# Nonemptiness and Compactness of Solutions Set for Nondifferentiable Multiobjective Optimization Problems

Authors: Xin-kun Wu; Jia-wei Chen; Yun-zhi Zou
Journal: Journal of Applied Mathematics
 (2011)

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## 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.


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## Body

## 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, letRn be the n-dimensional Euclidean space, and R+n={x=(x1,…,xn):xl≥0,l=1,…,n}. Let K be a nonempty convex subset of Rn, let H be a subset of K, and let intKH be the relative interior of H to K. Let f=(f1,f2,...,fn):Rn→Rn, η:Rn×Rn→Rn, and let C:K→2Rn such that, for each x∈K, C(x) is a closed convex cone, C(x)≠Rn, intC(x)≠∅, and R+n⊂C(x). The multiobjective optimization problem (for short, (MOP)) is defined as follows: (MOP)min⁡x∈K⁡f(x).We first recall some definitions and lemmas which are needed in the main results of this paper.Definition 2.1.
A pointx̅∈K is said to be a weakly efficient (weak minimum) solution of (MOP) if [f(K)-f(x̅)]∩(-intC(x̅))=∅.Definition 2.2.
A real-valued functionh:Rn→R is said to be locally Lipschitz with respect to η:Rn×Rn→Rn if, for each z0∈Rn, there exist a neighborhood N(z0) of z0 and a constant k>0 such that
(2.1)|h(x)-h(y)|≤k‖η(x,y)‖,∀x,y∈N(z0).Remark 2.3.
Ifη(x,y)=x-y, then the above definition reduces to that of local Lipschitz.Definition 2.4.
A set-valued functionT:Rn→2Rn is said to be locally bounded at z0∈Rn if there exist a neighborhood N(z0) of z0 and a constant k>0 such that
(2.2)‖w‖<k,∀z∈N(z0),∀w∈T(z).Definition 2.5.
LetT:Rn→2Rn be a set-valued function. The graph of T is defined as
(2.3)Gragh(T)={(x,y):x∈Rn,y∈T(x)}.
The inverse T-1 of T is defined by x∈T-1(y) if and only if y∈T(x).Definition 2.6.
LetX be a nonempty subset of topological vector space E. A set-valued mapping F:X→2E is called a KKM mapping if, for every finite subset {x1,x2,…,xn} of X,co{x1,x2,…,xn}⊂⋃i=1nF(xi), where co denotes the convex hull.Definition 2.7 (see [13]).
A real-valued functionh:Rn→R is said to be subinvex at z with respect to η:Rn×Rn→Rn if there exists ζ∈Rn, such that h(x)-h(z)≥〈ζ,η(x,z)〉, for any x∈Rn, 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∈Rn such that
(2.4)h(x)-h(z)≥〈ζ,η(x,z)〉,∀x∈Rn.Remark 2.8.
Ifh 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∈Rn:h(x)≥h(z)+〈ζ,x-z〉,∀x∈Rn}.Definition 2.9.
A vector-valued functiong:Rn→Rn is called strictly pseudosubinvex with respect to η if, for any x,u∈Rn and (ξ1,ξ2,…,ξn)∈∂ηg(u),
(2.5)g(x)-g(u)∈-int⁡C(u)⟹(〈ξ1,η(x,u)〉,〈ξ2,η(x,u)〉,…,〈ξn,η(x,u)〉)∈-int⁡R+n,
where ∂ηg(u)={(ξ1,ξ2,…,ξn)∈(Rn)n:gi(x)-gi(u)≥〈ξi,η(x,u)〉,∀x∈Rn}.Definition 2.10.
A vector-valued functiong:Rn→Rn is called pseudosubinvex with respect to η if, for any x,u∈Rn and (ξ1,ξ2,…,ξn)∈∂ηg(u),
(2.6)g(x)-g(u)∈-int⁡C(u)⟹(〈ξ1,η(x,u)〉,〈ξ2,η(x,u)〉,…,〈ξn,η(x,u)〉)∈-int⁡C(u),
where ∂ηg(u)={(ξ1,ξ2,…,ξn)∈(Rn)n:gi(x)-gi(u)≥〈ξi,η(x,u)〉,∀x∈Rn}.Remark 2.11.
