Nonemptiness and Compactness of Solutions Set for Nondifferentiable Multiobjective Optimization Problems

Abstract

A nondifferentiable multiobjective optimization problem with nonempty set constraints is considered, and the equivalence of weakly efficient solutions, the critical points for the nondifferentiable multiobjective optimization problems, and solutions for vector variational-like inequalities is established under some suitable conditions. Nonemptiness and compactness of the solutions set for the nondifferentiable multiobjective optimization problems are proved by using the FKKM theorem and a fixed-point theorem.

1. Introduction

The weak minimum (weakly efficient, weak Pareto) solution is an essential concept in mathematical models, economics, decision theory, optimal control, and game theory. For readers’ reference, we refer to [1–11] and the references therein.

In [5], Garzón et al. studied some relationships among the weakly efficient solutions, the critical points of optimization problems, and the solutions of vector variational-like inequalities with differentiable functions. In [12], Mishra and Wang extended the work of Garzón et al. [5] to nonsmooth case. In [9], Lee et al. investigated the existence of solutions of vector optimization problems with differentiable functions. In [7], Kazmi considered the relationship between the weakly efficient solutions of a vector optimization problem and the solutions of a vector variational-like inequality with preinvex and Frechet differentiable functions. For more related work in this interesting area, we refer to [4, 10].

Motivated and inspired by the works mentioned above, we consider nondifferentiable multiobjective optimization problems (MOPs) with nonempty set constraints. The relationship among weakly efficient solutions, critical points of (MOP), and solutions of the vector variational-like inequalities (for short, (VVLI)) is presented under subinvexity, strictly pseudosubinvexity, and pseudosubinvexity conditions. By using the FKKM theorem and a fixed-point theorem, we prove the nonemptiness and compactness of solutions set for (MOP). The results presented in this paper extend the corresponding results of [5, 7, 9, 12, 13].

2. Preliminaries

Throughout this paper, without other specifications, let 𝑅^𝑛 be the 𝑛-dimensional Euclidean space, and 𝑅^𝑛₊={𝑥=(𝑥₁,…,𝑥_𝑛)∶𝑥_𝑙≥0,𝑙=1,…,𝑛}. Let 𝐾 be a nonempty convex subset of 𝑅^𝑛, let 𝐻 be a subset of 𝐾, and let int_𝐾𝐻 be the relative interior of 𝐻 to 𝐾. Let 𝑓=(𝑓₁,𝑓₂,...,𝑓_𝑛)∶𝑅^𝑛→𝑅^𝑛, 𝜂∶𝑅^𝑛×𝑅^𝑛→𝑅^𝑛, and let 𝐶∶𝐾→2^𝑅𝑛 such that, for each 𝑥∈𝐾, 𝐶(𝑥) is a closed convex cone, 𝐶(𝑥)≠𝑅^𝑛, int𝐶(𝑥)≠∅, and 𝑅^𝑛₊⊂𝐶(𝑥). The multiobjective optimization problem (for short, (MOP)) is defined as follows: min_𝑥∈𝐾𝑓(𝑥). (MOP)

We first recall some definitions and lemmas which are needed in the main results of this paper.

Definition 2.1. A point 𝑥∈𝐾 is said to be a weakly efficient (weak minimum) solution of (MOP) if [𝑓(𝐾)−𝑓(𝑥)]∩(−int𝐶(𝑥))=∅.

Definition 2.2. A real-valued function ℎ∶𝑅^𝑛→𝑅 is said to be locally Lipschitz with respect to 𝜂∶𝑅^𝑛×𝑅^𝑛→𝑅^𝑛 if, for each 𝑧₀∈𝑅^𝑛, there exist a neighborhood 𝑁(𝑧₀) of 𝑧₀ and a constant 𝑘>0 such that ℎ(𝑥)−ℎ𝑦≤𝑘𝜂𝑥,𝑦,∀𝑥,𝑦∈𝑁𝑧₀. (2.1)

Remark 2.3. If 𝜂(𝑥,𝑦)=𝑥−𝑦, then the above definition reduces to that of local Lipschitz.

Definition 2.4. A set-valued function 𝑇∶𝑅^𝑛→2^𝑅𝑛 is said to be locally bounded at 𝑧₀∈𝑅^𝑛 if there exist a neighborhood 𝑁(𝑧₀) of 𝑧₀ and a constant 𝑘>0 such that ‖𝑤‖<𝑘,∀𝑧∈𝑁𝑧₀,∀𝑤∈𝑇(𝑧). (2.2)

Definition 2.5. Let 𝑇∶𝑅^𝑛→2^𝑅𝑛 be a set-valued function. The graph of 𝑇 is defined as Gragh(𝑇)=𝑥,𝑦∶𝑥∈𝑅^𝑛,𝑦∈𝑇(𝑥). (2.3) The inverse 𝑇⁻¹ of 𝑇 is defined by 𝑥∈𝑇⁻¹(𝑦) if and only if 𝑦∈𝑇(𝑥).

Definition 2.6. Let 𝑋 be a nonempty subset of topological vector space 𝐸. A set-valued mapping 𝐹∶𝑋→2^𝐸 is called a KKM mapping if, for every finite subset {𝑥₁,𝑥₂,…,𝑥_𝑛} of 𝑋,co{𝑥₁,𝑥₂,…,𝑥_𝑛}⊂⋃^𝑛_𝑖=1𝐹(𝑥_𝑖), where co denotes the convex hull.

Definition 2.7 (see [13]). A real-valued function ℎ∶𝑅^𝑛→𝑅 is said to be subinvex at 𝑧 with respect to 𝜂∶𝑅^𝑛×𝑅^𝑛→𝑅^𝑛 if there exists 𝜁∈𝑅^𝑛, such that ℎ(𝑥)−ℎ(𝑧)≥⟨𝜁,𝜂(𝑥,𝑧)⟩, for any 𝑥∈𝑅^𝑛, where 𝜁 is called the 𝜂-subgradient of ℎ at 𝑧. The 𝜂-subdifferential of ℎ at 𝑧, denoted 𝜕^𝜂ℎ(𝑧), is the set of all 𝜁∈𝑅^𝑛 such that ℎ(𝑥)−ℎ(𝑧)≥𝜁,𝜂(𝑥,𝑧),∀𝑥∈𝑅^𝑛. (2.4)

Remark 2.8. If ℎ is locally Lipschitz, then the subinvexity of ℎ with respect to 𝜂 collapses to the invexity of ℎ with respect to 𝜂 in the sense of Clarke's generalized directional derivative with respect to 𝜂 [1]. If 𝜂(𝑥,𝑦)=𝑥−𝑦 for any 𝑥,𝑦∈𝐾, then the 𝜂-subdifferential 𝜕^𝜂ℎ(𝑧) reduces to the subdifferential 𝜕ℎ(𝑧) in the sense of convex analysis, where 𝜕ℎ(𝑧)={𝜁∈𝑅^𝑛∶ℎ(𝑥)≥ℎ(𝑧)+⟨𝜁,𝑥−𝑧⟩,∀𝑥∈𝑅^𝑛}.

