Sufficiency and Duality in Nonsmooth Multiobjective Programming Problem under Generalized Univex Functions

Abstract

We consider a nonsmooth multiobjective programming problem where the functions involved are nondifferentiable. The class of univex functions is generalized to a far wider class of (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-V-type I univex functions. Then, through various nontrivial examples, we illustrate that the class introduced is new and extends several known classes existing in the literature. Based upon these generalized functions, Karush-Kuhn-Tucker type sufficient optimality conditions are established. Further, we derive weak, strong, converse, and strict converse duality theorems 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. Consequently, various generalizations of convex functions have been introduced in the literature (see Hanson [1], Vial [2], Hanson and Mond [3], Jeyakumar and Mond [4], Hanson et al. [5], Liang et al. [6], and Gulati et al. [7]).

Nonsmooth optimization provides analytical tools for studying optimization problems involving functions that are not differentiable in the usual sense. Several nonlinear analysis problems arise from areas of optimization theory, game theory, differential equations, mathematical physics, convex analysis, and nonlinear functional analysis. For a nondifferentiable multiobjective programming problem, there exists a generalization of invexity to locally Lipschitz functions with gradients replaced by the Clarke generalized subgradient. Instead of Clarke generalized subgradient, Ye [8] used the concept of directional derivative to define the class of 𝑑 invex functions. Also, he derived necessary and sufficient optimality conditions taking functions 𝑓^′(𝑥_𝑜;𝑦) and 𝑔^′_𝐽(𝑥_𝑜;𝑦) to be convex. However, Antczak [9] considered the directional derivatives of objective and constraint functions to be preinvex and derived duality results for Wolfe type, Mond-Weir type, and mixed type dual programs. Mishra and Noor [10] extended the class of functions to 𝑑-𝑉-type I functions and obtained sufficient optimality and duality results for Mond-Weir type multiobjective dual program. Nahak and Mohapatra [11] obtained duality results for multiobjective programming problem under (𝑑-𝜌-𝜂-𝜃) invexity assumptions. Slimani and Radjef [12] introduced a far wider class of nondifferentiable functions called 𝑑_𝐼-𝑉-type I functions in which each component is directionally differentiable in its own direction instead of the same direction and established sufficient optimality and duality results.

On the other hand, Bector et al. [13] generalized the notion of convexity to univex functions. Rueda et al. [14] obtained optimality and duality results for several mathematical programs by combining the concepts of type I and univex functions. Mishra [15] obtained optimality results and saddle point results for multiobjective programs under generalized type I univex functions. Generalizing the functions, Mishra et al. [16] obtained duality results for a nondifferentiable multiobjective programming problem under generalized 𝑑-univexity. As an extension, Ahmad [17] introduced a new class of 𝑑_𝐼-𝑉-type I univex functions which was generalized to a class of (𝑑_𝐼-𝜌-𝜎)-𝑉-type I univex functions by Kharbanda et al. [18].

In this paper, we introduce a new generalized class of (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex functions which generalizes the class of functions introduced by Kharbanda et al. [18], Ahmad [17], Slimani and Radjef [12], Mishra and Noor [10], Mishra et al. [16], Antczak [9], Suneja and Srivastava [19], and Ye [8]. Further, we establish weak, strong, converse, and strict converse duality results for Mond-Weir type multiobjective dual program.

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, and let 𝑥_𝑜 be an arbitrary point of 𝐷 and ℎ∶𝐷→𝑅,𝑓∶𝐷→𝑅^𝑚, 𝜙∶𝑅→𝑅, 𝑏∶𝐷×𝐷→𝑅₊. Also, we denote 𝑅^𝑚_≥={𝑦∶𝑦∈𝑅^𝑚 and 𝑦≥0},  𝑅^𝑘_≧(>)={𝑦∶𝑦∈𝑅^𝑘 and 𝑦≧0   (𝑦>0)} and 𝑖=1,𝑚={1,2,…,𝑚},  𝑗=1,𝑘={1,2,…,𝑘}.

Definition 1 (Weir and Mond [20] and Weir and Jeyakumar [21]). The function ℎ is called preinvex on 𝐷 if, for all 𝑥,𝑥_𝑜∈𝐷, there exists a vector function 𝜂 such that ∀𝜆∈[0,1],  𝑥_𝑜+𝜆𝜂(𝑥,𝑥_𝑜)∈𝐷, one has 𝜆ℎ(𝑥)+(1−𝜆)ℎ𝑥_𝑜≧ℎ𝑥_𝑜+𝜆𝜂𝑥,𝑥_𝑜. (1)

Definition 2 (Mititelu [22]). The set 𝐷 is said to be invex at 𝑥_𝑜 with respect to 𝜂, if, for each 𝑥∈𝐷, 𝑥_𝑜+𝜆𝜂𝑥,𝑥_𝑜∈𝐷,∀𝜆∈[0,1]. (2)𝐷is said to be an invex set with respect to 𝜂, if 𝐷 is invex at each 𝑥_𝑜∈𝐷 with respect to same 𝜂.

Definition 3 (Antczak [9]). Let 𝐷⊆𝑅^𝑛 be an invex set. An 𝑚-dimensional vector-valued function 𝜓∶𝐷→𝑅^𝑚 is said to be preinvex with respect to 𝜂, if each of its components is preinvex on 𝐷 with respect to the same function 𝜂.

Definition 4 (Clarke [23]). The function ℎ is said to be locally Lipschitz at 𝑥_𝑜∈𝐷, if there exist a neighbourhood 𝑣(𝑥_𝑜) of 𝑥_𝑜 and a constant 𝑘>0 such that ℎ𝑦−ℎ(𝑥)≦𝑘𝑦−𝑥,∀𝑥,𝑦∈𝑣𝑥_𝑜, (3) where ‖⋅‖ denotes the Euclidean norm. Also, one says that ℎ is locally Lipschitz on 𝐷 if it is locally Lipschitz at every point of 𝐷.

Definition 5 (Bector et al. [13]). A differentiable function ℎ is said to be univex at 𝑥_𝑜 with respect to 𝜙,𝜂,𝑏 if, ∀𝑥∈𝐷, one has 𝑏𝑥,𝑥_𝑜𝜙ℎ(𝑥)−ℎ𝑥_𝑜≧∇ℎ(𝑥_𝑜)^𝑇𝜂𝑥,𝑥_𝑜. (4)

Definition 6 (Mishra et al. [16]). Let 𝐷⊆𝑅^𝑛 be a nonempty open set. The function 𝑓 is called 𝑑-univex at 𝑥_𝑜∈𝐷 with respect to 𝜙,𝜂,𝑏 if it is directionally differentiable at 𝑥_𝑜 such that, for any 𝑥∈𝐷, 𝑏𝑥,𝑥_𝑜𝜙𝑓_𝑖(𝑥)−𝑓_𝑖𝑥_𝑜≧𝑓^′_𝑖𝑥_𝑜;𝜂𝑥,𝑥_𝑜,∀𝑖=1,𝑚, (5) where 𝑓^′_𝑖(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜)) denotes the directional derivative of 𝑓_𝑖 at 𝑥_𝑜 in the direction 𝜂(𝑥,𝑥_𝑜): 𝑓^′_𝑖𝑥_𝑜;𝜂𝑥,𝑥_𝑜=lim_𝜆→0+𝑓_𝑖𝑥_𝑜+𝜆𝜂𝑥,𝑥_𝑜−𝑓_𝑖𝑥_𝑜𝜆. (6) If the above inequalities are satisfied at any point 𝑥_𝑜∈𝐷, then 𝑓 is said to be 𝑑-univex on 𝐷 with respect to 𝜂.

Definition 7 (Slimani and Radjef [12]). The function ℎ is said to be semidirectionally differentiable at 𝑥_𝑜∈𝐷 in the direction 𝜂(𝑥,𝑥_𝑜) if its directional derivative ℎ^′(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜)) exists finite for all 𝑥∈𝐷.