SinceR+n⊂C(x), it is clear that (1)
ifg:Rn→Rn is strictly pseudosubinvex with respect to η, then it is pseudosubinvex with respect to η;(2)
ifgi:Rn→R,i=1,2,…,n, are subinvex with respect to η, then g=(g1,g2,…,gn) is pseudosubinvex with respect to η.Definition 2.12.
A pointy∈K is called a critical point of (MOP) if there exists λ=(λ1,λ2,…,λn)∈R+n, with λi>0 for some i∈{1,2,…,n}, such that
(2.7)∑i=1nλiξi=0,∀ξ=(ξ1,ξ2,…,ξn)∈∂ηf(y).Lemma 2.13 (see [14](FKKM theorem)).
LetQ be a nonempty subset of Hausdorff topological vector space X. Let G:Q→2X 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]).
LetX be a nonempty and convex subset of Hausdorff topological vector space E, and let S,T:X→2X be two set-valued maps such that, for each x∈X,coS(x)⊂T(x),S(x)≠∅,X=⋃{intXS-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 x̅∈B, such that z∈intXS-1(x̅), then there exists z̅∈X such that z̅∈T(z̅).We consider the following vector variational-like inequality (for short,(VVLI)): find x̅∈K such that for any x∈K, there exists ξ=(ξ1,ξ2,…,ξn)∈∂ηf(x̅) such that (2.8)(〈ξ1,η(x,x̅)〉,〈ξ2,η(x,x̅)〉,…,〈ξn,η(x,x̅)〉)∉-int⁡C(x̅).


## 3. Relationships between (MOP) and (VVLI)

In this section, we will investigate the properties ofη-subdifferential ∂ηh of the function h:Rn→R, and the relationships among weakly efficient solutions, critical points of (MOP), and the solutions of (VVILP).Theorem 3.1.
Leth:Rn→R be subinvex with respect to η, then the following statements are true: (i)
for eachz∈Rn, ∂ηh(z) is a nonempty closed-convex subset of Rn,(ii)
if for eachx,z∈Rn,η(x,z)+η(z,x)=0, then ∂ηh is η-monotone, that is, for any x1,x2∈Rn,
(3.1)〈ζ-ξ,η(x1,x2)〉≥0,∀ζ∈∂ηh(x1),ξ∈∂ηh(x2),(iii)
ifh is locally Lipschitz with respect to η, η is continuous in the second argument, and η(x,x)=0, for any x∈Rn, then Graph(∂ηh) is closed, and ∂ηh(·) is upper semicontinuous,(iv)
ifη is an open map, η(x,x)=0 for any x∈Rn, and h is locally Lipschitz with respect to η, then ∂ηh(·) is locally bounded on Rn.Proof.
Assertions (i), (ii), and (iv) are shown in [13]. We only need to prove assertion (iii). Let (xn,yn)∈Graph(∂ηh), with xn→x0 and yn→y0. Since
(3.2)∂ηh(xj)≐{ζ∈Rn:h(x)-h(xj)≥〈ζ,η(x,xj)〉,∀x∈Rn},i=1,2,…,n,
and h is locally Lipschitz with respect to η, there exist a neighborhood N(x0) of x0 and a constant k>0 such that
(3.3)|h(x)-h(z)|≤k‖η(x,z)‖,∀x,z∈N(x0).
Then there exists n0 such that xn∈N(x0) for all n>n0, and so
(3.4)|h(x0)-h(xn)|≤k‖η(x0,xn)‖.
Consequently, we have h(xn)→h(x0) as n→∞. It follows from
(3.5)h(x)-h(xn)≥〈yn,η(x,xn)〉,∀x∈Rn
that
(3.6)h(x)-h(x0)≥〈y0,η(x,x0)〉,∀x∈Rn,
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.