Definition 2.9. A vector-valued function 𝑔∶𝑅^𝑛→𝑅^𝑛 is called strictly pseudosubinvex with respect to 𝜂 if, for any 𝑥,𝑢∈𝑅^𝑛 and (𝜉₁,𝜉₂,…,𝜉_𝑛)∈𝜕^𝜂𝑔(𝑢), 𝑔(𝑥)−𝑔(𝑢)∈−int𝐶(𝑢)⟹⟨𝜉₁,𝜂(𝑥,𝑢)⟩,⟨𝜉₂,𝜂(𝑥,𝑢)⟩,…,⟨𝜉_𝑛,𝜂(𝑥,𝑢)⟩∈−int𝑅^𝑛₊, (2.5) where 𝜕^𝜂𝑔(𝑢)={(𝜉₁,𝜉₂,…,𝜉_𝑛)∈(𝑅^𝑛)^𝑛∶𝑔_𝑖(𝑥)−𝑔_𝑖(𝑢)≥⟨𝜉_𝑖,𝜂(𝑥,𝑢)⟩,∀𝑥∈𝑅^𝑛}.

Definition 2.10. A vector-valued function 𝑔∶𝑅^𝑛→𝑅^𝑛 is called pseudosubinvex with respect to 𝜂 if, for any 𝑥,𝑢∈𝑅^𝑛 and (𝜉₁,𝜉₂,…,𝜉_𝑛)∈𝜕^𝜂𝑔(𝑢), 𝑔(𝑥)−𝑔(𝑢)∈−int𝐶(𝑢)⟹𝜉₁,𝜂(𝑥,𝑢),𝜉₂,𝜂(𝑥,𝑢),…,𝜉_𝑛,𝜂(𝑥,𝑢)∈−int𝐶(𝑢), (2.6) where 𝜕^𝜂𝑔(𝑢)={(𝜉₁,𝜉₂,…,𝜉_𝑛)∈(𝑅^𝑛)^𝑛∶𝑔_𝑖(𝑥)−𝑔_𝑖(𝑢)≥⟨𝜉_𝑖,𝜂(𝑥,𝑢)⟩,∀𝑥∈𝑅^𝑛}.

Remark 2.11. Since 𝑅^𝑛₊⊂𝐶(𝑥), it is clear that (1)if 𝑔∶𝑅^𝑛→𝑅^𝑛 is strictly pseudosubinvex with respect to 𝜂, then it is pseudosubinvex with respect to 𝜂;(2)if 𝑔_𝑖∶𝑅^𝑛→𝑅,𝑖=1,2,…,𝑛, are subinvex with respect to 𝜂, then 𝑔=(𝑔₁,𝑔₂,…,𝑔_𝑛) is pseudosubinvex with respect to 𝜂.

Definition 2.12. A point 𝑦∈𝐾 is called a critical point of (MOP) if there exists 𝜆=(𝜆₁,𝜆₂,…,𝜆_𝑛)∈𝑅^𝑛₊, with 𝜆_𝑖>0 for some 𝑖∈{1,2,…,𝑛}, such that ^𝑛_𝑖=1𝜆_𝑖𝜉_𝑖=0,∀𝜉=𝜉₁,𝜉₂,…,𝜉_𝑛∈𝜕^𝜂𝑓𝑦. (2.7)

Lemma 2.13 (see [14](FKKM theorem)). Let 𝑄 be a nonempty subset of Hausdorff topological vector space 𝑋. Let 𝐺∶𝑄→2^𝑋 be a KKM mapping such that for any 𝑥∈𝑄,𝐺(𝑥) is closed and 𝐺(𝑥₀) is compact for some 𝑥₀∈𝑄, then there exists 𝑥^∗∈𝑄 such that 𝑥^∗∈𝐺(𝑦) for all 𝑦∈𝑄, that is, ⋂_𝑥∈𝑄𝐺(𝑥)≠∅.

Lemma 2.14 (see [13]). Let 𝑋 be a nonempty and convex subset of Hausdorff topological vector space 𝐸, and let 𝑆,𝑇∶𝑋→2^𝑋 be two set-valued maps such that, for each 𝑥∈𝑋,co𝑆(𝑥)⊂𝑇(𝑥),𝑆(𝑥)≠∅,𝑋=⋃{int_𝑋𝑆⁻¹(𝑧)∶𝑧∈𝑋}. If there exist a nonempty compact convex set 𝐵⊂𝑋 and a nonempty compact set 𝐷⊂𝑋 such that, for each 𝑧∈𝑋⧵𝐷, there exists 𝑥∈𝐵, such that 𝑧∈int_𝑋𝑆⁻¹(𝑥), then there exists 𝑧∈𝑋 such that 𝑧∈𝑇(𝑧).

We consider the following vector variational-like inequality (for short, (VVLI)): find 𝑥∈𝐾 such that for any 𝑥∈𝐾, there exists 𝜉=(𝜉₁,𝜉₂,…,𝜉_𝑛)∈𝜕^𝜂𝑓(𝑥) such that 𝜉₁,𝜂(𝑥,𝑥),𝜉₂,𝜂(𝑥,𝑥),…,𝜉_𝑛,𝜂(𝑥,𝑥)∉−int𝐶(𝑥). (2.8)

3. Relationships between (MOP) and (VVLI)

In this section, we will investigate the properties of 𝜂-subdifferential 𝜕^𝜂ℎ of the function ℎ∶𝑅^𝑛→𝑅, and the relationships among weakly efficient solutions, critical points of (MOP), and the solutions of (VVILP).

Theorem 3.1. Let ℎ∶𝑅^𝑛→𝑅 be subinvex with respect to 𝜂, then the following statements are true: (i)for each 𝑧∈𝑅^𝑛, 𝜕^𝜂ℎ(𝑧) is a nonempty closed-convex subset of 𝑅^𝑛, (ii)if for each 𝑥,𝑧∈𝑅^𝑛,𝜂(𝑥,𝑧)+𝜂(𝑧,𝑥)=0, then 𝜕^𝜂ℎ is 𝜂-monotone, that is, for any 𝑥₁,𝑥₂∈𝑅^𝑛, 𝜁−𝜉,𝜂𝑥₁,𝑥₂≥0,∀𝜁∈𝜕^𝜂ℎ𝑥₁,𝜉∈𝜕^𝜂ℎ𝑥₂, (3.1)(iii)if ℎ is locally Lipschitz with respect to 𝜂, 𝜂 is continuous in the second argument, and 𝜂(𝑥,𝑥)=0, for any 𝑥∈𝑅^𝑛, then Graph(𝜕^𝜂ℎ) is closed, and 𝜕^𝜂ℎ(⋅) is upper semicontinuous, (iv)if 𝜂 is an open map, 𝜂(𝑥,𝑥)=0 for any 𝑥∈𝑅^𝑛, and ℎ is locally Lipschitz with respect to 𝜂, then 𝜕^𝜂ℎ(⋅) is locally bounded on 𝑅^𝑛.