Definition 8 (Slimani and Radjef [12]). Let 𝐷⊆𝑅^𝑛 be a nonempty open set. The function 𝑓 is called 𝑑_𝐼-invex at 𝑥_𝑜∈𝐷 with respect to (𝜂_𝑖)_𝑖=1,𝑚, if, for any 𝑥∈𝐷, 𝑓_𝑖(𝑥)−𝑓_𝑖𝑥_𝑜≧𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜,∀𝑖=1,𝑚, (7) where 𝑓_𝑖 is semidirectionally differentiable at 𝑥_𝑜 in direction 𝜂_𝑖∶𝐷×𝐷→𝑅^𝑛, for 𝑖=1,𝑚.

We consider the following nonlinear multiobjective programming problem: (MP)Minimize𝑓(𝑥)=𝑓₁(𝑥),𝑓₂(𝑥),…,𝑓_𝑚(𝑥)subjectto𝑔(𝑥)≦0, (8) where 𝑥∈𝐷 and the functions 𝑓∶𝐷→𝑅^𝑚, 𝑔∶𝐷→𝑅^𝑘, and 𝐷 is a nonempty open subset of 𝑅^𝑛. Let 𝑋={𝑥∈𝐷∶𝑔(𝑥)≦0} be the set of feasible solutions of (MP). For 𝑥_𝑜∈𝐷, if we denote 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, (9) then 𝐽𝑥_𝑜∪𝐽̃𝑥_𝑜∪𝐽𝑥_𝑜={1,2,…,𝑘}. (10)

Now we define a new class of (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex functions where 𝜑∶𝐷×𝐷×𝑅→𝑅 is a functional which for any 𝑥,𝑥_𝑜∈𝐷 satisfies the following properties:(i)𝜑(𝑥,𝑥_𝑜;𝑎₁+𝑎₂)≦𝜑(𝑥,𝑥_𝑜;𝑎₁)+𝜑(𝑥,𝑥_𝑜;𝑎₂), for all 𝑎₁,𝑎₂∈𝑅, (subadditive in third argument);(ii)𝜑(𝑥,𝑥_𝑜;𝛼𝑎)=𝛼𝜑(𝑥,𝑥_𝑜;𝑎), for all 𝛼∈𝑅,  𝛼≧0 and for all 𝑎∈𝑅, (positive homogeneous in third argument);(iii)𝜑(𝑥,𝑥_𝑜;𝑎)>0, for all 𝑎>0.

Let the functions 𝑓∶𝐷→𝑅^𝑚  and  𝑔∶𝐷→𝑅^𝑘 where 𝑓_𝑖 and 𝑔_𝑗 are semidirectionally differentiable functions in the directions 𝜂_𝑖∶𝑋×𝐷→𝑅^𝑛 and 𝜃_𝑗∶𝑋×𝐷→𝑅^𝑛 for 𝑖=1,𝑚 and 𝑗=1,𝑘. Also, let 𝛼,𝛼 be the vectors in 𝑅^𝑚+𝑘 whose components are the functions 𝛼¹_𝑖,𝛼²_𝑗∶𝑋×𝐷→𝑅₊⧵{0},𝛼¹_𝑖,𝛼²_𝑗∶𝑋×𝐷→𝑅₊⧵{0}, respectively, for 𝑖=1,𝑚, 𝑗=1,𝑘, while 𝜌∈𝑅^𝑚+𝑘 and 𝜌∈𝑅² whose components are in 𝑅 and 𝜎∶𝑋×𝐷→𝑅^𝑛;  𝑏_𝑜 and 𝑏₁ are nonnegative functions defined on 𝑋×𝐷, 𝜙_𝑜∶𝑅→𝑅, and 𝜙₁∶𝑅→𝑅.

Definition 9. (𝑓,𝑔) is said to be (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑥_𝑜∈𝐷 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_𝑗=1,𝑘 if for all 𝑥∈𝑋𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜𝑓_𝑖(𝑥)−𝑓_𝑖𝑥_𝑜≧𝜑𝑥,𝑥_𝑜;𝛼¹_𝑖𝑥,𝑥_𝑜𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜+𝜌¹_𝑖𝜎(𝑥,𝑥_𝑜)²,∀𝑖=1,𝑚,−𝑏₁𝑥,𝑥_𝑜𝜙₁𝑔_𝑗𝑥_𝑜≧𝜑𝑥,𝑥_𝑜;𝛼²_𝑗𝑥,𝑥_𝑜𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜+𝜌²_𝑗𝜎(𝑥,𝑥_𝑜)²,∀𝑗=1,𝑘. (11)

If the inequalities in 𝑓 are strict (whenever 𝑥≠𝑥_𝑜), then (𝑓,𝑔) is said to be semistrictly (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑥_𝑜 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_𝑗=1,𝑘.

Remark 10. (i) If in the above definition,  𝜑(𝑥,𝑥_𝑜;𝛼¹_𝑖(𝑥,𝑥_𝑜)𝑓^′_𝑖(𝑥_𝑜;𝜂_𝑖(𝑥,𝑥_𝑜))) = 𝛼¹_𝑖(𝑥,𝑥_𝑜)𝑓^′_𝑖(𝑥_𝑜;𝜂_𝑖(𝑥,𝑥_𝑜)), 𝜑(𝑥,𝑥_𝑜;𝛼²_𝑗(𝑥,𝑥_𝑜)𝑔^′_𝑗(𝑥_𝑜;𝜃_𝑗(𝑥,𝑥_𝑜))) = 𝛼²_𝑗(𝑥,𝑥_𝑜)𝑔^′_𝑗(𝑥_𝑜;𝜃_𝑗(𝑥,𝑥_𝑜)), for all 𝑖=1,𝑚 and 𝑗=1,𝑘, then we obtain the definition of (𝑑_𝐼-𝜌-𝜎)-𝑉-type I univex function given by Kharbanda et al. [18]. Also, if in addition, we take 𝜌¹_𝑖,𝜌²_𝑗=0, then the above definition reduces to definition of 𝑑_𝐼-𝑉-type I univex function introduced by Ahmad [17].
(ii) If 𝜑 is same as in (i) and 𝜌¹_𝑖,𝜌²_𝑗=0,  𝛼¹_𝑖(𝑥,𝑥_𝑜)=𝛼²_𝑗(𝑥,𝑥_𝑜)=1, for all 𝑖=1,𝑚 and  𝑗=1,𝑘 and 𝑏_𝑜(𝑥,𝑥_𝑜)=𝑏₁(𝑥,𝑥_𝑜)=1, 𝜙_𝑜(𝑡)=𝑡, and 𝜙₁(𝑡)=𝑡, then the above definition becomes definition of 𝑑_𝐼-𝑉-type I function introduced by Slimani and Radjef [12].
(iii) If 𝑓 and 𝑔 are differentiable functions and 𝜑(𝑥,𝑥_𝑜;𝛼¹_𝑖(𝑥,𝑥_𝑜)𝑓^′_𝑖(𝑥_𝑜;𝜂_𝑖(𝑥,𝑥_𝑜))) = 𝛼¹_𝑖(𝑥,𝑥_𝑜)[∇𝑓_𝑖(𝑥_𝑜)]^𝑇𝜂(𝑥,𝑥_𝑜), 𝜑(𝑥,𝑥_𝑜;𝛼²_𝑗(𝑥,𝑥_𝑜)𝑔^′_𝑗(𝑥_𝑜;𝜃_𝑗(𝑥,𝑥_𝑜))) = 𝛼²_𝑗(𝑥,𝑥_𝑜)[∇𝑔_𝑗(𝑥_𝑜)]^𝑇𝜂(𝑥,𝑥_𝑜), and 𝜌¹_𝑖,𝜌²_𝑗=0,  𝑖=1,𝑚 and  𝑗=1,𝑘  and  𝑏_𝑜(𝑥,𝑥_𝑜)=𝑏₁(𝑥,𝑥_𝑜)=1,  𝜙_𝑜(𝑡)=𝑡,  𝜙₁(𝑡)=𝑡, then above definition reduces to 𝑉-type I functions given by Hanson et al. [5]. Also, if, in addition, we take 𝛼¹_𝑖(𝑥,𝑥_𝑜)=𝛼²_𝑗(𝑥,𝑥_𝑜)=1, then we get the definition of type I function defined by Hanson and Mond [3].
(iv) If, in the above definition,  𝜑(𝑥,𝑥_𝑜;𝛼¹_𝑖(𝑥,𝑥_𝑜)𝑓^′_𝑖(𝑥_𝑜;𝜂_𝑖(𝑥,𝑥_𝑜))) = 𝛼¹_𝑖(𝑥,𝑥_𝑜)𝑓^′_𝑖(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜)), 𝜑(𝑥,𝑥_𝑜;𝛼²_𝑗(𝑥,𝑥_𝑜)𝑔^′_𝑗(𝑥_𝑜;𝜃_𝑗(𝑥,𝑥_𝑜))) = 𝛼²_𝑗(𝑥,𝑥_𝑜)𝑔^′_𝑗(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜)), and 𝜌¹_𝑖,𝜌²_𝑗=0, 𝑖=1,𝑚, and 𝑗=1,𝑘 and 𝑏_𝑜(𝑥,𝑥_𝑜)=𝑏₁(𝑥,𝑥_𝑜)=1, 𝜙_𝑜(𝑡)=𝑡, and 𝜙₁(𝑡)=𝑡, then we obtain the definition of 𝑑-type I function introduced by Suneja and Srivastava [19].