Letf be pseudosubinvex with respect to η. If y̅∈K is a solution of (VVLI), then y̅ is a weakly efficient solution of (MOP).Proof.
Lety̅∈K be a solution of (VVLI). If y̅ is not a weakly efficient solution of (MOP), then there exists x̅∈K such that
(3.7)f(x̅)-f(y̅)∈-int⁡C(y̅).
Since f is pseudosubinvex with respect to η, we have
(3.8)(〈ξ1η(x̅,y̅)〉,〈ξ2,η(x̅,y̅)〉,…,〈ξn,η(x̅,y̅)〉)∈-int⁡C(y̅),∀(ξ1,ξ2,…,ξn)∈∂ηf(y̅),
which contradicts the assumption. This completes the proof.Corollary 3.3.
Letfi,i=1,2,…,n be subinvex with respect to η. If y̅∈K is a solution of (VVLI), then y̅ is a weakly efficient solution of (MOP).Theorem 3.4.
Letη 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 fi,i=1,2,…,n be subinvex and locally Lipschitz with respect to η. If y̅∈K is a weakly efficient solution of (MOP), then y̅ is a solution of (VVLI).Proof..
Lety̅∈K be a weakly efficient solution of (MOP). If y̅ is not a solution of (VVLI), then
(3.9)y̅∉{y∈K:∀x∈K,∃ξi∈∂ηfi(y),(〈ξ1,η(x,y)〉,…,〈ξ1,η(x,y)〉)∉-int⁡C(y)}.
We assert that
(3.10){y∈K:∀x∈K,∃ξi∈∂ηfi(y),(〈ξ1,η(x,y)〉,…,〈ξn,η(x,y)〉)∉-int⁡C(y)}={y∈K:∀x∈K,∀μi∈∂ηfi(x),(〈μ1,η(x,y)〉,…,〈μn,η(x,y)〉)∉-int⁡C(y)}.
Let
(3.11)z̅∈{y∈K:∀x∈K,∃ξi∈∂ηfi(y),(〈ξ1,η(x,y)〉,…,〈ξn,η(x,y)〉)∉}-int⁡C(y)}.
Suppose to the contrary that
(3.12)z̅∉{y∈K:∀x∈K,∀μi∈∂ηfi(x),(〈μ1,η(x,y)〉,...,〈μn,η(x,y)〉)∉-int⁡C(y)},
then there exist x̂∈K and μ̂i∈∂ηfi(x̂) such that
(3.13)(〈μ̂1,η(x̂,z̅)〉,…,〈μ̂n,η(x̂,z̅)〉)∈-int⁡C(z̅).
Since ∂ηfi is η-monotone, we have
(3.14)〈μi-ξi,η(x̂,z̅)〉≥0,∀μi∈∂ηfi(x̂),ξi∈∂ηfi(z̅),
that is,
(3.15)〈μi,η(x̂,z̅)〉-〈ξi,η(x̂,z̅)〉∈R+,i=1,2,…,n.
As a consequence,
(3.16)(〈μ̂1,η(x̂,z̅)〉,…,〈μ̂n,η(x̂,z̅)〉)-(〈ξ1,η(x̂,z̅)〉,…,〈ξn,η(x̂,z̅)〉)∈R+n,
and it follows that
(3.17)(〈ξ1,η(x̂,z̅)〉,…,〈ξn,η(x̂,z̅)〉)∈(〈μ̂1,η(x̂,z̅)〉,…,〈μ̂n,η(x̂,z̅)〉)-R+n⊂-int⁡C(z̅)-R+n⊂-int⁡C(z̅),
which contradicts the assumption.