Proof. Assertions (i), (ii), and (iv) are shown in [13]. We only need to prove assertion (iii). Let (𝑥_𝑛,𝑦_𝑛)∈Graph(𝜕^𝜂ℎ), with 𝑥_𝑛→𝑥₀ and 𝑦_𝑛→𝑦₀. Since 𝜕^𝜂ℎ𝑥_𝑗≐𝜁∈𝑅^𝑛∶ℎ(𝑥)−ℎ𝑥_𝑗≥𝜁,𝜂𝑥,𝑥_𝑗,∀𝑥∈𝑅^𝑛,𝑖=1,2,…,𝑛, (3.2) and ℎ is locally Lipschitz with respect to 𝜂, there exist a neighborhood 𝑁(𝑥₀) of 𝑥₀ and a constant 𝑘>0 such that |ℎ(𝑥)−ℎ(𝑧)|≤𝑘𝜂(𝑥,𝑧),∀𝑥,𝑧∈𝑁𝑥₀. (3.3) Then there exists 𝑛₀ such that 𝑥_𝑛∈𝑁(𝑥₀) for all 𝑛>𝑛₀, and so ℎ𝑥₀−ℎ𝑥_𝑛≤𝑘𝜂𝑥₀,𝑥_𝑛. (3.4) Consequently, we have ℎ(𝑥_𝑛)→ℎ(𝑥₀) as 𝑛→∞. It follows from ℎ(𝑥)−ℎ𝑥_𝑛≥𝑦_𝑛,𝜂𝑥,𝑥_𝑛,∀𝑥∈𝑅^𝑛 (3.5) that ℎ(𝑥)−ℎ𝑥₀≥⟨𝑦₀,𝜂𝑥,𝑥₀⟩,∀𝑥∈𝑅^𝑛, (3.6) that is, 𝑦₀∈𝜕^𝜂ℎ(𝑥₀). Hence, (𝑥₀,𝑦₀)∈Graph(𝜕^𝜂ℎ). In view of (i), we have that 𝜕^𝜂ℎ(⋅) is upper semicontinuous. This completes the proof.

Theorem 3.2. Let 𝑓 be pseudosubinvex with respect to 𝜂. If 𝑦∈𝐾 is a solution of (VVLI), then 𝑦 is a weakly efficient solution of (MOP).

Proof. Let 𝑦∈𝐾 be a solution of (VVLI). If 𝑦 is not a weakly efficient solution of (MOP), then there exists 𝑥∈𝐾 such that 𝑓(𝑥)−𝑓𝑦∈−int𝐶𝑦. (3.7) Since 𝑓 is pseudosubinvex with respect to 𝜂, we have 𝜉₁𝜂𝑥,𝑦,𝜉₂,𝜂𝑥,𝑦,…,𝜉_𝑛,𝜂𝑥,𝑦∈−int𝐶𝑦,∀𝜉₁,𝜉₂,…,𝜉_𝑛∈𝜕^𝜂𝑓𝑦, (3.8) which contradicts the assumption. This completes the proof.

Corollary 3.3. Let 𝑓_𝑖,𝑖=1,2,…,𝑛 be subinvex with respect to 𝜂. If 𝑦∈𝐾 is a solution of (VVLI), then 𝑦 is a weakly efficient solution of (MOP).

Theorem 3.4. Let 𝜂 be an open map, and continuous and affine in the first argument, let 𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0, for any 𝑥,𝑦∈𝐾. Let 𝑓_𝑖,𝑖=1,2,…,𝑛 be subinvex and locally Lipschitz with respect to 𝜂. If 𝑦∈𝐾 is a weakly efficient solution of (MOP), then 𝑦 is a solution of (VVLI).

