11. (𝑓,𝑔) is said to be aweak quasistrictly-pseudo (𝑑-𝜌-𝜂-𝜃)-type I univex at 𝑥_𝑜∈𝐷 if for all 𝑥∈𝑋𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜𝑓(𝑥)−𝑓𝑥_𝑜≤0⟹𝑓^′𝑥_𝑜;𝜂𝑥,𝑥_𝑜≦−𝜌¹𝜃𝑥,𝑥_𝑜²,𝑏₁𝑥,𝑥_𝑜𝜙₁𝑔𝑥_𝑜≧0⟹𝑔^′𝑥_𝑜;𝜂𝑥,𝑥_𝑜≤−𝜌²𝜃𝑥,𝑥_𝑜². (2.12)

Remark 2.12. In the above definitions, if 𝑓 and 𝑔 are differentiable functions such that 𝑓′(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜))=[▿𝑓(𝑥_𝑜)]^𝑇𝜂(𝑥,𝑥_𝑜); 𝑔′(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜))=[▿𝑔(𝑥_𝑜)]^𝑇𝜂(𝑥,𝑥_𝑜); 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,…,𝜇_𝑝)∈𝑅^𝑝 and 𝜆=(𝜆₁,…,𝜆_{𝐽(𝑥𝑜)})∈𝐽(𝑥_𝑜).

Theorem 3.1. Suppose there exists a feasible solution 𝑥_𝑜 for (MP), vector functions 𝜂∶𝑋×𝐷→𝑅^𝑛 and vectors 𝜇>𝟎 and 𝜆≧𝟎, such that (i)∑^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜))+∑_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝑔^′_𝑗(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜))≧0, (ii)(𝑓,𝑔_{𝐽(𝑥𝑜)}) is a strong pseudo-quasi (𝑑-𝜌-𝜂-𝜃)-type I univex at 𝑥_𝑜,(iii)for any 𝑢∈𝑅^𝑝, 𝑢≤0⇒𝜙_𝑜(𝑢)≤0 and 𝑣∈𝑅^{𝐽(𝑥𝑜)},𝑣≧0⇒𝜙₁(𝑣)≧0;𝑏_𝑜(𝑥,𝑥_𝑜)>0, 𝑏₁(𝑥,𝑥_𝑜)≧0,(iv)∑^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖+∑_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝜌²_𝑗≧0, then 𝑥_𝑜 is an efficient solution of (MP).

Proof. Suppose 𝑥_𝑜 is not an efficient solution of (MP), then there exists 𝑥∈𝑋 such that 𝑓(𝑥)≤𝑓(𝑥_𝑜).
Since 𝑔_𝑗(𝑥_𝑜)=0,𝑗∈𝐽(𝑥_𝑜), therefore by hypothesis (iii), we get 𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜𝑓(𝑥)−𝑓𝑥_𝑜𝑏≤0,₁𝑥,𝑥_𝑜𝜙₁𝑔_{𝐽(𝑥𝑜)}𝑥_𝑜≧0. (3.1) which using hypothesis (ii) yields 𝑓^′𝑥_𝑜;𝜂𝑥,𝑥_𝑜≤−𝜌¹𝜃𝑥,𝑥_𝑜²,𝑔^′_𝐽𝑥𝑜𝑥_𝑜;𝜂𝑥,𝑥_𝑜≦−𝜌²_𝐽𝑥𝑜𝜃𝑥,𝑥_𝑜². (3.2) Also 𝜇>0 and 𝜆≧0, so, we get ^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂𝑥,𝑥_𝑜<−^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖𝜃𝑥,𝑥_𝑜²,_{𝑥𝑗∈𝐽𝑜}𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜂𝑥,𝑥_𝑜≦−_{𝑥𝑗∈𝐽𝑜}𝜆_𝑗𝜌²_𝑗𝜃𝑥,𝑥_𝑜². (3.3) Adding the above inequalities, we obtain ^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂𝑥,𝑥_𝑜+_{𝑥𝑗∈𝐽𝑜}𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜂𝑥,𝑥_𝑜<−^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖+_{𝑥𝑗∈𝐽𝑜}𝜆_𝑗𝜌²_𝑗𝜃𝑥,𝑥_𝑜²≦0Byhypothesis(iv), (3.4) which contradicts hypothesis (i). Hence the proof.