On the other hand, let(3.18)y′∈{y∈K:∀x∈K,∀μi∈∂ηfi(x),(〈μ1,η(x,y)〉,…,〈μn,η(x,y)〉)∉-int⁡C(y)},
and let rm∈(0,1] such that rm→0. For any given x∈K, set xm=(1-rm)y′+rmx, for any x∈K. Then xm→y′ as m→∞. Since η is affine in the first argument, and η(x,x)=0, we have
(3.19)(〈μ1m,η(xm,y′)〉,…,〈μnm,η(xm,y′)〉)=rm(〈μ1m,η(x,y′)〉,…,〈μnm,η(x,y′)〉)∉-int⁡C(y′),∀μim∈∂ηfi(xm),
that is,
(3.20)(〈μ1m,η(x,y′)〉,…,〈μnm,η(x,y′)〉)∉-int⁡C(y′).
Since ∂ηfi is locally bounded, there exist a neighborhood N(y′) of y′, and k>0 such that for any z∈N(y′), and ψi∈∂ηfi(z), we have
(3.21)‖ψi‖≤k,i=1,2,…,n.
Then there exists m0, such that xm∈N(y′) for any m≥m0, and so
(3.22)‖μim‖≤k,i=1,2,…,n.
Consequently, {μim} has a convergent subsequence. Without loss of generality, let μim→μi′. By Theorem 3.1 (iii), we have μi′∈∂ηfi(y′). Since Rn∖intC(y′) is closed, it follows that
(3.23)y′∈{y∈K:∀x∈K,∃ξi∈∂ηfi(y),(〈ξ1,η(x,y)〉,…,〈ξn,η(x,y)〉)∉-int⁡C(y)}.
Thus, it follows from (3.9) that
(3.24)y̅∉{y∈K:∀x∈K,∀μi∈∂ηfi(x),(〈μ1,η(x,y)〉,...,〈μn,η(x,y)〉)∉-int⁡C(y)},
that is, there exist x̅∈K and ζ̅i∈∂ηfi(x̅) such that
(3.25)(〈ζ̅1,η(x̅,y̅)〉,…,〈ζ̅n,η(x̅,y̅)〉)∈-int⁡C(y̅).
Since
(3.26)fi(y̅)-fi(x̅)≥〈ζ̅i,η(y̅,x̅)〉,η(y̅,x̅)+η(x̅,y̅)=0,
we have
(3.27)(f1(x̅)-f1(y̅),…,fn(x̅)-fn(y̅))-(〈ζ̅1,η(x̅,y̅)〉,…,〈ζ̅n,η(x̅,y̅)〉)∈-R+n.
Thus,
(3.28)f(x̅)-f(y̅)∈-R+n-int⁡C(y̅)⊂-int⁡C(y̅),
that is,
(3.29)f(x̅)-f(y̅)∈-int⁡C(y̅),
which contradicts the assumption. This completes the proof.Theorem 3.5.
Letfi,i=1,2,…,n be subinvex with respect to η, and let f be strictly pseudosubinvex with respect to η. If y̅∈K is a critical point of (MOP), then y̅ is a weakly efficient solution of (MOP).Proof.
Lety̅∈K be a critical point of (MOP). If y̅ is not the weakly efficient solution of (MOP), then there exists x̅∈K such that
(3.30)f(x̅)-f(y̅)∈-int⁡C(y̅).
By the strict pseudosubinvexity of f with respect to η, one has
(3.31)〈ξ,η(x̅,y̅)〉∈-int⁡R+n,∀ξ=(ξ1,ξ2,…,ξn)∈∂ηf(y̅),
that is,
(3.32)(〈ξ1,η(x̅,y̅)〉,…,〈ξn,η(x̅,y̅)〉)∈-int⁡R+n.
Thus, we have
(3.33)〈ξi,η(x̅,y̅)〉<0,i=1,2,…,n.
Since y̅∈K is a critical point of (MOP), there is λ=(λ1,λ2,…,λn),i∈1,2,…,n, with λi>0 for some i∈{1,2,…,n}, such that
(3.34)∑i=1nλiξi=0.