Proof.. Let 𝑦∈𝐾 be a weakly efficient solution of (MOP). If 𝑦 is not a solution of (VVLI), then 𝑦∉𝑦∈𝐾∶∀𝑥∈𝐾,∃𝜉_𝑖∈𝜕^𝜂𝑓_𝑖𝑦,𝜉₁,𝜂𝑥,𝑦,…,𝜉₁,𝜂𝑥,𝑦∉−int𝐶𝑦. (3.9) We assert that 𝑦∈𝐾∶∀𝑥∈𝐾,∃𝜉_𝑖∈𝜕^𝜂𝑓_𝑖𝑦,𝜉₁,𝜂𝑥,𝑦,…,𝜉_𝑛𝑦=,𝜂𝑥,𝑦∉−int𝐶𝑦∈𝐾∶∀𝑥∈𝐾,∀𝜇_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥),𝜇₁,𝜂𝑥,𝑦,…,𝜇_𝑛,𝜂𝑥,𝑦∉−int𝐶𝑦. (3.10) Let 𝑧∈𝑦∈𝐾∶∀𝑥∈𝐾,∃𝜉_𝑖∈𝜕^𝜂𝑓_𝑖𝑦,𝜉₁,𝜂𝑥,𝑦,…,𝜉_𝑛,𝜂𝑥,𝑦∉−int𝐶𝑦. (3.11) Suppose to the contrary that 𝑧∉𝑦∈𝐾∶∀𝑥∈𝐾,∀𝜇_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥),𝜇₁,𝜂𝑥,𝑦,...,𝜇_𝑛,𝜂𝑥,𝑦∉−int𝐶𝑦, (3.12) then there exist 𝑥∈𝐾 and 𝜇_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥) such that 𝜇₁,𝜂(𝑥,𝑧),…,𝜇_𝑛,𝜂(𝑥,𝑧)∈−int𝐶(𝑧). (3.13) Since 𝜕^𝜂𝑓_𝑖 is 𝜂-monotone, we have 𝜇_𝑖−𝜉_𝑖,𝜂(𝑥,𝑧)≥0,∀𝜇_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥),𝜉_𝑖∈𝜕^𝜂𝑓_𝑖(𝑧), (3.14) that is, 𝜇_𝑖,𝜂(𝑥,𝑧)−𝜉_𝑖,𝜂(𝑥,𝑧)∈𝑅₊,𝑖=1,2,…,𝑛. (3.15) As a consequence, 𝜇₁,𝜂(𝑥,𝑧),…,𝜇_𝑛,𝜂(𝑥,𝑧)−𝜉₁,𝜂(𝑥,𝑧),…,𝜉_𝑛,𝜂(𝑥,𝑧)∈𝑅^𝑛₊, (3.16) and it follows that 𝜉₁,𝜂(𝑥,𝑧),…,𝜉_𝑛,𝜂(𝑥,𝑧)∈𝜇₁,𝜂(𝑥,𝑧),…,𝜇_𝑛,𝜂(𝑥,𝑧)−𝑅^𝑛₊⊂−int𝐶(𝑧)−𝑅^𝑛₊⊂−int𝐶(𝑧), (3.17) which contradicts the assumption.
On the other hand, let 𝑦′∈𝑦∈𝐾∶∀𝑥∈𝐾,∀𝜇_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥),𝜇₁,𝜂𝑥,𝑦,…,𝜇_𝑛,𝜂𝑥,𝑦∉−int𝐶𝑦, (3.18) and let 𝑟_𝑚∈(0,1] such that 𝑟_𝑚→0. For any given 𝑥∈𝐾, set 𝑥_𝑚=(1−𝑟_𝑚)𝑦′+𝑟_𝑚𝑥, for any 𝑥∈𝐾. Then 𝑥_𝑚→𝑦′ as 𝑚→∞. Since 𝜂 is affine in the first argument, and 𝜂(𝑥,𝑥)=0, we have 𝜇^𝑚₁,𝜂𝑥_𝑚,𝑦′,…,𝜇^𝑚_𝑛,𝜂𝑥_𝑚,𝑦^′=𝑟_𝑚𝜇^𝑚₁,𝜂𝑥,𝑦′⟩,…,⟨𝜇^𝑚_𝑛,𝜂𝑥,𝑦′∉−int𝐶𝑦′,∀𝜇^𝑚_𝑖∈𝜕^𝜂𝑓_𝑖𝑥_𝑚, (3.19) that is, 𝜇^𝑚₁,𝜂𝑥,𝑦^′,…,𝜇^𝑚_𝑛,𝜂𝑥,𝑦^′∉−int𝐶𝑦′. (3.20) Since 𝜕^𝜂𝑓_𝑖 is locally bounded, there exist a neighborhood 𝑁(𝑦′) of 𝑦^′, and 𝑘>0 such that for any 𝑧∈𝑁(𝑦′), and 𝜓_𝑖∈𝜕^𝜂𝑓_𝑖(𝑧), we have 𝜓_𝑖≤𝑘,𝑖=1,2,…,𝑛. (3.21) Then there exists 𝑚₀, such that 𝑥_𝑚∈𝑁(𝑦′) for any 𝑚≥𝑚₀, and so 𝜇^𝑚_𝑖≤𝑘,𝑖=1,2,…,𝑛. (3.22) Consequently, {𝜇^𝑚_𝑖} has a convergent subsequence. Without loss of generality, let 𝜇^𝑚_𝑖→𝜇^′_𝑖. By Theorem 3.1 (iii), we have 𝜇^′_𝑖∈𝜕^𝜂𝑓_𝑖(𝑦′). Since 𝑅^𝑛⧵int𝐶(𝑦′) is closed, it follows that 𝑦′∈𝑦∈𝐾∶∀𝑥∈𝐾,∃𝜉_𝑖∈𝜕^𝜂𝑓_𝑖𝑦,𝜉₁,𝜂𝑥,𝑦,…,𝜉_𝑛,𝜂𝑥,𝑦∉−int𝐶𝑦. (3.23) Thus, it follows from (3.9) that 𝑦∉𝑦∈𝐾∶∀𝑥∈𝐾,∀𝜇_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥),𝜇₁,𝜂𝑥,𝑦,...,𝜇_𝑛,𝜂𝑥,𝑦∉−int𝐶𝑦, (3.24) that is, there exist 𝑥∈𝐾 and 𝜁_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥) such that 𝜁₁,𝜂𝑥,𝑦,…,𝜁_𝑛,𝜂𝑥,𝑦∈−int𝐶𝑦. (3.25) Since 𝑓_𝑖𝑦−𝑓_𝑖(𝑥)≥𝜁_𝑖,𝜂𝑦,𝑥,𝜂𝑦,𝑥+𝜂𝑥,𝑦=0, (3.26) we have 𝑓₁(𝑥)−𝑓₁𝑦,…,𝑓_𝑛(𝑥)−𝑓_𝑛𝑦−𝜁₁,𝜂𝑥,𝑦,…,𝜁_𝑛,𝜂𝑥,𝑦∈−𝑅^𝑛₊. (3.27) Thus, 𝑓(𝑥)−𝑓𝑦∈−𝑅^𝑛₊−int𝐶𝑦⊂−int𝐶𝑦, (3.28) that is, 𝑓(𝑥)−𝑓𝑦∈−int𝐶𝑦, (3.29) which contradicts the assumption. This completes the proof.

Theorem 3.5. Let 𝑓_𝑖,𝑖=1,2,…,𝑛 be subinvex with respect to 𝜂, and let 𝑓 be strictly pseudosubinvex with respect to 𝜂. If 𝑦∈𝐾 is a critical point of (MOP), then 𝑦 is a weakly efficient solution of (MOP).

Proof. Let 𝑦∈𝐾 be a critical point of (MOP). If 𝑦 is not the weakly efficient solution of (MOP), then there exists 𝑥∈𝐾 such that 𝑓(𝑥)−𝑓𝑦∈−int𝐶𝑦. (3.30) By the strict pseudosubinvexity of 𝑓 with respect to 𝜂, one has 𝜉,𝜂𝑥,𝑦∈−int𝑅^𝑛₊,∀𝜉=𝜉₁,𝜉₂,…,𝜉_𝑛∈𝜕^𝜂𝑓𝑦, (3.31) that is, 𝜉₁,𝜂𝑥,𝑦,…,𝜉_𝑛,𝜂𝑥,𝑦∈−int𝑅^𝑛₊. (3.32) Thus, we have 𝜉_𝑖,𝜂𝑥,𝑦<0,𝑖=1,2,…,𝑛. (3.33) Since 𝑦∈𝐾 is a critical point of (MOP), there is 𝜆=(𝜆₁,𝜆₂,…,𝜆_𝑛),𝑖∈1,2,…,𝑛, with 𝜆_𝑖>0 for some 𝑖∈{1,2,…,𝑛}, such that ^𝑛_𝑖=1𝜆_𝑖𝜉_𝑖=0. (3.34) Set 𝑆_𝜆=∑^𝑛_𝑖=1𝜆_𝑖>0, then ^𝑛_𝑖=1𝜆_𝑖𝑆_𝜆=1,^𝑛_𝑖=1𝜆_𝑖𝑆_𝜆𝜉_𝑖=0, (3.35) and so 0=^𝑛_𝑖=1𝜆_𝑖𝑆_𝜆𝜉_𝑖,𝜂𝑥,𝑦<0,𝑖=1,2,…,𝑛, (3.36) which is a contradiction. This completes the proof.

Remark 3.6. If for each 𝑦∈𝐾,𝐶(𝑦)=𝑅^𝑛₊, then the following statements are true in the sense of Clarke’s generalized directional derivative [12]: (1)all critical points of (MOP) are weakly efficient solutions of (MOP) if and only if 𝑓 is strictly pseudoinvex with respect to 𝜂;(2)if 𝑓 is strictly pseudoinvex with respect to 𝜂, and locally Lipschitz, then the critical points, the weakly efficient solutions of (MOP), and the solutions of (VVLI) are equivalent.