Definition 11. (𝑓,𝑔) is said to be quasi (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑥_𝑜∈𝐷 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_𝑗=1,𝑘, if for some vectors 𝜇∈𝑅^𝑚_≧,  𝜆∈𝑅^𝑘_≧ and for all 𝑥∈𝑋𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜^𝑚_𝑖=1𝜇_𝑖𝛼¹_𝑖𝑥,𝑥_𝑜𝑓_𝑖(𝑥)−𝑓_𝑖𝑥_𝑜≦0⟹𝜑𝑥,𝑥_𝑜;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜≦−𝜌¹𝜎(𝑥,𝑥_𝑜)²,𝑏₁𝑥,𝑥_𝑜𝜙₁^𝑘_𝑗=1𝜆_𝑗𝛼²_𝑗𝑥,𝑥_𝑜𝑔_𝑗𝑥_𝑜≧0⟹𝜑𝑥,𝑥_𝑜;^𝑘_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜≦−𝜌²𝜎(𝑥,𝑥_𝑜)². (12)

If the second (implied) inequality in 𝑓 is strict (𝑥≠𝑥_𝑜), then (𝑓,𝑔) is said to be semistrictly quasi (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑥_𝑜 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_𝑗=1,𝑘.

Definition 12. (𝑓,𝑔) is said to be pseudo (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑥_𝑜∈𝐷 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_𝑗=1,𝑘, if for some vectors 𝜇∈𝑅^𝑚_≧,  𝜆∈𝑅^𝑘_≧ and for all 𝑥∈𝑋𝜑𝑥,𝑥_𝑜;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜≧−𝜌¹𝜎(𝑥,𝑥_𝑜)²⟹𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜^𝑚_𝑖=1𝜇_𝑖𝛼¹_𝑖𝑥,𝑥_𝑜𝑓_𝑖(𝑥)−𝑓_𝑖𝑥_𝑜𝜑≧0,𝑥,𝑥_𝑜;^𝑘_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜≧−𝜌²𝜎𝑥,𝑥_𝑜²⟹𝑏₁𝑥,𝑥_𝑜𝜙₁^𝑘_𝑗=1𝜆_𝑗𝛼²_𝑗𝑥,𝑥_𝑜𝑔_𝑗𝑥_𝑜≦0. (13)

If the second (implied) inequality in 𝑓 (resp., 𝑔) is strict (𝑥≠𝑥_𝑜), then (𝑓,𝑔) is said to be semistrictly pseudo (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex in 𝑓 (resp., 𝑔) and if the second (implied) inequalities in 𝑓 and 𝑔 are both strict, then (𝑓,𝑔) is said to be strictly pseudo (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑥_𝑜 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_𝑗=1,𝑘.

Definition 13. (𝑓,𝑔) is said to be quasi-pseudo (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑥_𝑜∈𝐷 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_𝑗=1,𝑘, if for some vectors 𝜇∈𝑅^𝑚_≧,  𝜆∈𝑅^𝑘_≧ and for all 𝑥∈𝑋𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜^𝑚_𝑖=1𝜇_𝑖𝛼¹_𝑖𝑥,𝑥_𝑜𝑓_𝑖(𝑥)−𝑓_𝑖𝑥_𝑜≦0⟹𝜑𝑥,𝑥_𝑜;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜≦−𝜌¹𝜎(𝑥,𝑥_𝑜)²,𝜑𝑥,𝑥_𝑜;^𝑘_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜≧−𝜌²𝜎(𝑥,𝑥_𝑜)²⟹𝑏₁𝑥,𝑥_𝑜𝜙₁^𝑘_𝑗=1𝜆_𝑗𝛼²_𝑗𝑥,𝑥_𝑜𝑔_𝑗𝑥_𝑜≦0. (14)

If the second (implied) inequality in 𝑔 is strict (𝑥≠𝑥_𝑜), then (𝑓,𝑔) is said to be quasistrictly pseudo (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑥_𝑜 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_𝑗=1,𝑘.

Definition 14. (𝑓,𝑔) is said to be pseudo-quasi (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑥_𝑜∈𝐷 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_𝑗=1,𝑘, if for some vectors 𝜇∈𝑅^𝑚_≧,  𝜆∈𝑅^𝑘_≧ and for all 𝑥∈𝑋𝜑𝑥,𝑥_𝑜;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜≧−𝜌¹𝜎(𝑥,𝑥_𝑜)²⟹𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜^𝑚_𝑖=1𝜇_𝑖𝛼¹_𝑖𝑥,𝑥_𝑜𝑓_𝑖(𝑥)−𝑓_𝑖𝑥_𝑜≧0,𝑏₁𝑥,𝑥_𝑜𝜙₁^𝑘_𝑗=1𝜆_𝑗𝛼²_𝑗𝑥,𝑥_𝑜𝑔_𝑗𝑥_𝑜≧0⟹𝜑𝑥,𝑥_𝑜;^𝑘_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜≦−𝜌²𝜎(𝑥,𝑥_𝑜)². (15)(16)

If the second (implied) inequality in 𝑓 is strict (𝑥≠𝑥_𝑜), then (𝑓,𝑔) is said to be strictly pseudo-quasi (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑥_𝑜 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_𝑗=1,𝑘.

3. Illustration

In this section, we give some nontrivial examples which illustrate that the class of functions introduced in this paper is nonempty.