Theorem 3.2. Suppose there exists a feasible solution 𝑥_𝑜 for (MP), vector functions 𝜂∶𝑋×𝐷→𝑅^𝑛 and vectors 𝜇≥𝟎 and 𝜆≧𝟎, such that (i)∑^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜))+∑_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝑔^′_𝑗(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜))≧0, (ii)(𝑓,𝑔_{𝐽(𝑥𝑜)}) is a weak strictly-pseudo-quasi (𝑑-𝜌-𝜂-𝜃)-type I univex at 𝑥_𝑜, (iii)for any 𝑢∈𝑅^𝑝,𝑢≤0⇒𝜙_𝑜(𝑢)≤0 and 𝑣∈𝑅^{𝐽(𝑥𝑜)}, 𝑣≧0⇒𝜙₁(𝑣)≧0; 𝑏_𝑜(𝑥,𝑥_𝑜)>0, 𝑏₁(𝑥,𝑥_𝑜)≧0,(iv)∑^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖+∑_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝜌²_𝑗≧0,  then 𝑥_𝑜 is an efficient solution of (MP).

Proof. Suppose 𝑥_𝑜 is not an efficient solution of (MP), then there exists 𝑥∈𝑋 such that 𝑓(𝑥)≤𝑓(𝑥_𝑜).
As 𝑔_𝑗(𝑥_𝑜)=0, 𝑗∈𝐽(𝑥_𝑜), so, hypothesis (iii) yields 𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜𝑓(𝑥)−𝑓𝑥_𝑜𝑏≤0,₁𝑥,𝑥_𝑜𝜙₁𝑔_{𝐽(𝑥𝑜)}𝑥_𝑜≧0. (3.5) By hypothesis (ii), the above inequalities imply 𝑓^′𝑥_𝑜;𝜂𝑥,𝑥_𝑜<−𝜌¹𝜃𝑥,𝑥_𝑜²,𝑔^′_𝐽𝑥𝑜𝑥_𝑜;𝜂𝑥,𝑥_𝑜≦−𝜌²_{𝐽(𝑥𝑜)}‖𝜃𝑥,𝑥_𝑜‖². (3.6) Since 𝜇≥0 and 𝜆≧0, we get ^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂𝑥,𝑥_𝑜<−^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖𝜃𝑥,𝑥_𝑜²,_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜂𝑥,𝑥_𝑜≦−_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝜌²_𝑗𝜃𝑥,𝑥_𝑜². (3.7) Adding the above inequalities, we obtain ^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂𝑥,𝑥_𝑜+_{𝑥𝑗∈𝐽𝑜}𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜂𝑥,𝑥_𝑜<−^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖+_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝜌²_𝑗𝜃(𝑥,𝑥_𝑜)²≦0usinghypothesis(iv), (3.8) which contradicts hypothesis (i). Hence the proof.

Theorem 3.3. Suppose there exists a feasible solution 𝑥_𝑜 for (MP), vector functions 𝜂∶𝑋×𝐷→𝑅^𝑛 and vectors 𝜇≥𝟎 and 𝜆≧𝟎, such that (i)∑^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜))+∑_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝑔^′_𝑗(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜))≧0, (ii)(𝑓,𝑔_{𝐽(𝑥𝑜)}) is a weak strictly-pseudo (𝑑-𝜌-𝜂-𝜃)-type I univex at 𝑥_𝑜,(iii)for any 𝑢∈𝑅^𝑝,𝑢≤0⇒𝜙_𝑜(𝑢)≤0 and 𝑣∈𝑅^{𝐽(𝑥𝑜)}, 𝑣≧0⇒𝜙₁(𝑣)≧0; 𝑏_𝑜(𝑥,𝑥_𝑜)>0, 𝑏₁(𝑥,𝑥_𝑜)≧0,(iv)∑^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖+∑_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝜌²_𝑗≧0, then 𝑥_𝑜 is an efficient solution of (MP).