4. Existence of Weakly Efficient Solutions for (MOP)

In this section, we present several existence theorems for (MOP), by using the FKKM theorem and a fixed-point theorem.

Theorem 4.1. Let 𝐾⊂𝑅^𝑛 be nonempty convex. Suppose that the following conditions are satisfied: (i)𝜂(𝑥,𝑥)=0 for any 𝑥∈𝐾, (ii)𝑥↦𝜂(𝑥,⋅) is affine, 𝑦↦𝜂(⋅,𝑦) is continuous, (iii)the set-valued function 𝑊∶𝐾→2^𝑅𝑛 is given by 𝑊(𝑥)=𝑅^𝑛⧵(−int𝐶(𝑥)) for any 𝑥∈𝐾 such that Graph(𝑊) is closed, (iv)𝑓_𝑖∶𝑅^𝑛→𝑅,𝑖=1,2,…,𝑛 are subinvex and locally Lipschitz with respect to 𝜂, (v)there exists a nonempty closed bounded set 𝐷⊂𝐾 such that, for each 𝑦∈𝐾⧵𝐷, there exists 𝑥∈𝐷, and for any 𝜁_𝑖∈𝜕^𝜂𝑓_𝑖(𝑦),𝑖∈{1,2,…,𝑛}, such that 𝜁₁,𝜂𝑥,𝑦,𝜁₂,𝜂𝑥,𝑦,…,𝜁₁,𝜂𝑥,𝑦∈−int𝐶𝑦, (4.1) then the solutions set of (MOP) is nonempty compact.

Proof. Define a set-valued mapping 𝐺∶𝐾→2^𝐷 by 𝐺(𝑧)=𝑦∈𝐷∶∃𝜉_𝑖∈𝜕^𝜂𝑓_𝑖𝑦,𝜉₁,𝜂𝑧,𝑦,𝜉₂,𝜂𝑧,𝑦,…,𝜉_𝑛𝑦,𝜂𝑧,𝑦∉−int𝐶,∀𝑧∈𝐾. (4.2)
For any finite set {𝑦₁,𝑦₂,…,𝑦_𝑛}⊂𝐾, let 𝑁≐co(𝐷∪{𝑦₁,𝑦₂,…,𝑦_𝑛}), then 𝑁⊂𝐾 is a compact convex set. Define another set-valued mapping 𝐹∶𝑁→2^𝑁 by, 𝐹(𝑧)=𝑦∈𝑁∶∃𝜉_𝑖∈𝜕^𝜂𝑓_𝑖𝑦,𝜉₁,𝜂𝑧,𝑦,𝜉₂,𝜂𝑧,𝑦,…,𝜉_𝑛𝑦,𝜂𝑧,𝑦∉−int𝐶,∀𝑧∈𝑁. (4.3) Obviously, 𝑧∈𝐹(𝑧) for any 𝑧∈𝑁, that is, 𝐹(𝑧) is nonempty. Let {𝑦_𝜆}_𝜆∈Δ⊂𝐹(𝑧) be a net such that 𝑦_𝜆⟶𝑦₀∈𝑁,𝜉^𝜆_𝑖∈𝜕^𝜂𝑓_𝑖𝑦_𝜆. (4.4) From Theorem 3.1(iii), we conclude that Graph(𝜕^𝜂𝑓_𝑖) is closed. Therefore, there exists 𝜉⁰_𝑖∈𝜕^𝜂𝑓_𝑖(𝑦₀), such that 𝜉^𝜆_𝑖→𝜉⁰_𝑖,𝑖=1,2,…,𝑛. It follows from the closedness of Graph(𝜕^𝜂𝑓_𝑖) that 𝜉⁰₁,𝜂𝑧,𝑦₀,𝜉⁰₂,𝜂𝑧,𝑦₀,…,𝜉⁰_𝑛,𝜂𝑧,𝑦₀∉−int𝐶𝑦₀. (4.5) Thus, 𝑦₀∈𝐹(𝑧), that is, 𝐹(𝑧) is closed, and so 𝐹(𝑧) is compact, since 𝑁 is compact.
It is easy to prove that, for any finite set {𝑢₁,𝑢₂,…,𝑢_𝑚}⊂𝑁,co{𝑢₁,𝑢₂,…,𝑢_𝑚}⊂⋃^𝑚_𝑖=1𝐹(𝑢_𝑖). In fact, if there exists 𝜆_𝑖≥0,𝑖=1,2,…,𝑚, with Σ^𝑚_𝑖=1𝜆_𝑖=1 such that 𝑢=^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖∉^𝑚_𝑖=1𝐹𝑢_𝑖, (4.6) then for any 𝜁_𝑖∈𝜕^𝜂𝑓_𝑖(𝑢), 𝜁₁,𝜂𝑢_𝑖,𝑢,𝜁₂,𝜂𝑢_𝑖,𝑢,…,𝜁_𝑛,𝜂𝑢_𝑖,𝑢∈−int𝐶(𝑢). (4.7) By the convexity of 𝐶, ^𝑚_𝑖=1𝜆_𝑖⟨𝜁₁,𝜂𝑢_𝑖,𝑢⟩,⟨𝜁₂,𝜂𝑢_𝑖,𝑢⟩,...,⟨𝜁_𝑛,𝜂𝑢_𝑖,𝑢⟩∈−int𝐶(𝑢). (4.8) Since 𝜂 is affine in the first argument, we have 𝜁₁,𝜂^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖,𝑢,𝜁₂,𝜂^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖,𝑢,…,𝜁_𝑛,𝜂^𝑚_𝑖=1𝜆_𝑖𝑢_𝑖,𝑢∈−int𝐶(𝑢). (4.9) Now from the assumption that 𝜂(𝑥,𝑥)=0 for any 𝑥∈𝐾, we get 0∈−int𝐶(𝑢), which is a contradiction. Therefore, 𝐹 is a KKM mapping. By Lemma 2.13, there exists 𝑧₀∈𝑁 such that 𝑧₀∈_𝑧∈𝑁𝐹(𝑧), (4.10) that is, ∀𝑧∈𝑁,∃𝜉_𝑖∈𝜕^𝜂𝑓_𝑖𝑧₀,𝜉₁,𝜂𝑧,𝑧₀,𝜉₂,𝜂𝑧,𝑧₀,…,𝜉_𝑛,𝜂𝑧,𝑧₀∉−int𝐶𝑧₀. (4.11) From assumptions, we have 𝑧₀∈𝐷, and moreover, 𝑧₀∈⋂^𝑛_𝑖=1𝐺(𝑦_𝑖), that is, {𝐺(𝑦)}_𝑦∈𝐾 has the finite intersection property. Consequently, {𝐺(𝑦)}_𝑦∈𝐾≠∅, that is, there exists 𝑦∈𝐾 such that for any 𝑥∈𝐾, there exists 𝜉_𝑖∈𝜕^𝜂𝑓_𝑖(𝑦),𝑖=1,2,…,𝑛, such that 𝜉₁,𝜂𝑥,𝑦,𝜉₂,𝜂𝑥,𝑦,…,𝜉_𝑛,𝜂𝑥,𝑦∉−int𝐶𝑦. (4.12) From Theorem 3.1, 𝑦 is a weakly efficient solution of (MOP).
Denote the solutions set of (MOP) by 𝐽. Let {𝑥_𝑚}⊂𝐽, such that 𝑥_𝑚→𝑥₀, then for any 𝑥∈𝐾, there exists 𝜉^𝑚_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥_𝑚),𝑖=1,2,…,𝑛, such that 𝜉^𝑚₁,𝜂𝑥,𝑥_𝑚,𝜉^𝑚₂,𝜂𝑥,𝑥_𝑚,…,𝜉^𝑚_𝑛,𝜂𝑥,𝑥_𝑚∉−int𝐶𝑥_𝑚. (4.13) From Theorem 3.1, we have that 𝜕^𝜂𝑓_𝑖(⋅) is locally bounded on 𝑅^𝑛 and Graph(𝜕^𝜂𝑓_𝑖) is closed. Thus, there exists 𝜉⁰_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥₀) such that 𝜉^𝑚𝑙_𝑖→𝜉⁰_𝑖, where {𝜉^𝑚𝑙_𝑖}⊂{𝜉^𝑚_𝑖}. Since 𝜂 is continuous in the second argument and Graph(𝑊) is closed, we have 𝜉⁰₁,𝜂𝑥,𝑥₀,𝜉⁰₂,𝜂𝑥,𝑥₀,…,𝜉⁰_𝑛,𝜂𝑥,𝑥₀∉−int𝐶𝑥₀. (4.14) Thus, 𝑥₀∈𝐽, that is, 𝐽 is closed. From assumptions, we get 𝐽⊂𝐷, and so 𝐽 is compact. This completes the proof.