Set Sλ=∑i=1nλi>0, then
(3.35)∑i=1nλiSλ=1,∑i=1nλiSλξi=0,
and so
(3.36)0=〈∑i=1nλiSλξi,η(x̅,y̅)〉<0,i=1,2,…,n,
which is a contradiction. This completes the proof.Remark 3.6.
If for eachy∈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)
iff 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.
LetK⊂Rn 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 functionW:K→2Rn is given by W(x)=Rn∖(-intC(x)) for any x∈K such that Graph(W) is closed,(iv)
fi:Rn→R,i=1,2,…,n are subinvex and locally Lipschitz with respect to η,(v)
there exists a nonempty closed bounded setD⊂K such that, for each y∈K∖D, there exists x̅∈D, and for any ζi∈∂ηfi(y),i∈{1,2,…,n}, such that
(4.1)(〈ζ1,η(x̅,y)〉,〈ζ2,η(x̅,y)〉,…,〈ζ1,η(x̅,y)〉)∈-int⁡C(y), then the solutions set of (MOP) is nonempty compact.Proof.
Define a set-valued mappingG:K→2D by
(4.2)G(z)={y∈D:∃ξi∈∂ηfi(y),(〈ξ1,η(z,y)〉,〈ξ2,η(z,y)〉,…,〈ξn,η(z,y)〉)∉-int⁡C(y)},∀z∈K.
For any finite set{y1,y2,…,yn}⊂K, let N≐co(D∪{y1,y2,…,yn}), then N⊂K is a compact convex set. Define another set-valued mapping F:N→2N by,
(4.3)F(z)={y∈N:∃ξi∈∂ηfi(y),(〈ξ1,η(z,y)〉,〈ξ2,η(z,y)〉,…,〈ξn,η(z,y)〉)∉-int⁡C(y)},∀z∈N.
Obviously, z∈F(z) for any z∈N, that is, F(z) is nonempty. Let {yλ}λ∈Δ⊂F(z) be a net such that
(4.4)yλ⟶y0∈N,ξiλ∈∂ηfi(yλ).
From Theorem 3.1(iii), we conclude that Graph(∂ηfi) is closed. Therefore, there exists ξi0∈∂ηfi(y0), such that ξiλ→ξi0,i=1,2,…,n. It follows from the closedness of Graph(∂ηfi) that
(4.5)(〈ξ10,η(z,y0)〉,〈ξ20,η(z,y0)〉,…,〈ξn0,η(z,y0)〉)∉-int⁡C(y0).
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,…,um}⊂N,co{u1,u2,…,um}⊂⋃i=1mF(ui). In fact, if there exists λi≥0,i=1,2,…,m, with Σi=1mλi=1 such that
(4.6)u̅=∑i=1mλiui∉⋃i=1mF(ui),
then for any ζi∈∂ηfi(u̅),
(4.7)(〈ζ1,η(ui,u̅)〉,〈ζ2,η(ui,u̅)〉,…,〈ζn,η(ui,u̅)〉)∈-int⁡C(u̅).
By the convexity of C,
(4.8)∑i=1mλi(〈ζ1,η(ui,u̅)〉,〈ζ2,η(ui,u̅)〉,...,〈ζn,η(ui,u̅)〉)∈-int⁡C(u̅).
Since η is affine in the first argument, we have
(4.9)(〈ζ1,η(∑i=1mλiui,u̅)〉,〈ζ2,η(∑i=1mλiui,u̅)〉,…,〈ζn,η(∑i=1mλiui,u̅)〉)∈-int⁡C(u̅).
Now from the assumption that η(x,x)=0 for any x∈K, we get 0∈-intC(u̅), which is a contradiction. Therefore, F is a KKM mapping. By Lemma 2.13, there exists z0∈N such that
(4.10)z0∈⋂z∈NF(z),
that is,
(4.11)∀z∈N,∃ξi∈∂ηfi(z0),(〈ξ1,η(z,z0)〉,〈ξ2,η(z,z0)〉,…,〈ξn,η(z,z0)〉)∉-int⁡C(z0).