Proof. Suppose 𝑥_𝑜 is not an efficient solution of (MP), then there exists 𝑥∈𝑋 such that 𝑓(𝑥)≤𝑓(𝑥_𝑜).
As 𝑔_𝑗(𝑥_𝑜)=0, 𝑗∈𝐽(𝑥_𝑜), so hypothesis (iii) implies 𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜𝑓(𝑥)−𝑓𝑥_𝑜𝑏≤0,₁𝑥,𝑥_𝑜𝜙₁𝑔_{𝐽(𝑥𝑜)}𝑥_𝑜≧0. (3.9) Since hypothesis (ii) holds, above inequalities imply 𝑓^′𝑥_𝑜;𝜂𝑥,𝑥_𝑜<−𝜌¹𝜃𝑥,𝑥_𝑜²,𝑔^′_𝐽𝑥𝑜𝑥_𝑜;𝜂𝑥,𝑥_𝑜<−𝜌²_{𝐽(𝑥𝑜)}𝜃(𝑥,𝑥_𝑜)². (3.10) Also 𝜇≥0 and 𝜆≧0, so we obtain^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂𝑥,𝑥_𝑜<−^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖𝜃(𝑥,𝑥_𝑜)²,_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜂𝑥,𝑥_𝑜≦−_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝜌²_𝑗𝜃(𝑥,𝑥_𝑜)². (3.11) On adding and using hypothesis (iv), above inequalities yield ^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂𝑥,𝑥_𝑜+_{𝑥𝑗∈𝐽𝑜}𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜂𝑥,𝑥_𝑜<−^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖+_{𝑥𝑗∈𝐽𝑜}𝜆_𝑗𝜌²_𝑗𝜃𝑥,𝑥_𝑜²≦0 (3.12) which contradicts hypothesis (i). Hence the proof.

Theorem 3.4. Suppose there exists a feasible solution 𝑥_𝑜 for (MP), vector functions 𝜂∶𝑋×𝐷→𝑅^𝑛 and vectors 𝜇≧𝟎 and 𝜆>𝟎, such that (i)∑^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜))+∑_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝑔^′_𝑗(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜))≧0, (ii)(𝑓,𝑔_{𝐽(𝑥𝑜)}) is weak quasi-strictly-pseudo (𝑑-𝜌-𝜂-𝜃)-type I univex at 𝑥_𝑜, (iii)for any 𝑢∈𝑅^𝑝,𝑢≤0⇒𝜙_𝑜(𝑢)≤0 and 𝑣∈𝑅^{𝐽(𝑥𝑜)},𝑣≧0⇒𝜙₁(𝑣)≧0;𝑏_𝑜(𝑥,𝑥_𝑜)>0,𝑏₁(𝑥,𝑥_𝑜)≧0,(iv)∑^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖+∑_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝜌²_𝑗≧0, then 𝑥_𝑜 is an efficient solution of (MP).

Proof. Suppose 𝑥_𝑜 is not an efficient solution of (MP), then there exists 𝑥∈𝑋 such that 𝑓(𝑥)≤𝑓(𝑥_𝑜).
Since 𝑔_𝑗(𝑥_𝑜)=0, 𝑗∈𝐽(𝑥_𝑜), therefore hypothesis (iii) yields 𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜𝑓(𝑥)−𝑓𝑥_𝑜𝑏≤0,₁𝑥,𝑥_𝑜𝜙₁𝑔_{𝐽(𝑥𝑜)}𝑥_𝑜≧0. (3.13) By hypothesis (ii), we get 𝑓′𝑥_𝑜;𝜂𝑥,𝑥_𝑜≦−𝜌¹𝜃(𝑥,𝑥_𝑜)²,𝑔^′_𝐽𝑥𝑜𝑥_𝑜;𝜂𝑥,𝑥_𝑜≤−𝜌²_𝐽𝑥𝑜𝜃𝑥,𝑥_𝑜². (3.14) Also 𝜇≧0 and 𝜆>0, so, we obtain ^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂𝑥,𝑥_𝑜≦−^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖𝜃𝑥,𝑥_𝑜²,_{𝑥𝑗∈𝐽𝑜}𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜂𝑥,𝑥_𝑜<−_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝜌²_𝑗𝜃(𝑥,𝑥_𝑜)². (3.15)(3.16)
On adding and using hypothesis (iv), above inequalities yield ^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂𝑥,𝑥_𝑜+_{𝑥𝑗∈𝐽𝑜}𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜂𝑥,𝑥_𝑜<−^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖+_{𝑥𝑗∈𝐽𝑜}𝜆_𝑗𝜌²_𝑗𝜃𝑥,𝑥_𝑜²≦0, (3.17) 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 (i)𝑥_𝑜 is a weakly efficient solution of (MP), (ii)𝑔_𝑗 is continuous at 𝑥_𝑜 for 𝑗∈𝐽(̃𝑥_𝑜),(iii)there exists a vector functions 𝜂∶𝑋×𝐷→𝑅^𝑛,(iv)for all 𝑖=1,𝑝 and 𝑗∈𝐽(𝑥_𝑜), 𝑓_𝑖 and 𝑔_𝑗 are directionally differentiable at 𝑥_𝑜 and the functions 𝑓^′_𝑖(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜)),𝑖=1,𝑝 and 𝑔^′_𝑗(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜)),𝑗∈𝐽(𝑥_𝑜) are preinvex functions of 𝑥 on 𝑋, (v)the function 𝑔 satisfies the generalized Slater's constraint qualification at 𝑥_𝑜, then there exists 𝜇∈𝑅^𝑝_⩾ and 𝜆∈𝑅^𝑘_≧ such that ^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂𝑥,𝑥_𝑜+^𝑘_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜂𝑥,𝑥_𝑜𝜆≧0∀𝑥∈𝑋,_𝑗𝑔_𝑗𝑥_𝑜=0,𝑗=1,𝑘. (3.18)