Example 15. Let 𝑓∶𝑅²→𝑅 and 𝑔∶𝑅²→𝑅 be defined by 𝑓𝑥₁,𝑥₂=𝑥²₁+𝑥²₂+𝑥₁+4if𝑥₁≠0,𝑥₂≠04if𝑥₁=0,𝑥₂=08+𝑥²₁+𝑥²₂𝑔𝑥else,₁,𝑥₂=−2−4𝑥²₁−6𝑥²₂if𝑥₁≠0,𝑥₂≠00else. (17) Let 𝜂(𝑥,𝑦)=(1+𝑥²₁,2+𝑦²₂), 𝜃(𝑥,𝑦)=(0,𝑥₂), 𝜑(𝑥,𝑦;𝑎)=(|𝑎|/2)(1+𝑥²₁𝑦²₂), 𝜎(𝑥,𝑦)=(𝑥₁+𝑦₁,1+𝑦₂), and 𝛼¹(𝑥,𝑦)=8,  𝛼²(𝑥,𝑦)=4 where 𝑥=(𝑥₁,𝑥₂), 𝑦=(𝑦₁,𝑦₂).
Also, let 𝑏_𝑜(𝑥,𝑦)=1/2, 𝑏₁(𝑥,𝑦)=4, 𝜙_𝑜(𝑡)=4𝑡, 𝜙₁(𝑡)=4𝑡, 𝜌¹=−4, and 𝜌²=−2.
The set 𝑋 of feasible solutions of problem is nonempty. Clearly, 𝑓 and 𝑔 are semidirectionally differentiable at 𝑥_𝑜=(0,0) with 𝑓^′(𝑥_𝑜;𝜂(𝑥,𝑥_𝑜))=1+𝑥²₁ and 𝑔^′(𝑥_𝑜;𝜃(𝑥,𝑥_𝑜))=0.
It is easy to see that for all 𝑥∈𝑋𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜𝑓(𝑥)−𝑓𝑥_𝑜≧𝜑𝑥,𝑥_𝑜;𝛼¹𝑥,𝑥_𝑜𝑓^′𝑥_𝑜;𝜂𝑥,𝑥_𝑜+𝜌¹𝜎(𝑥,𝑥_𝑜)²,−𝑏₁𝑥,𝑥_𝑜𝜙₁𝑔𝑥_𝑜≧𝜑𝑥,𝑥_𝑜;𝛼²𝑥,𝑥_𝑜𝑔^′𝑥_𝑜;𝜃𝑥,𝑥_𝑜+𝜌²𝜎(𝑥,𝑥_𝑜)². (18) Therefore (𝑓,𝑔) is (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I function at 𝑥_𝑜.
However, if we take 𝑥=(1,1), then (i)𝑏_𝑜(𝑥,𝑥)𝜙_𝑜𝑓(𝑥)−𝑓𝑥_𝑜<𝛼¹𝑥,𝑥_𝑜𝑓^′𝑥_𝑜;𝜂𝑥,𝑥_𝑜+𝜌¹𝜎𝑥,𝑥_𝑜²,𝑏_𝑜(𝑥,𝑥)𝜙_𝑜𝑓(𝑥)−𝑓𝑥_𝑜<𝛼¹𝑥,𝑥_𝑜𝑓^′𝑥_𝑜;𝜂𝑥,𝑥_𝑜. (19)
Thus (𝑓,𝑔) is neither (𝑑_𝐼-𝜌-𝜎)-𝑉-type I univex function given by Kharbanda et al. [18] nor 𝑑_𝐼-𝑉-type I univex function at 𝑥_𝑜 as given by Ahmad [17].
Hence the above example clearly illustrates that the class of (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex functions is more generalized than the class of (𝑑_𝐼-𝜌-𝜎)-𝑉-type I univex functions and the class of 𝑑_𝐼-𝑉-type I univex functions.
Next we show that (𝑓,𝑔) is pseudo-quasi (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex function but not (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex function.

Example 16. Let 𝑓∶𝑅²→𝑅³ and 𝑔∶𝑅²→𝑅² be defined by 𝑓₁𝑥₁,𝑥₂=4𝑥²₁−2𝑥₂+4𝑥²₂if𝑥₁≠0,𝑥₂≠09+9𝑥²₂if𝑥₁=0,𝑥₂𝑓≠00else,₂𝑥₁,𝑥₂=2𝑥²₂𝑥²₁+6+𝑥₂if𝑥₁≠0,𝑥₂𝑓≠00else,₃𝑥₁,𝑥₂=1+6𝑥²₁if𝑥₁≠0,𝑥₂=02+3𝑥²₂if𝑥₁=0,𝑥₂𝑔≠00else,₁𝑥₁,𝑥₂=0if𝑥₁=0or𝑥₂=0−1+𝑥²₁𝑔else,₂𝑥₁,𝑥₂=2−𝑥²₁if𝑥₁≠0,𝑥₂=02−6𝑥²₂if𝑥₁=0,𝑥₂≠0−1else. (20) Let 𝜂₁(𝑥,𝑦)=(1+𝑥²₁+𝑦²₁,1+𝑥²₂), 𝜂₂(𝑥,𝑦)=(0,𝑥₁+𝑥₂), 𝜂₃(𝑥,𝑦)=(4+𝑥⁴₁,2+𝑥²₂+𝑦²₂), 𝜃₁(𝑥,𝑦)=(1+𝑥₁,0), 𝜃₂(𝑥,𝑦)=(2+𝑥⁴₁,2+2𝑥²₂), 𝜑(𝑥,𝑦;𝑎)=|𝑎|(𝑥²₁+𝑦²₁), 𝑏_𝑜(𝑥,𝑦)=4, 𝑏₁(𝑥,𝑦)=4, 𝜙_𝑜(𝑡)=𝑡/2,  𝜙₁(𝑡)=4𝑡, and 𝜎(𝑥,𝑦)=(𝑥₁+𝑦₁,𝑥₂+𝑦₂).
Also, let 𝛼¹₁(𝑥,𝑦)=2, 𝛼¹₂(𝑥,𝑦)=1, 𝛼¹₃(𝑥,𝑦)=1, 𝛼²₁(𝑥,𝑦)=1, 𝛼²₂(𝑥,𝑦)=3, 𝜌¹=1, 𝜌²=−4, 𝜌¹₁=4, 𝜌¹₂=1, 𝜌¹₃=−6, 𝜌²₁=−4, 𝜌²₂=−6, 𝜇₁=𝜇₂=1/2,  𝜇₃=0, and 𝜆₁=1,  𝜆₂=0 where 𝑥=(𝑥₁,𝑥₂), 𝑦=(𝑦₁,𝑦₂), 𝛼¹_𝑖(𝑥,𝑦)=1/𝛼¹_𝑖(𝑥,𝑦), and 𝛼²_𝑗(𝑥,𝑦)=1/𝛼²_𝑗(𝑥,𝑦), 𝑖=1,2,3, 𝑗=1,2.
The set 𝑋 of feasible solutions of problem is nonempty. Clearly, 𝑓₁,𝑓₂,𝑓₃ and 𝑔₁,𝑔₂ are semidirectionally differentiable at 𝑥_𝑜=(0,0) with 𝑓^′₁(𝑥_𝑜;𝜂₁(𝑥,𝑥_𝑜))=−2(1+𝑥²₂),  𝑓^′₂(𝑥_𝑜;𝜂₂(𝑥,𝑥_𝑜))=0, 𝑓^′₃(𝑥_𝑜;𝜂₃(𝑥,𝑥_𝑜))=0, 𝑔^′₁(𝑥_𝑜;𝜃₁(𝑥,𝑥_𝑜))=0, and 𝑔^′₂(𝑥_𝑜;𝜃₂(𝑥,𝑥_𝑜))=0.
It is easy to see that for all 𝑥∈𝑋𝜑𝑥,𝑥_𝑜;³_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜+𝜌¹𝜎(𝑥,𝑥_𝑜)²=𝑥²₂𝑥²₁+𝑥²₂+2𝑥²₁≧0⟹𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜³_𝑖=1𝜇_𝑖𝛼¹_𝑖𝑥,𝑥_𝑜𝑓_𝑖(𝑥)−𝑓_𝑖𝑥_𝑜𝑏≧0,₁𝑥,𝑥_𝑜𝜙₁²_𝑗=1𝜆_𝑗𝛼²_𝑗𝑥,𝑥_𝑜𝑔_𝑗𝑥_𝑜=0≧0⟹𝜑𝑥,𝑥_𝑜;²_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜+𝜌²𝜎𝑥,𝑥_𝑜²=−4𝑥²₁+𝑥²₂≦0. (21) Therefore (𝑓,𝑔) is pseudo-quasi (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex function at 𝑥_𝑜.
However, for the above defined problem, if we take (i)𝑥=(1,1), 𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜𝑓₁(𝑥)−𝑓₁𝑥_𝑜<𝜑𝑥,𝑥_𝑜;𝛼¹₁𝑥,𝑥_𝑜𝑓^′₁𝑥_𝑜;𝜂₁𝑥,𝑥_𝑜+𝜌¹₁𝜎𝑥,𝑥_𝑜²; (22)(ii)𝑥=(1,0), 𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜𝑓₂(𝑥)−𝑓₂𝑥_𝑜<𝜑𝑥,𝑥_𝑜;𝛼¹₂𝑥,𝑥_𝑜𝑓^′₂𝑥_𝑜;𝜂₂𝑥,𝑥_𝑜+𝜌¹₂𝜎(𝑥,𝑥_𝑜)². (23)So (𝑓,𝑔) is not (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex function at 𝑥_𝑜.