From assumptions, we have z0∈D, and moreover, z0∈⋂i=1nG(yi), that is, {G(y)}y∈K has the finite intersection property. Consequently, {G(y)}y∈K≠∅, that is, there exists y̅∈K such that for any x∈K, there exists ξ̅i∈∂ηfi(y̅),i=1,2,…,n, such that
(4.12)(〈ξ̅1,η(x,y̅)〉,〈ξ̅2,η(x,y̅)〉,…,〈ξ̅n,η(x,y̅)〉)∉-int⁡C(y̅).
From Theorem 3.1, y̅ is a weakly efficient solution of (MOP).
Denote the solutions set of (MOP) by J. Let {xm}⊂J, such that xm→x0, then for any x∈K, there exists ξim∈∂ηfi(xm),i=1,2,…,n, such that
(4.13)(〈ξ1m,η(x,xm)〉,〈ξ2m,η(x,xm)〉,…,〈ξnm,η(x,xm)〉)∉-int⁡C(xm).
From Theorem 3.1, we have that ∂ηfi(·) is locally bounded on Rn and Graph(∂ηfi) is closed. Thus, there exists ξi0∈∂ηfi(x0) such that ξiml→ξi0, where {ξiml}⊂{ξim}. Since η is continuous in the second argument and Graph(W) is closed, we have
(4.14)(〈ξ10,η(x,x0)〉,〈ξ20,η(x,x0)〉,…,〈ξn0,η(x,x0)〉)∉-int⁡C(x0).
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 ..
Letn=2,K=[0,+∞)×[0,1],η(x,y)=(x1-y1,0),andf(x)=(x1,x1) for any x,y∈K, and C(x)≡R+2 for any x∈K, where x=(x1,x2),y=(y1,y2). Let D=[0,1]×[0,1]. For each y∈(0,+∞)×[0,1],
(4.15)f1(x)-f1(y)=x1-y1≥〈ζ1,η(x,y)〉=ζ11(x1-y1),∀ζ1=(ζ11,ζ12)∈∂ηf1(y),
then
(4.16)(1-ζ11)(x1-y1)≥0,
and so ζ11=1. Since
(4.17)f2(x)-f2(y)=x1-y1≥〈ζ2,η(x,y)〉=ζ21(x1-y1),∀ζ2=(ζ21,ζ22)∈∂ηf2(y),
we have ζ21=1. Therefore,
(4.18)∂ηf(y)={ζ=(ζ1,ζ2):ζ1=(1,a),ζ2=(1,b),∀a,b∈R}.
Similarly, one has
(4.19)∂ηf(0)={ζ=(ζ1,ζ2):ζ1=(c,a),ζ2=(d,b),c≤1,d≤1,∀a,b∈R}.
Consequently, for any y∈K∖D,x̅∈D,ζ∈∂ηf(y) such that
(4.20)(〈ζ1,η(x̅,y)〉,〈ζ2,η(x̅,y)〉)=(x̅1-y1,x̅1-y1)∈-int⁡C(y).
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]).
Letfi,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.
LetK⊂Rn 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 functionW:K→2Rn is given by W(x)=Rn∖(-intC(x)) for any x∈K such that Graph(W) is closed,(iv)
fi:Rn→R,i=1,2,…,n are subinvex and locally Lipschitz with respect to η,(v)
there existsy0∈K, such that {x∈K:∃ζi∈∂ηfi(x),i=1,2,…,n,(〈ζ1,η(y0,y)〉,〈ζ2,η(y0,y)〉,…,〈ζn,η(y0,y)〉)∉-intC(x)} is closed and bounded, then the solutions set of (MOP) is nonempty compact.Proof.
Let(4.21)D={x∈K:∃ζi∈∂ηfi(x),i=1,2,…,n,(〈ζ1,η(y0,x)〉,…,〈ζn,η(y0,x)〉)∉-int⁡C(x)}.