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 𝑔_𝜆=(𝜆₁𝑔₁,…,𝜆_𝑘𝑔_𝑘). Max𝑓𝑦subjectto^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑦,𝜂𝑥,𝑦+^𝑘_𝑗=1𝜆_𝑗𝑔^′_𝑗𝜆𝑦,𝜂𝑥,𝑦≧0∀𝑥∈𝑋,_𝑗𝑔_𝑗𝑦≧0,𝑗=1,𝑘, (MWD) where 𝑦∈𝐷,  𝜇∈𝑅^𝑝_⩾,  𝜆∈𝑅^𝑘_≧,  𝜂∶𝑋×𝐷→𝑅^𝑛. 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 (i)(𝑓,𝑔_𝜆) is a weak strictly-pseudo-quasi (𝑑-𝜌-𝜂-𝜃)-type I univex at 𝑦, (ii)for any 𝑢∈𝑅^𝑝,𝑢≤0⇒𝜙_𝑜(𝑢)≤0 and 𝑣∈𝑅^𝑘,𝑣≧0⇒𝜙₁(𝑣)≧0;𝑏_𝑜(𝑥,𝑦)>0,𝑏₁(𝑥,𝑦)≧0,(iii)∑^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖+∑^𝑘_𝑗=1𝜌²_𝑗≧0, then 𝑓(𝑥)≰𝑓𝑦. (4.1)

Proof. Suppose to the contrary that 𝑓(𝑥)≤𝑓𝑦. (4.2) Since 𝜆_𝑗𝑔_𝑗(𝑦)≧0, 𝑗=1,𝑘, hypothesis (ii) yields 𝑏_𝑜𝑥,𝑦𝜙_𝑜𝑓(𝑥)𝑦𝑏−𝑓≤0,₁𝑥,𝑦𝜙₁𝑔_𝜆𝑦≧0. (4.3) As hypothesis (i) holds, therefore the above inequalities imply 𝑓^′𝑦;𝜂𝑥,𝑦<−𝜌¹𝜃(𝑥,𝑦)²,𝑔^′_𝜆𝑦;𝜂𝑥,𝑦≦−𝜌²𝜃(𝑥,𝑦)². (4.4) Also 𝜇∈𝑅^𝑝_⩾, so, we obtain ^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑦;𝜂𝑥,𝑦<−^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖𝜃(𝑥,𝑦)²,_𝑘𝑗=1𝜆_𝑗𝑔^′_𝑗𝑦;𝜂𝑥,𝑦≦−^𝑘_𝑗=1𝜌²_𝑗𝜃𝑥,𝑦². (4.5) On adding above inequalities and using hypothesis (iii), we get ^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑦;𝜂𝑥,𝑦+^𝑘_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑦;𝜂𝑥,𝑦<−^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖+^𝑘_𝑗=1𝜌²_𝑗𝜃(𝑥,𝑦)²≦0, (4.6) 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 𝜇_𝑖>0, 𝑖=1,𝑝. If (i)(𝑓,𝑔_𝜆) is a strong pseudo-quasi (𝑑-𝜌-𝜂-𝜃)-type I univex at 𝑦, (ii)for any 𝑢∈𝑅^𝑝,𝑢≤0⇒𝜙_𝑜(𝑢)≤0 and 𝑣∈𝑅^𝑘,𝑣≧0⇒𝜙₁(𝑣)≧0;𝑏_𝑜(𝑥,𝑦)>0,𝑏₁(𝑥,𝑦)≧0,(iii)∑^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖+∑^𝑘_𝑗=1𝜌²_𝑗≧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 (i)(𝑓,𝑔_𝜆) is weak strictly-pseudo (𝑑-𝜌-𝜂-𝜃)-type I univex at 𝑦, (ii)for any 𝑢∈𝑅^𝑝,𝑢≤0⇒𝜙_𝑜(𝑢)≤0 and 𝑣∈𝑅^𝑘,𝑣≧0⇒𝜙₁(𝑣)≧0;𝑏_𝑜(𝑥,𝑦)>0,𝑏₁(𝑥,𝑦)≧0,(iii)∑^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖+∑^𝑘_𝑗=1𝜌²_𝑗≧0, then 𝑓(𝑥)≰𝑓(𝑦).