4. Sufficient Optimality Conditions

In this section, we discuss some sufficient optimality conditions for a point to be an efficient solution of (MP) under newly defined class of (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex functions.

Theorem 17. Suppose there exist a feasible solution 𝑥_𝑜 of (MP) and vector functions 𝜂_𝑖∶𝑋×𝐷→𝑅^𝑛,  𝑖=1,𝑚,  𝜃_𝑗∶𝑋×𝐷→𝑅^𝑛,  𝑗∈𝐽(𝑥_𝑜), and scalars 𝜇_𝑖>0,  𝑖=1,𝑚, and 𝜆_𝑗≧0,  𝑗∈𝐽(𝑥_𝑜) such that(i)∑^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖(𝑥_𝑜;𝜂_𝑖(𝑥,𝑥_𝑜))+∑_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝑔^′_𝑗(𝑥_𝑜;𝜃_𝑗(𝑥,𝑥_𝑜))≧0, ∀𝑥∈𝑋,(ii)for any 𝑢∈𝑅, 𝑢<0⇒𝜙_𝑜(𝑢)<0 and 𝑢≧0⇒𝜙₁(𝑢)≧0, 𝑏_𝑜(𝑥,𝑥_𝑜)>0, 𝑏₁(𝑥,𝑥_𝑜)≧0,(iii)(𝑓,𝑔_{𝐽(𝑥𝑜)}) is pseudo-quasi (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑥_𝑜 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_{𝑗∈𝐽(𝑥𝑜)}, and(iv)𝜌¹+𝜌²≧0, then 𝑥_𝑜 is an efficient solution of (MP).

Proof. Suppose that 𝑥_𝑜 is not an efficient solution of (MP). Then there exists an 𝑥∈𝑋 of (MP) such that 𝑓(𝑥)≤𝑓(𝑥_𝑜).
As 𝜇_𝑖>0,  𝛼¹_𝑖(𝑥,𝑥_𝑜)>0,  𝑖=1,𝑚, therefore ^𝑚_𝑖=1𝜇_𝑖𝛼¹_𝑖𝑥,𝑥_𝑜𝑓_𝑖(𝑥)−𝑓_𝑖𝑥_𝑜<0. (24) Also 𝑔_𝑗(𝑥_𝑜)=0,  𝜆_𝑗≧0,  𝛼²_𝑗(𝑥,𝑥_𝑜)>0,  𝑗∈𝐽(𝑥_𝑜) imply _{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝛼²_𝑗𝑥,𝑥_𝑜𝑔_𝑗𝑥_𝑜=0. (25) Since hypothesis (ii) holds, therefore inequality (24) and equality (25) become 𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜^𝑚_𝑖=1𝜇_𝑖𝛼¹_𝑖𝑥,𝑥_𝑜𝑓_𝑖(𝑥)−𝑓_𝑖𝑥_𝑜𝑏<0,₁𝑥,𝑥_𝑜𝜙_{1𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝛼²_𝑗𝑥,𝑥_𝑜𝑔_𝑗𝑥_𝑜≧0. (26) Using hypothesis (iii), we obtain 𝜑𝑥,𝑥_𝑜;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜<−𝜌¹𝜎(𝑥,𝑥_𝑜)²,𝜑𝑥,𝑥_𝑜;_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜≦−𝜌²𝜎(𝑥,𝑥_𝑜)². (27) The above inequalities along with subadditivity of 𝜑 yield 𝜑𝑥,𝑥_𝑜;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜+_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜<−𝜌¹+𝜌²𝜎𝑥,𝑥_𝑜²≦0(usinghypothesis(iv)). (28) But as hypothesis (i) holds and 𝜑(𝑥,𝑥_𝑜;0)=0 and 𝜑(𝑥,𝑥_𝑜;𝑎)>0 for 𝑎>0, therefore 𝜑𝑥,𝑥_𝑜;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜+_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜≧0. (29) Thus we get a contradiction and hence the proof.

Theorem 18. Suppose there exist a feasible solution 𝑥_𝑜 of (MP) and vector functions 𝜂_𝑖∶𝑋×𝐷→𝑅^𝑛,  𝑖=1,𝑚, and 𝜃_𝑗∶𝑋×𝐷→𝑅^𝑛,  𝑗∈𝐽(𝑥_𝑜), and scalars 𝜇_𝑖≧0, 𝑖=1,𝑚, and 𝜆_𝑗≧0,  𝑗∈𝐽(𝑥_𝑜) satisfying(i)∑^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖(𝑥_𝑜;𝜂_𝑖(𝑥,𝑥_𝑜))+∑_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝑔^′_𝑗(𝑥_𝑜;𝜃_𝑗(𝑥,𝑥_𝑜))≧0, for all 𝑥∈𝑋,(ii)for any 𝑢∈𝑅, 𝑢≦0⇒𝜙_𝑜(𝑢)≦0 and 𝑢≧0⇒𝜙₁(𝑢)≧0, 𝑏_𝑜(𝑥,𝑥_𝑜)≧0, 𝑏₁(𝑥,𝑥_𝑜)≧0,(iii)(𝑓,𝑔_{𝐽(𝑥𝑜)}) is strictly pseudo (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑥_𝑜 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_{𝑗∈𝐽(𝑥𝑜)}, and(iv)𝜌¹+𝜌²≧0, then 𝑥_𝑜 is an efficient solution of (MP).

Proof. Suppose that 𝑥_𝑜 is not an efficient solution of (MP). Then there exists 𝑥∈𝑋 of (MP) such that 𝑓(𝑥)≤𝑓(𝑥_𝑜).
As 𝜇_𝑖≧0, 𝑖=1,𝑚, 𝛼¹_𝑖(𝑥,𝑥_𝑜)>0, 𝑖=1,𝑚, and 𝑔_𝑗(𝑥_𝑜)=0, 𝜆_𝑗≧0,  𝛼²_𝑗(𝑥,𝑥_𝑜)>0, 𝑗∈𝐽(𝑥_𝑜), therefore ^𝑚_𝑖=1𝜇_𝑖𝛼¹_𝑖𝑥,𝑥_𝑜𝑓_𝑖(𝑥)−𝑓_𝑖𝑥_𝑜≦0,_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝛼²_𝑗𝑥,𝑥_𝑜𝑔_𝑗𝑥_𝑜=0. (30) Using hypothesis (ii), we obtain 𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜^𝑚_𝑖=1𝜇_𝑖𝛼¹_𝑖𝑥,𝑥_𝑜𝑓_𝑖(𝑥)−𝑓_𝑖𝑥_𝑜𝑏≦0,₁𝑥,𝑥_𝑜𝜙_{1𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝛼²_𝑗𝑥,𝑥_𝑜𝑔_𝑗𝑥_𝑜≧0. (31) Since (𝑓,𝑔_{𝐽(𝑥𝑜)}) is strictly pseudo (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑥_𝑜, therefore, the above inequalities yield 𝜑𝑥,𝑥_𝑜;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜<−𝜌¹𝜎(𝑥,𝑥_𝑜)²,𝜑𝑥,𝑥_𝑜;_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜<−𝜌²𝜎(𝑥,𝑥_𝑜)². (32) Using subadditivity of 𝜑 and hypothesis (iv), we get 𝜑𝑥,𝑥_𝑜;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜+_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜<−𝜌¹+𝜌²𝜎𝑥,𝑥_𝑜²≦0. (33) But hypothesis (i) and properties of 𝜑 imply 𝜑𝑥,𝑥_𝑜;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜+_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜≧0, (34) which leads to a contradiction. Hence 𝑥_𝑜 is an efficient solution of (MP).