Example 4.2 .. Let 𝑛=2,𝐾=[0,+∞)×[0,1],𝜂(𝑥,𝑦)=(𝑥₁−𝑦₁,0),and𝑓(𝑥)=(𝑥₁,𝑥₁) for any 𝑥,𝑦∈𝐾, and 𝐶(𝑥)≡𝑅²₊ for any 𝑥∈𝐾, where 𝑥=(𝑥₁,𝑥₂),𝑦=(𝑦₁,𝑦₂). Let 𝐷=[0,1]×[0,1]. For each 𝑦∈(0,+∞)×[0,1], 𝑓₁(𝑥)−𝑓₁𝑦=𝑥₁−𝑦₁≥𝜁₁,𝜂𝑥,𝑦=𝜁¹₁𝑥₁−𝑦₁,∀𝜁₁=𝜁¹₁,𝜁²₁∈𝜕^𝜂𝑓₁𝑦, (4.15) then 1−𝜁¹₁𝑥₁−𝑦₁≥0, (4.16) and so 𝜁¹₁=1. Since 𝑓₂(𝑥)−𝑓₂𝑦=𝑥₁−𝑦₁≥⟨𝜁₂,𝜂𝑥,𝑦⟩=𝜁¹₂𝑥₁−𝑦₁,∀𝜁₂=𝜁¹₂,𝜁²₂∈𝜕^𝜂𝑓₂𝑦, (4.17) we have 𝜁¹₂=1. Therefore, 𝜕^𝜂𝑓𝑦=𝜁=𝜁₁,𝜁₂∶𝜁₁=(1,𝑎),𝜁₂=(1,𝑏),∀𝑎,𝑏∈𝑅. (4.18) Similarly, one has 𝜕^𝜂𝑓(0)=𝜁=𝜁₁,𝜁₂∶𝜁₁=(𝑐,𝑎),𝜁₂=(𝑑,𝑏),𝑐≤1,𝑑≤1,∀𝑎,𝑏∈𝑅. (4.19) Consequently, for any 𝑦∈𝐾⧵𝐷,𝑥∈𝐷,𝜁∈𝜕^𝜂𝑓(𝑦) such that 𝜁₁,𝜂𝑥,𝑦,𝜁₂,𝜂𝑥,𝑦=𝑥₁−𝑦₁,𝑥₁−𝑦₁∈−int𝐶𝑦. (4.20) Then it is easy to check that all assumptions in Theorem 4.1 hold and 𝐽={(0,𝑣)∶0≤𝑣≤1}.

Corollary 4.3 (see [9]). Let 𝑓_𝑖,𝑖=1,2,…,𝑛 be invex in the sense of Clarke's generalized directional derivative with respect to 𝜂 and locally Lipschitz. Suppose that other conditions are the same as in Theorem 4.1, then the solutions set of (MOP) is nonempty compact.

Corollary 4.4. Let 𝐾⊂𝑅^𝑛 be nonempty convex. Suppose that the following conditions are satisfied: (i)𝜂(𝑥,𝑥)=0 for any 𝑥∈𝐾, (ii)𝑥↦𝜂(𝑥,⋅) is affine and 𝑦↦𝜂(⋅,𝑦) is continuous, (iii)the set-valued function 𝑊∶𝐾→2^𝑅𝑛 is given by 𝑊(𝑥)=𝑅^𝑛⧵(−int𝐶(𝑥)) for any 𝑥∈𝐾 such that Graph(𝑊) is closed, (iv)𝑓_𝑖∶𝑅^𝑛→𝑅,𝑖=1,2,…,𝑛 are subinvex and locally Lipschitz with respect to 𝜂, (v)there exists 𝑦₀∈𝐾, such that {𝑥∈𝐾∶∃𝜁_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥),𝑖=1,2,…,𝑛,(⟨𝜁₁,𝜂(𝑦₀,𝑦)⟩,⟨𝜁₂,𝜂(𝑦₀,𝑦)⟩,…,⟨𝜁_𝑛,𝜂(𝑦₀,𝑦)⟩)∉−int𝐶(𝑥)} is closed and bounded, then the solutions set of (MOP) is nonempty compact.

Proof. Let 𝐷=𝑥∈𝐾∶∃𝜁_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥),𝑖=1,2,…,𝑛,𝜁₁,𝜂𝑦₀,𝑥,…,𝜁_𝑛,𝜂𝑦₀,𝑥∉−int𝐶(𝑥). (4.21) Obviously, 𝑦₀∈𝐷, and moreover, 𝐷 is nonempty closed bounded. Therefore, for any 𝑦∈𝐾⧵𝐷, there exists 𝑦₀∈𝐷 such that for any 𝜁_𝑖∈𝜕^𝜂𝑓_𝑖(𝑦),𝑖=1,2,…,𝑛, 𝜁₁,𝜂𝑦₀,𝑦,𝜁₂,𝜂𝑦₀,𝑦,…,⟨𝜁_𝑛,𝜂𝑦₀,𝑦⟩∈−int𝐶𝑦. (4.22) By Theorem 4.1, the solutions set of (MOP) is nonempty compact. This completes the proof.