Obviously, y0∈D, and moreover, D is nonempty closed bounded. Therefore, for any y∈K∖D, there exists y0∈D such that for any ζi∈∂ηfi(y),i=1,2,…,n,
(4.22)(〈ζ1,η(y0,y)〉,〈ζ2,η(y0,y)〉,…,〈ζn,η(y0,y)〉)∈-int⁡C(y).
By Theorem 4.1, the solutions set of (MOP) is nonempty compact. This completes the proof.Theorem 4.5.
LetK⊂Rn 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 functionW:K→2Rn is given by W(x)=Rn∖(-intC(x)) for any x∈K such that Graph(W) is closed,(iv)
fi:Rn→R,i=1,2,…,n are subinvex and locally Lipschitz with respect to η,(v)
there existsy0∈K such that for any x∈K and ζi∈∂ηfi(x),i=1,2,…,n, such that
(4.23)lim⁡∥x∥→∞〈ζi,η(y0,x)〉<0,i=1,2,…,n, then the solutions set of (MOP) is nonempty compact.Proof.
Since there existsy0∈K, such that for any ζi∈∂ηfi(x),i=1,2,…,n, and x∈K such that
(4.24)lim⁡∥x∥→∞〈ζi,η(y0,x)〉<0,i=1,2,…,n,
there exists ci>0,i=1,2,…,n, such that for each x∈K, with ∥x∥>ci, such that
(4.25)〈ζi,η(y0,x)〉<0,i=1,2,…,n.
Taking d=max{c1+ε,c2+ε,…,cn+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 ζi∈∂ηfi(x),i=1,2,…,n, such that
(4.26)(〈ζ1,η(y0,x)〉,…,〈ζn,η(y0,x)〉)∈-int⁡R+n⊂-int⁡C(x).
By Theorem 4.1, the solutions set of (MOP) is nonempty compact. This completes the proof.Theorem 4.6.
LetK⊂Rn 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 functionW:K→2Rn is given by W(x)=Rn∖(-intC(x)) for any x∈K such that Graph(W) is closed,(iv)
fi:Rn→R,i=1,2,…,n are subinvex and locally Lipschitz with respect to η,(v)
there exist a nonempty closed bounded setB⊂K and a nonempty bounded closed set D⊂K such that, for each y∈K∖D, there exist x̅∈B and ζi∈∂ηfi(y),i∈{1,2,…,n} such that
(4.27)(〈ζ̅1,η(x̅,y)〉,…,〈ζ̅n,η(x̅,y)〉)∈-int⁡C(y) then the solutions set of (MOP) is nonempty compact.Proof.
Define set-valued mappingG:K→2K by
(4.28)G(x)={y∈K:∀ζi∈∂ηfi(x),(〈ζ1,η(x,y)〉,…,〈ζn,η(x,y)〉)∉-int⁡C(y)},∀x∈K.
Obviously, x∈G(x) for any x∈K, that is, G(x)≠∅.
For any net{yλ}λ∈Δ⊂G(x), with yλ→y0, we have, for any ζi∈∂ηfi(x),
(4.29)(〈ζ1,η(x,yλ)〉,…,〈ζn,η(x,yλ)〉)∉-int⁡C(yλ),
that is,
(4.30)(〈ζ1,η(x,yλ)〉,…,〈ζn,η(x,yλ)〉)∈W(yλ).
In view of the continuity ofη with respect to the second argument and closedness of Graph(W), we obtain
(4.31)(〈ζ1,η(x,y0)〉,…,〈ζn,η(x,y0)〉)∈W(y0),
and so
(4.32)(〈ζ1,η(x,y0)〉,…,〈ζn,η(x,y0)〉)∉-int⁡C(y0).
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
(4.33){y∈K:∀x∈K,∃ξi∈∂ηfi(y),(〈ξ1,η(x,y)〉,…,〈ξn,η(x,y)〉)∉-int⁡C(y)}=∅.