Corollary 4.4. Let 𝑥_𝑜 and (𝑦_𝑜,𝜇,𝜆,𝜂) be the feasible solutions for (MP) and (MWD), respectively, such that 𝑓(𝑥_𝑜)=𝑓(𝑦_𝑜). If the weak duality holds between (MP) and (MWD) for all feasible solutions of two problems, then 𝑥_𝑜 is efficient for (MP) and (𝑦_𝑜,𝜇,𝜆,𝜂) is efficient for (MWD).

Proof. Suppose that 𝑥_𝑜 is not efficient for (MP), then for some 𝑥∈𝑋𝑓(𝑥)≤𝑓𝑥_𝑜=𝑓𝑦_𝑜. (4.7) which contradicts weak duality theorems as (𝑦_𝑜,𝜇,𝜆,𝜂) is feasible for (MWD) and 𝑥 is feasible for (MP). So, 𝑥_𝑜 is efficient for (MP). Similarly (𝑦_𝑜,𝜇,𝜆,𝜂) is efficient for (MWD).

Theorem 4.5 (Strong Duality). Let 𝑥_𝑜 be a weakly efficient solution of (MP), 𝑔_𝑗 is continuous at 𝑥_𝑜 for 𝑗∈𝐽(̃𝑥_𝑜), 𝑓,𝑔 are directionally differentiable at 𝑥_𝑜 with 𝑓^′_𝑖(𝑥_𝑜,𝜂(𝑥,𝑥_𝑜)), and 𝑔^′_𝑗(𝑥_𝑜,𝜂(𝑥,𝑥_𝑜)) as preinvex functions on 𝑋. Also if 𝑔 satisfies the generalized Slater's constraint qualification at 𝑥_𝑜, then ∃ 𝜇∈𝑅^𝑝_⩾,𝜆∈𝑅^𝑘_≧ such that (𝑥_𝑜,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 (𝑥_𝑜,𝜇,𝜆,𝜂) is an efficient solution of (MWD).

Proof. Since 𝑥_𝑜 is a weakly efficient solution of (MP), therefore by Theorem 3.5, there exists 𝜇∈𝑅^𝑝_⩾,𝜆∈𝑅^𝑘_≧ such that ^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜,𝜂𝑥,𝑥_𝑜+^𝑘_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑥_𝑜,𝜂𝑥,𝑥_𝑜𝜆≧0∀𝑥∈𝑋,_𝑗𝑔_𝑗𝑥_𝑜=0,𝑗=1,𝑘. (4.8)
It follows that (𝑥_𝑜,𝜇,𝜆,𝜂)∈𝑊 and therefore feasible for (MWD). Clearly objective function values of (MP) and (MWD) are equal at optimal points.
Suppose (𝑥_𝑜,𝜇,𝜆,𝜂) is not an efficient solution for (MWD). Then ∃(𝑦,𝜇,𝜆,𝜂)∈𝑊 such that 𝑓(𝑥_𝑜)≤𝑓(𝑦), which contradicts weak duality theorems. Therefore (𝑥_𝑜,𝜇,𝜆,𝜂) is an efficient solution of (MWD). Hence the proof.