Theorem 4.5. Let 𝐾⊂𝑅^𝑛 be a nonempty convex set. Suppose that the following conditions are satisfied: (i)𝜂(𝑦,𝑥)+𝜂(𝑥,𝑦)=0 for any 𝑥,𝑦∈𝐾, (ii)𝑥↦𝜂(𝑥,⋅) is affine and 𝑦↦𝜂(⋅,𝑦) is continuous, (iii)the set-valued function 𝑊∶𝐾→2^𝑅𝑛 is given by 𝑊(𝑥)=𝑅^𝑛⧵(−int𝐶(𝑥)) for any 𝑥∈𝐾 such that Graph(𝑊) is closed, (iv)𝑓_𝑖∶𝑅^𝑛→𝑅,𝑖=1,2,…,𝑛 are subinvex and locally Lipschitz with respect to 𝜂, (v)there exists 𝑦₀∈𝐾 such that for any 𝑥∈𝐾 and 𝜁_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥),𝑖=1,2,…,𝑛, such that lim_{‖𝑥‖→∞}𝜁_𝑖,𝜂𝑦₀,𝑥<0,𝑖=1,2,…,𝑛, (4.23) then the solutions set of (MOP) is nonempty compact.

Proof. Since there exists 𝑦₀∈𝐾, such that for any 𝜁_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥),𝑖=1,2,…,𝑛, and 𝑥∈𝐾 such that lim_{‖𝑥‖→∞}𝜁_𝑖,𝜂𝑦₀,𝑥<0,𝑖=1,2,…,𝑛, (4.24) there exists 𝑐_𝑖>0,𝑖=1,2,…,𝑛, such that for each 𝑥∈𝐾, with ‖𝑥‖>𝑐_𝑖, such that 𝜁_𝑖,𝜂𝑦₀,𝑥<0,𝑖=1,2,…,𝑛. (4.25) Taking 𝑑=max{𝑐₁+𝜀,𝑐₂+𝜀,…,𝑐_𝑛+𝜀,‖𝑦₀‖+𝜀}, where 𝜀>0, and let 𝐷={𝑥∈𝐾∶‖𝑥‖≤𝑑}. Clearly, 𝐷 is closed bounded. Then for any 𝑥∈𝐾⧵𝐷, there exists 𝑦₀∈𝐷, and for any 𝜁_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥),𝑖=1,2,…,𝑛, such that 𝜁₁,𝜂𝑦₀,𝑥,…,𝜁_𝑛,𝜂𝑦₀,𝑥∈−int𝑅^𝑛₊⊂−int𝐶(𝑥). (4.26) By Theorem 4.1, the solutions set of (MOP) is nonempty compact. This completes the proof.

Theorem 4.6. Let 𝐾⊂𝑅^𝑛 be nonempty convex. Suppose that the following conditions are satisfied: (i)𝜂(𝑦,𝑥)+𝜂(𝑥,𝑦)=0 for any 𝑥,𝑦∈𝐾, (ii)𝜂 is an open mapping, 𝑥↦𝜂(𝑥,⋅) is affine, and 𝑦↦𝜂(⋅,𝑦) is continuous, (iii)the set-valued function 𝑊∶𝐾→2^𝑅𝑛 is given by 𝑊(𝑥)=𝑅^𝑛⧵(−int𝐶(𝑥)) for any 𝑥∈𝐾 such that Graph(𝑊) is closed, (iv)𝑓_𝑖∶𝑅^𝑛→𝑅,𝑖=1,2,…,𝑛 are subinvex and locally Lipschitz with respect to 𝜂, (v)there exist a nonempty closed bounded set 𝐵⊂𝐾 and a nonempty bounded closed set 𝐷⊂𝐾 such that, for each 𝑦∈𝐾⧵𝐷, there exist 𝑥∈𝐵 and 𝜁_𝑖∈𝜕^𝜂𝑓_𝑖(𝑦),𝑖∈{1,2,…,𝑛} such that 𝜁₁,𝜂𝑥,𝑦,…,𝜁_𝑛,𝜂𝑥,𝑦∈−int𝐶𝑦 (4.27) then the solutions set of (MOP) is nonempty compact.