In order to illustrate the result obtained, we will give an example of a multiobjective optimization problem in which the efficient solution will be obtained by the application of Theorem 18.

Example 19. Let Minimize𝑓(𝑥)=𝑓₁(𝑥),𝑓₂(𝑥)subjectto𝑔(𝑥)=𝑔₁(𝑥),𝑔₂(𝑥)≦0, (35) where 𝑓∶𝑅²→𝑅² and 𝑔∶𝑅²→𝑅² be defined as 𝑓₁𝑥₁,𝑥₂=2+𝑥²₂+8𝑥₁if𝑥₁≠0,𝑥₂≠00if𝑥₁=0,𝑥₂𝑥=0₂₁+𝑥²₂𝑓else,₂𝑥₁,𝑥₂=𝑥²₁+𝑥²₂−4𝑥₁if𝑥₁≠0,𝑥₂≠00if𝑥₁=0,𝑥₂=04+2𝑥²₁+3𝑥⁴₂𝑔else,₁𝑥₁,𝑥₂=−𝑥⁴₁+1if𝑥₁≠0,𝑥₂=00if𝑥₁=0,𝑥₂=04𝑥₁−𝑥²₁−3𝑥²₂𝑔else,₂𝑥₁,𝑥₂=2𝑥₁−3𝑥⁴₁if𝑥₁≠0,𝑥₂=00if𝑥₁=0,𝑥₂=0−𝑥²₂−𝑥²₁+2else. (36) Let 𝜂₁(𝑥,𝑦)=(1+𝑥²₁+𝑦²₁,0), 𝜂₂(𝑥,𝑦)=(1+𝑥²₁,2+2𝑥²₂), 𝜃₁(𝑥,𝑦)=(1+𝑥²₁,2+𝑥⁴₂), 𝜃₂(𝑥,𝑦)=(2+2𝑥²₁,0), 𝜑(𝑥,𝑦;𝑎)=|𝑎|(1+𝑥²₁𝑦²₁), 𝑏_𝑜(𝑥,𝑦)=4, 𝑏₁(𝑥,𝑦)=4, 𝜙_𝑜(𝑡)=𝑡/2, 𝜙₁(𝑡)=2𝑡, and 𝜎(𝑥,𝑦)=(𝑥₁+𝑦₁,1+𝑦₂).
Also, let 𝛼¹₁(𝑥,𝑦)=4, 𝛼¹₂(𝑥,𝑦)=2, 𝛼²₁(𝑥,𝑦)=2, 𝛼²₂(𝑥,𝑦)=1, 𝜌¹=4, 𝜌²=−4, 𝜌¹₁=0, 𝜌¹₂=4, 𝜌²₁=−2, 𝜌²₂=−5, 𝜇₁=𝜇₂=1/2, 𝜆₁=1/2, and 𝜆₂=0 where 𝑥=(𝑥₁,𝑥₂),  𝑦=(𝑦₁,𝑦₂),𝛼¹_𝑖(𝑥,𝑦)=1/𝛼¹_𝑖(𝑥,𝑦), and  𝛼²_𝑗(𝑥,𝑦)=1/𝛼²_𝑗(𝑥,𝑦), 𝑖=1,2,  𝑗=1,2.
The set 𝑋 of feasible solutions of problem is nonempty. Clearly, 𝑓₁,𝑓₂ and 𝑔₁,𝑔₂ are semidirectionally differentiable at 𝑥_𝑜=(0,0) with 𝑓^′₁(𝑥_𝑜;𝜂₁(𝑥,𝑥_𝑜))=0,  𝑓^′₂(𝑥_𝑜;𝜂₂(𝑥,𝑥_𝑜))=−4(1+𝑥²₁),  𝑔^′₁(𝑥_𝑜;𝜃₁(𝑥,𝑥_𝑜))=4(1+𝑥²₁), and 𝑔^′₂(𝑥_𝑜;𝜃₂(𝑥,𝑥_𝑜))=4(1+𝑥²₁).
It is easy to see that for all 𝑥∈𝑋⧵{𝑥_𝑜}𝜑𝑥,𝑥_𝑜;²_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜≧−𝜌¹𝜎(𝑥,𝑥_𝑜)²⟹𝑏_𝑜𝑥,𝑥_𝑜𝜙_𝑜²_𝑖=1𝜇_𝑖𝛼¹_𝑖𝑥,𝑥_𝑜𝑓_𝑖(𝑥)−𝑓_𝑖𝑥_𝑜𝑏>0,₁𝑥,𝑥_𝑜𝜙₁²_𝑗=1𝜆_𝑗𝛼²_𝑗𝑥,𝑥_𝑜𝑔_𝑗𝑥_𝑜≧0⟹𝜑𝑥,𝑥_𝑜;²_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜<−𝜌²𝜎(𝑥,𝑥_𝑜)². (37) Hence (𝑓,𝑔_{𝐽(𝑥𝑜)}) is strictly pseudo (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex function at 𝑥_𝑜.
Also, hypotheses (i), (ii), and (iv) of Theorem 18 are clearly satisfied and it follows that 𝑥_𝑜 is an efficient solution of the above defined multiobjective optimization problem, whereas it will be impossible to apply for this purpose the sufficient optimality conditions given in Kharbanda et al. [18], Ahmad [17], Slimani and Radjef [12], Mishra and Noor [10], Mishra et al. [16], Antczak [9], Suneja and Srivastava [19], and Ye [8].

Theorem 20. Suppose there exist a feasible solution 𝑥_𝑜 of (MP) and vector functions 𝜂_𝑖∶𝑋×𝐷→𝑅^𝑛,  𝑖=1,𝑚, and 𝜃_𝑗∶𝑋×𝐷→𝑅^𝑛,  𝑗∈𝐽(𝑥_𝑜), and scalars 𝜇_𝑖≧0,  𝑖=1,𝑚, and 𝜆_𝑗≧0,  𝑗∈𝐽(𝑥_𝑜), satisfying(i)∑^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖(𝑥_𝑜;𝜂_𝑖(𝑥,𝑥_𝑜))+∑_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝑔^′_𝑗(𝑥_𝑜;𝜃_𝑗(𝑥,𝑥_𝑜))≧0, for all 𝑥∈𝑋,(ii)for any 𝑢∈𝑅, 𝑢≦0⇒𝜙_𝑜(𝑢)≦0 and 𝑢≧0⇒𝜙₁(𝑢)≧0,  𝑏_𝑜(𝑥,𝑥_𝑜)≧0, 𝑏₁(𝑥,𝑥_𝑜)≧0, and(iii)𝜌¹+𝜌²≧0. Also, if either the fact that(a)(𝑓,𝑔_{𝐽(𝑥𝑜)}) is semistrictly quasi (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑥_𝑜 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_{𝑗∈𝐽(𝑥𝑜)}, or(b)(𝑓,𝑔_{𝐽(𝑥𝑜)}) is quasistrictly pseudo (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑥_𝑜 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_{𝑗∈𝐽(𝑥𝑜)}, or(c)(𝑓,𝑔_{𝐽(𝑥𝑜)}) is strictly pseudo-quasi (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑥_𝑜 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_{𝑗∈𝐽(𝑥𝑜)} holds,then 𝑥_𝑜 is an efficient solution of (MP).

Proof. If (a) or (c) holds, then proceeding as in previous theorem we get 𝜑𝑥,𝑥_𝑜;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜<−𝜌¹𝜎(𝑥,𝑥_𝑜)²,𝜑𝑥,𝑥_𝑜;_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜≦−𝜌²𝜎(𝑥,𝑥_𝑜)². (38) And if (b) holds, we get 𝜑𝑥,𝑥_𝑜;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜≦−𝜌¹𝜎(𝑥,𝑥_𝑜)²,𝜑𝑥,𝑥_𝑜;_{𝑗∈𝐽(𝑥𝑜)}𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜<−𝜌²𝜎(𝑥,𝑥_𝑜)². (39) The remaining part of proof runs on the lines of the proof of Theorem 18.

Now, following Antczak [9] and Slimani and Radjef [12], we state the following necessary optimality conditions.