From the proof of Theorem 3.4, we have
(4.34){y∈K:∀x∈K,∀μi∈∂ηfi(x),(〈μ1,η(x,y)〉,...,〈μn,η(x,y)〉)∉-int⁡C(y)}=∅.
Therefore, for each y∈K,{x∈K:y∉G(x)}≠∅.
Define set-valued mappingsF1,F2:K→2K by, respectively,
(4.35)F1(y)={x∈K:∃ζi∈∂ηfi(x),(〈ζ1,η(x,y)〉,…,〈ζn,η(x,y)〉)∈-int⁡C(y)},∀y∈K,F2(y)={x∈K:∀ξi∈∂ηfi(y),(〈ξ1,η(x,y)〉,…,〈ξn,η(x,y)〉)∈-int⁡C(y)},∀y∈K.
Obviously, for each y∈K,F1(y)≠∅.
For any finite set{x1,x2,…,xm}⊂F1(y), there exist ζij∈∂ηfi(xj),j=1,2,…,m,i=1,2,…,n such that
(4.36)(〈ζ1j,η(xj,y)〉,〈ζ2j,η(xj,y)〉,…,〈ζnj,η(xj,y)〉)∈-int⁡C(y).
By the η-monotonicity of ∂ηfi, we have, for each j∈{1,2,…,m},
(4.37)〈ζij,η(xj,y)〉-〈ξi,η(xj,y)〉≥0,∀ξi∈∂ηfi(y).
It follows that
(4.38)(〈ζ1j,η(xj,y)〉,…,〈ζnj,η(xj,y)〉)-(〈ξ1,η(xj,y)〉,…,〈ξn,η(xj,y)〉)∈R+n⊂C(y).
Since η(x,y)+η(y,x)=0, for any x,y∈K, we get
(4.39)(〈ξ1,η(y,xj)〉,…,〈ξn,η(y,xj)〉)-(〈ζ1j,η(y,xj)〉,…,〈ζnj,η(y,xj)〉)∈C(y),
that is,
(4.40)(〈ξ1,η(y,xj)〉,…,〈ξn,η(y,xj)〉)∈(〈ζ1j,η(y,xj)〉,…,〈ζnj,η(y,xj)〉)+C(y)⊂int⁡C(y)+C(y)⊂int⁡C(y).
Therefore, for any αj≥0,Σj=1mαj=1,Σj=1mαj(〈ξ1,η(y,xj)〉,…,〈ξn,η(y,xj)〉)∈intC(y), and thus
(4.41)(〈ξ1,∑j=1mαjη(xj,y)〉,…,〈ξn,∑j=1mαjη(xj,y)〉)∈-int⁡C(y).
Since η is affine in the first argument, it follows that
(4.42)(〈ξ1,η(∑j=1mαjxj,y)〉,…,〈ξn,η(∑j=1mαjxj,y)〉)∈-int⁡C(y).
We have Σj=1mαjxj∈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, intKF1-1(x)=F1-1(x). From (4.34), we have
(4.43)⋃x∈K{y∈K:∃ζi∈∂ηfi(x),(〈ζ1,η(x,y)〉,…,〈ζn,η(x,y)〉)∈-int⁡C(y)}=K,
that is,
(4.44)⋃x∈KF1-1(x)=K.
Furthermore,
(4.45)K=⋃x∈KF1-1(x)=⋃x∈KintKF1-1(x).
From assumption(v), one has
(4.46)y∈F1-1(x̅)=intKF1-1(x̅).
By Lemma 2.14, there exists y̅∈K such that y̅∈F2(y̅). Since η(x,y)+η(y,x)=0 for any x,y∈K, we have η(y̅,y̅)=0. Thus,
(4.47)∀ξi∈∂ηfi(y̅),(〈ξ1,η(y̅,y̅)〉,…,〈ξn,η(y̅,y̅)〉)∈-int⁡C(y̅),
that is, 0∈-intC(y̅), 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 thatJ 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.



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