Proof. Define set-valued mapping 𝐺∶𝐾→2^𝐾 by 𝐺(𝑥)=𝑦∈𝐾∶∀𝜁_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥),𝜁₁,𝜂𝑥,𝑦,…,𝜁_𝑛,𝜂𝑥,𝑦∉−int𝐶𝑦,∀𝑥∈𝐾. (4.28) Obviously, 𝑥∈𝐺(𝑥) for any 𝑥∈𝐾, that is, 𝐺(𝑥)≠∅.
For any net {𝑦_𝜆}_𝜆∈Δ⊂𝐺(𝑥), with 𝑦_𝜆→𝑦₀, we have, for any 𝜁_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥), 𝜁₁,𝜂𝑥,𝑦_𝜆,…,𝜁_𝑛,𝜂𝑥,𝑦_𝜆∉−int𝐶𝑦_𝜆, (4.29) that is, 𝜁₁,𝜂𝑥,𝑦_𝜆,…,𝜁_𝑛,𝜂𝑥,𝑦_𝜆∈𝑊𝑦_𝜆. (4.30)
In view of the continuity of 𝜂 with respect to the second argument and closedness of Graph(𝑊), we obtain 𝜁₁,𝜂𝑥,𝑦₀,…,𝜁_𝑛,𝜂𝑥,𝑦₀∈𝑊𝑦₀, (4.31) and so 𝜁₁,𝜂𝑥,𝑦₀,…,𝜁_𝑛,𝜂𝑥,𝑦₀∉−int𝐶𝑦₀. (4.32) Consequently, 𝑦₀∈𝐺(𝑥), that is, 𝐺(𝑥) is closed. By Theorem 3.4, we only need to prove that (VVLI) has a solution.
Suppose to the contrary that (VVLI) has no solution, then 𝑦∈𝐾∶∀𝑥∈𝐾,∃𝜉_𝑖∈𝜕^𝜂𝑓_𝑖𝑦,𝜉₁,𝜂𝑥,𝑦,…,𝜉_𝑛,𝜂𝑥,𝑦∉−int𝐶𝑦=∅. (4.33) From the proof of Theorem 3.4, we have 𝑦∈𝐾∶∀𝑥∈𝐾,∀𝜇_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥),𝜇₁,𝜂𝑥,𝑦,...,𝜇_𝑛,𝜂𝑥,𝑦∉−int𝐶𝑦=∅. (4.34) Therefore, for each 𝑦∈𝐾,{𝑥∈𝐾∶𝑦∉𝐺(𝑥)}≠∅.
Define set-valued mappings 𝐹₁,𝐹₂∶𝐾→2^𝐾 by, respectively, 𝐹₁𝑦=𝑥∈𝐾∶∃𝜁_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥),𝜁₁,𝜂𝑥,𝑦,…,𝜁_𝑛𝑦𝐹,𝜂𝑥,𝑦∈−int𝐶,∀𝑦∈𝐾,₂𝑦=𝑥∈𝐾∶∀𝜉_𝑖∈𝜕^𝜂𝑓_𝑖𝑦,𝜉₁,𝜂𝑥,𝑦,…,𝜉_𝑛,𝜂𝑥,𝑦∈−int𝐶𝑦,∀𝑦∈𝐾. (4.35) Obviously, for each 𝑦∈𝐾,𝐹₁(𝑦)≠∅.
For any finite set {𝑥₁,𝑥₂,…,𝑥_𝑚}⊂𝐹₁(𝑦), there exist 𝜁^𝑗_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥_𝑗),𝑗=1,2,…,𝑚,𝑖=1,2,…,𝑛 such that 𝜁^𝑗₁,𝜂𝑥_𝑗,𝑦,𝜁^𝑗₂,𝜂𝑥_𝑗,𝑦,…,𝜁^𝑗_𝑛,𝜂𝑥_𝑗,𝑦∈−int𝐶𝑦. (4.36) By the 𝜂-monotonicity of 𝜕^𝜂𝑓_𝑖, we have, for each 𝑗∈{1,2,…,𝑚}, 𝜁^𝑗_𝑖,𝜂𝑥_𝑗,𝑦−𝜉_𝑖,𝜂𝑥_𝑗,𝑦≥0,∀𝜉_𝑖∈𝜕^𝜂𝑓_𝑖𝑦. (4.37) It follows that 𝜁^𝑗₁,𝜂𝑥_𝑗,𝑦,…,𝜁^𝑗_𝑛,𝜂𝑥_𝑗,𝑦−𝜉₁,𝜂𝑥_𝑗,𝑦,…,𝜉_𝑛,𝜂𝑥_𝑗,𝑦∈𝑅^𝑛₊⊂𝐶𝑦. (4.38) Since 𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0, for any 𝑥,𝑦∈𝐾, we get 𝜉₁,𝜂𝑦,𝑥_𝑗,…,𝜉_𝑛,𝜂𝑦,𝑥_𝑗−𝜁^𝑗₁,𝜂𝑦,𝑥_𝑗,…,𝜁^𝑗_𝑛,𝜂𝑦,𝑥_𝑗∈𝐶𝑦, (4.39) that is, 𝜉₁,𝜂𝑦,𝑥_𝑗,…,𝜉_𝑛,𝜂𝑦,𝑥_𝑗∈𝜁^𝑗₁,𝜂𝑦,𝑥_𝑗,…,𝜁^𝑗_𝑛,𝜂𝑦,𝑥_𝑗𝑦𝑦𝑦𝑦.+𝐶⊂int𝐶+𝐶⊂int𝐶 (4.40) Therefore, for any 𝛼_𝑗≥0,Σ^𝑚_𝑗=1𝛼_𝑗=1,Σ^𝑚_𝑗=1𝛼_𝑗(⟨𝜉₁,𝜂(𝑦,𝑥_𝑗)⟩,…,⟨𝜉_𝑛,𝜂(𝑦,𝑥_𝑗)⟩)∈int𝐶(𝑦), and thus 𝜉₁,^𝑚_𝑗=1𝛼_𝑗𝜂𝑥_𝑗,𝑦,…,𝜉_𝑛,^𝑚_𝑗=1𝛼_𝑗𝜂𝑥_𝑗,𝑦∈−int𝐶𝑦. (4.41) Since 𝜂 is affine in the first argument, it follows that 𝜉₁,𝜂^𝑚_𝑗=1𝛼_𝑗𝑥_𝑗,𝑦,…,𝜉_𝑛,𝜂^𝑚_𝑗=1𝛼_𝑗𝑥_𝑗,𝑦∈−int𝐶𝑦. (4.42) We have Σ^𝑚_𝑗=1𝛼_𝑗𝑥_𝑗∈𝐹₂(𝑦), that is, co(𝐹₁(𝑦))⊂𝐹₂(𝑦) for any 𝑦∈𝐾. In view of the closedness of 𝐺(𝑥), one has 𝐹₁⁻¹(𝑥)=𝐾⧵𝐺(𝑥) as an open set in 𝐾. Therefore, int_𝐾𝐹₁⁻¹(𝑥)=𝐹₁⁻¹(𝑥). From (4.34), we have _𝑥∈𝐾𝑦∈𝐾∶∃𝜁_𝑖∈𝜕^𝜂𝑓_𝑖(𝑥),𝜁₁,𝜂𝑥,𝑦,…,𝜁_𝑛,𝜂𝑥,𝑦∈−int𝐶𝑦=𝐾, (4.43) that is, _𝑥∈𝐾𝐹₁⁻¹(𝑥)=𝐾. (4.44) Furthermore, 𝐾=_𝑥∈𝐾𝐹₁⁻¹(𝑥)=_𝑥∈𝐾int_𝐾𝐹₁⁻¹(𝑥). (4.45)
From assumption (𝑣), one has 𝑦∈𝐹₁⁻¹(𝑥)=int_𝐾𝐹₁⁻¹(𝑥). (4.46) By Lemma 2.14, there exists 𝑦∈𝐾 such that 𝑦∈𝐹₂(𝑦). Since 𝜂(𝑥,𝑦)+𝜂(𝑦,𝑥)=0 for any 𝑥,𝑦∈𝐾, we have 𝜂(𝑦,𝑦)=0. Thus, ∀𝜉_𝑖∈𝜕^𝜂𝑓_𝑖𝑦,𝜉₁,𝜂𝑦,𝑦,…,𝜉_𝑛,𝜂𝑦,𝑦∈−int𝐶𝑦, (4.47) that is, 0∈−int𝐶(𝑦), which is a contradiction. Therefore, (VVILP) has a solution. From Theorems 3.2 and 3.4, (MOP) has a weakly efficient solution.
Similarly, we can show that 𝐽 is compact.This completes the proof.

Remark 4.7. As previously mentioned, the results presented in this paper extend some corresponding results in [5, 7, 9, 12, 13]. For instance, Theorems 4.1 and 4.6 extended the corresponding results given by Kazmi [7], and Lee et al. [13], Lee et al. [9] from preinvex functions to subinvex functions. Corollaries 4.3 and 4.4 generalized the results given by Lee et al. [13] from differentiable functions to nondifferentiable functions.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 60804065); academic award for excellent PhD candidates funded by Ministry of Education of China; the Fundamental Research Fund for the Central Universities (no. 201120102020004).