Theorem 21 (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 exist vector functions 𝜂_𝑖∶𝑋×𝐷→𝑅^𝑛, 𝑖=1,𝑚, and 𝜃_𝑗∶𝑋×𝐷→𝑅^𝑛,  𝑗∈𝐽(𝑥_𝑜), such that at 𝑥_𝑜∈𝐷 the following inequalities are satisfied with respect to 𝜂∶𝑋×𝐷→𝑅^𝑛: 𝑓^′_𝑖𝑥_𝑜;𝜂𝑥,𝑥_𝑜≦𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜,∀𝑥∈𝑋,∀𝑖=𝑔1,𝑚,_′𝑗𝑥_𝑜;𝜂𝑥,𝑥_𝑜≦𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜,∀𝑥∈𝑋,∀𝑗∈𝐽𝑥_𝑜, (40)(iv)for all 𝑖=1,𝑚 and 𝑗∈𝐽(𝑥_𝑜), 𝑓_𝑖 and 𝑔_𝑗 are semidirectionally differentiable at 𝑥_𝑜 and the functions 𝑓^′_𝑖(𝑥_𝑜;𝜂_𝑖(𝑥,𝑥_𝑜)), 𝑖=1,𝑚, and 𝑔^′_𝑗(𝑥_𝑜;𝜃_𝑗(𝑥,𝑥_𝑜)), 𝑗∈𝐽(𝑥_𝑜), are preinvex functions of 𝑥 on 𝑋,(v)the function 𝑔 satisfies 𝑑_𝐼-constraint qualification at 𝑥_𝑜 with respect to (𝜃_𝑗)_{𝑗∈𝐽(𝑥𝑜)}, then there exist 𝜇∈𝑅^𝑚_⩾ and 𝜆∈𝑅^𝑘_≧ such that ^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜+^𝑘_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜𝜆≧0,∀𝑥∈𝑋,_𝑗𝑔_𝑗𝑥_𝑜=0,∀𝑗=1,𝑘. (41)

5. 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. Consider (MWD)Max𝑓𝑦subjectto^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑦;𝜂_𝑖+𝑥,𝑦_𝑘𝑗=1𝜆_𝑗𝑔^′_𝑗𝑦;𝜃_𝑗𝑥,𝑦≧0,∀𝑥∈𝑋,𝜆_𝑗𝑔_𝑗𝑦≧0,𝑗=1,𝑘, (42)(43) where 𝑦∈𝐷, 𝜇∈𝑅^𝑚_⩾, 𝜆∈𝑅^𝑘_≧, 𝜂_𝑖∶𝑋×𝐷→𝑅^𝑛, for all 𝑖=1,𝑚, and 𝜃_𝑗∶𝑋×𝐷→𝑅^𝑛, for all 𝑗=1,𝑘. Let 𝑊 be the set of feasible points of (MWD).

Theorem 22 (weak duality). Let 𝑥 and (𝑦,𝜇,𝜆,(𝜂_𝑖)_𝑖=1,𝑚,(𝜃_𝑗)_𝑗=1,𝑘) be the feasible solutions of (MP) and (MWD), respectively, with 𝜇>0. If (i)(𝑓,𝑔) is pseudo-quasi (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑦 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_𝑗=1,𝑘,(ii)for any 𝑢∈𝑅,  𝑢<0⇒𝜙_𝑜(𝑢)<0 and 𝑢≧0⇒𝜙₁(𝑢)≧0, 𝑏_𝑜(𝑥,𝑦)>0, 𝑏₁(𝑥,𝑦)≧0, and(iii)𝜌¹+𝜌²≧0, then 𝑓(𝑥)≰𝑓𝑦. (44)

Proof. Suppose to the contrary that 𝑓(𝑥)⩽𝑓𝑦. (45) As 𝜇_𝑖>0, 𝛼¹_𝑖(𝑥,𝑦)>0, 𝑖=1,𝑚, we get ^𝑚_𝑖=1𝜇_𝑖𝛼¹_𝑖𝑥,𝑦𝑓_𝑖(𝑥)−𝑓_𝑖𝑦<0. (46) Since 𝜆_𝑗𝑔_𝑗(𝑦)≧0, 𝛼²_𝑗(𝑥,𝑦)>0, 𝑗=1,𝑘, therefore ^𝑘_𝑗=1𝜆_𝑗𝛼²_𝑗𝑥,𝑦𝑔_𝑗𝑦≧0. (47) Using hypothesis (ii), we get 𝑏_𝑜𝑥,𝑦𝜙_𝑜^𝑚_𝑖=1𝜇_𝑖𝛼¹_𝑖𝑥,𝑦𝑓_𝑖(𝑥)−𝑓_𝑖𝑦𝑏<0,₁𝑥,𝑦𝜙₁^𝑘_𝑗=1𝜆_𝑗𝛼²_𝑗𝑥,𝑦𝑔_𝑗𝑦≧0. (48) Since hypothesis (i) holds, therefore the above inequalities yield 𝜑𝑥,𝑦;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑦;𝜂_𝑖𝑥,𝑦<−𝜌¹𝜎(𝑥,𝑦)²,𝜑𝑥,𝑦;^𝑘_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑦;𝜃_𝑗𝑥,𝑦≦−𝜌²𝜎(𝑥,𝑦)². (49) Using subadditivity of 𝜑 and hypothesis (iii), we get 𝜑𝑥,𝑦;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑦;𝜂_𝑖𝑥,𝑦+^𝑘_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑦;𝜃_𝑗𝜌𝑥,𝑦<−¹+𝜌²𝜎𝑥,𝑦²≦0. (50) However, the feasible condition (42) and properties of 𝜑 imply 𝜑𝑥,𝑦;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑦;𝜂_𝑖𝑥,𝑦+^𝑘_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑦;𝜃_𝑗𝑥,𝑦≧0. (51) Thus we get a contradiction and hence the proof.

The proof of the following theorems runs on the lines of the proof of Theorem 22.

Theorem 23 (weak duality). Let 𝑥 and (𝑦,𝜇,𝜆,(𝜂_𝑖)_𝑖=1,𝑚,(𝜃_𝑗)_𝑗=1,𝑘) be the feasible solutions of (MP) and (MWD), respectively. If(i)(𝑓,𝑔) is pseudo-quasi (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑦 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_𝑗=1,𝑘,(ii)for any 𝑢∈𝑅, 𝑢<0⇒𝜙_𝑜(𝑢)<0 and 𝑢≧0⇒𝜙₁(𝑢)≧0, 𝑏_𝑜(𝑥,𝑦)>0, 𝑏₁(𝑥,𝑦)≧0, and(iii)𝜌¹+𝜌²≧0, then 𝑓(𝑥)≮𝑓𝑦. (52)

Theorem 24 (weak duality). Let 𝑥 and (𝑦,𝜇,𝜆,(𝜂_𝑖)_𝑖=1,𝑚,(𝜃_𝑗)_𝑗=1,𝑘) be the feasible solutions of (MP) and (MWD), respectively. If(i)(𝑓,𝑔) is quasistrictly pseudo (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑦 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_𝑗=1,𝑘,(ii)for any 𝑢∈𝑅, 𝑢≦0⇒𝜙_𝑜(𝑢)≦0 and 𝑢≧0⇒𝜙₁(𝑢)≧0,  𝑏_𝑜(𝑥,𝑦)≧0, 𝑏₁(𝑥,𝑦)≧0, and(iii)𝜌¹+𝜌²≧0, then 𝑓(𝑥)≰𝑓𝑦. (53)