Theorem 4.6 (Converse Duality). Let (𝑦_𝑜,𝜇,𝜆,𝜂) be a feasible solution of (MWD). If (i)(𝑓,𝑔_𝜆) is a weak strictly-pseudo-quasi (𝑑-𝜌-𝜂-𝜃)-type I univex at 𝑦_𝑜, (ii)for any 𝑢∈𝑅^𝑝,𝑢≤0⇒𝜙_𝑜(𝑢)≤0 and 𝑣∈𝑅^𝑘,𝑣≧0⇒𝜙₁(𝑣)≧0;   𝑏_𝑜(𝑥_𝑜,𝑦_𝑜)>0,𝑏₁(𝑥_𝑜,𝑦_𝑜)≧0,(iii)∑^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖+∑^𝑘_𝑗=1𝜌²_𝑗≧0, then 𝑦_𝑜 is an efficient solution of (MP).

Proof. Suppose that 𝑦_𝑜 is not an efficient solution of (MP). Then ∃𝑥_𝑜∈𝑋 such that 𝑓𝑥_𝑜≤𝑓𝑦₀. (4.9) Now proceeding as in Theorem 4.1 (Weak Duality), we obtain a contradiction. Hence 𝑦_𝑜 is an efficient solution of (MP).

Theorem 4.7 (Strict Converse Duality). Let 𝑥_𝑜 and (𝑦_𝑜,𝜇,𝜆,𝜂) be the feasible solutions of (MP) and (MWD), respectively. If (i)𝑓(𝑥_𝑜)≦𝑓(𝑦_𝑜), (ii)(𝑓,𝑔_𝜆) is a weak quasi-strictly-pseudo (𝑑-𝜌-𝜂-𝜃)-type I univex at 𝑦_𝑜, (iii)for any 𝑢∈𝑅^𝑝,𝑢≦0⇒𝜙_𝑜(𝑢)≤0 and 𝑣∈𝑅^𝑘,𝑣≧0⇒𝜙₁(𝑣)≧0;𝑏_𝑜(𝑥_𝑜,𝑦_𝑜)>0,𝑏₁(𝑥_𝑜,𝑦_𝑜)≧0,(iv)∑^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖+∑^𝑘_𝑗=1𝜌²_𝑗≧0, then 𝑥_𝑜=𝑦_𝑜.

Proof. Suppose 𝑥_𝑜≠𝑦_𝑜.
Since 𝑦_𝑜 is a feasible solution of (MWD), therefore by hypothesis (i) and hypothesis (iii), we get 𝑏_𝑜𝑥_𝑜,𝑦_𝑜𝜙_𝑜𝑓𝑥_𝑜−𝑓𝑦_𝑜𝑏≤0,₁𝑥_𝑜,𝑦_𝑜𝜙₁𝑔_𝜆𝑦_𝑜≧0. (4.10) By hypothesis (ii), we obtain 𝑓^′𝑦_𝑜;𝜂𝑥_𝑜,𝑦_𝑜≦−𝜌¹𝜃(𝑥_𝑜,𝑦_𝑜)²,𝑔^′_𝜆𝑦_𝑜;𝜂𝑥_𝑜,𝑦_𝑜≤−𝜌²𝜃𝑥_𝑜,𝑦_𝑜². (4.11) Since 𝜇∈𝑅^𝑝_⩾, therefore the above inequalities yield ^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑦_𝑜;𝜂𝑥_𝑜,𝑦_𝑜≦−^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖𝜃(𝑥_𝑜,𝑦_𝑜)²,_𝑘𝑗=1𝜆_𝑗𝑔^′_𝑗𝑦_𝑜;𝜂𝑥_𝑜,𝑦_𝑜<−^𝑘_𝑗=1𝜌²_𝑗𝜃𝑥_𝑜,𝑦_𝑜², (4.12)(4.13) which on adding gives ^𝑝_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑦_𝑜;𝜂𝑥_𝑜,𝑦_𝑜+^𝑘_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑦_𝑜;𝜂𝑥_𝑜,𝑦_𝑜<−^𝑝_𝑖=1𝜇_𝑖𝜌¹_𝑖+^𝑘_𝑗=1𝜌²_𝑗𝜃(𝑥_𝑜,𝑦_𝑜)²≦0usinghypothesis(iv), (4.14) which is a contradiction to feasibility of 𝑦_𝑜. Hence 𝑥_𝑜=𝑦_𝑜.