Theorem 25 (strong duality). Let 𝑥_𝑜 be a weakly efficient solution of (MP) and 𝑔_𝑗 is continuous at 𝑥_𝑜 for 𝑗∈𝐽(̃𝑥_𝑜). Also, the vector functions 𝜂_𝑖,  𝑖=1,𝑚 and 𝜃_𝑗,  𝑗∈𝐽(𝑥_𝑜) exist for which 𝑓^′_𝑖𝑥_𝑜;𝜂𝑥,𝑥_𝑜≦𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜,∀𝑥∈𝑋,∀𝑖=𝑔1,𝑚,_′𝑗𝑥_𝑜;𝜂𝑥,𝑥_𝑜≦𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜,∀𝑥∈𝑋,∀𝑗∈𝐽𝑥_𝑜 (54) at 𝑥_𝑜 and 𝑓_𝑖,𝑔_𝑗 are semidirectionally differentiable at 𝑥_𝑜 with 𝑓^′_𝑖(𝑥_𝑜;𝜂_𝑖(𝑥,𝑥_𝑜)) and 𝑔^′_𝑗(𝑥_𝑜;𝜃_𝑗(𝑥,𝑥_𝑜)) as preinvex functions on 𝑋. Also if 𝑔 satisfies 𝑑_𝐼-constraint qualification at 𝑥_𝑜, then ∃𝜇∈𝑅^𝑚_≥ and 𝜆∈𝑅^𝑘_≧ such that (𝑥_𝑜,𝜇,𝜆,(𝜂_𝑖)_𝑖=1,𝑚,(𝜃_𝑗)_𝑗=1,𝑘) is feasible for (MWD) and the objective function values of (MP) and (MWD) are equal. Moreover, if any weak duality holds, then (𝑥_𝑜,𝜇,𝜆,(𝜂_𝑖)_𝑖=1,𝑚,(𝜃_𝑗)_𝑗=1,𝑘) is a weakly efficient solution of (MWD).

Proof. Since 𝑥_𝑜 is a weakly efficient solution of (MP), therefore, by Theorem 21, there exist 𝜇∈𝑅^𝑚_⩾ and 𝜆∈𝑅^𝑘_≧ such that ^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑥_𝑜;𝜂_𝑖𝑥,𝑥_𝑜+^𝑘_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑥_𝑜;𝜃_𝑗𝑥,𝑥_𝑜𝜆≧0,∀𝑥∈𝑋,_𝑗𝑔_𝑗𝑥_𝑜=0,∀𝑗=1,𝑘. (55)
It follows that (𝑥_𝑜,𝜇,𝜆,(𝜂_𝑖)_𝑖=1,𝑚,(𝜃_𝑗)_𝑗=1,𝑘)∈𝑊 and therefore feasible for (MWD). Clearly, objective function values of (MP) and (MWD) are equal at these points.
Suppose (𝑥_𝑜,𝜇,𝜆,(𝜂_𝑖)_𝑖=1,𝑚,(𝜃_𝑗)_𝑗=1,𝑘) is not a weakly efficient solution for (MWD). Then ∃(𝑦,𝜇,𝜆,(𝜂_𝑖)_𝑖=1,𝑚,(𝜃_𝑗)_𝑗=1,𝑘)∈𝑊 such that 𝑓(𝑥_𝑜)<𝑓(𝑦) which contradicts weak duality theorems. Therefore (𝑥_𝑜,𝜇,𝜆,(𝜂_𝑖)_𝑖=1,𝑚,(𝜃_𝑗)_𝑗=1,𝑘) is a weakly efficient solution of (MWD).

Theorem 26 (converse duality). Let (𝑦_𝑜,𝜇,𝜆,(𝜂_𝑖)_𝑖=1,𝑚,(𝜃_𝑗)_𝑗=1,𝑘) be a feasible solution of (MWD). Suppose hypotheses of Theorem 24 hold at 𝑦_𝑜, 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⩽𝑓𝑦_𝑜. (56) Hence, by Theorem 24 (weak duality), we obtain a contradiction. Therefore 𝑦_𝑜 is an efficient solution of (MP).

Theorem 27 (strict converse duality). Let 𝑥_𝑜 and (𝑦_𝑜,𝜇,𝜆,(𝜂_𝑖)_𝑖=1,𝑚,(𝜃_𝑗)_𝑗=1,𝑘) be the feasible solutions of (MP) and (MWD), respectively. If (i)𝑓(𝑥_𝑜)≦𝑓(𝑦_𝑜), (ii)(𝑓,𝑔) is semistrictly quasi (𝜑,𝛼,𝜌,𝜎)-𝑑_𝐼-𝑉-type I univex at 𝑦_𝑜 with respect to (𝜂_𝑖)_𝑖=1,𝑚 and (𝜃_𝑗)_𝑗=1,𝑘,(iii)for any 𝑢∈𝑅,  𝑢≦0⇒𝜙_𝑜(𝑢)≦0 and 𝑢≧0⇒𝜙₁(𝑢)≧0, 𝑏_𝑜(𝑥_𝑜,𝑦_𝑜)≧0, 𝑏₁(𝑥_𝑜,𝑦_𝑜)≧0, and(iv)𝜌¹+𝜌²≧0, then 𝑥_𝑜=𝑦_𝑜.

Proof. Suppose 𝑥_𝑜≠𝑦_𝑜.
Since 𝜇∈𝑅^𝑚_≥, 𝛼¹_𝑖(𝑥_𝑜,𝑦_𝑜)>0, 𝑖=1,𝑚, and hypothesis (i) holds, therefore ^𝑚_𝑖=1𝜇_𝑖𝛼¹_𝑖𝑥_𝑜,𝑦_𝑜𝑓_𝑖𝑥_𝑜≦^𝑚_𝑖=1𝜇_𝑖𝛼¹_𝑖𝑥_𝑜,𝑦_𝑜𝑓_𝑖𝑦_𝑜. (57) As 𝛼²_𝑗(𝑥_𝑜,𝑦_𝑜)>0,  𝑗=1,𝑘, and 𝑦_𝑜 is feasible solution of (MWD), therefore ^𝑘_𝑗=1𝜆_𝑗𝛼²_𝑗𝑥_𝑜,𝑦_𝑜𝑔_𝑗𝑦_𝑜≧0. (58) Using hypothesis (iii), we obtain 𝑏_𝑜𝑥_𝑜,𝑦_𝑜𝜙_𝑜^𝑚_𝑖=𝑖𝜇_𝑖𝛼¹_𝑖𝑥_𝑜,𝑦_𝑜𝑓_𝑖𝑥_𝑜−𝑓_𝑖𝑦_𝑜𝑏≦0,₁𝑥_𝑜,𝑦_𝑜𝜙₁^𝑘_𝑗=1𝜆_𝑗𝛼²_𝑗𝑥_𝑜,𝑦_𝑜𝑔_𝑗𝑦_𝑜≧0. (59) Applying hypothesis (ii) to the above inequalities, we get 𝜑𝑥_𝑜,𝑦_𝑜;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑦_𝑜;𝜂_𝑖𝑥_𝑜,𝑦_𝑜<−𝜌¹𝜎(𝑥_𝑜,𝑦_𝑜)²,𝜑𝑥_𝑜,𝑦_𝑜;^𝑘_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑦_𝑜;𝜃_𝑗𝑥_𝑜,𝑦_𝑜≦−𝜌²𝜎(𝑥_𝑜,𝑦_𝑜)². (60) Using subadditivity of 𝜑 and hypothesis (iv), we get 𝜑𝑥_𝑜,𝑦_𝑜;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑦_𝑜;𝜂_𝑖𝑥_𝑜,𝑦_𝑜+^𝑘_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑦_𝑜;𝜃_𝑗𝑥_𝑜,𝑦_𝑜<0 (61) which is a contradiction as feasible condition (42) holds and 𝜑(𝑥_𝑜,𝑦_𝑜;0)=0 and 𝜑(𝑥_𝑜,𝑦_𝑜;𝑎)>0 for 𝑎>0 yield 𝜑𝑥_𝑜,𝑦_𝑜;^𝑚_𝑖=1𝜇_𝑖𝑓^′_𝑖𝑦_𝑜;𝜂_𝑖𝑥_𝑜,𝑦_𝑜+^𝑘_𝑗=1𝜆_𝑗𝑔^′_𝑗𝑦_𝑜;𝜃_𝑗𝑥_𝑜,𝑦_𝑜≧0. (62) Hence 𝑥_𝑜=𝑦_𝑜.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.