A Generalized Alternative Theorem of Partial and Generalized Cone Subconvexlike Set-Valued Maps and Its Applications in Linear Spaces

Abstract

We first introduce a new notion of the partial and generalized cone subconvexlike set-valued map and give an equivalent characterization of the partial and generalized cone subconvexlike set-valued map in linear spaces. Secondly, a generalized alternative theorem of the partial and generalized cone subconvexlike set-valued map was presented. Finally, Kuhn-Tucker conditions of set-valued optimization problems were established in the sense of globally proper efficiency.

1. Introduction

Generalized convexity plays an important role in set-valued optimization. The generalization of convexity from vector-valued maps to set-valued maps happened in the 1970s. Borwein [1] and Giannessi [2] introduced and studied the cone convexity of set-valued maps. Based on Borwein and Giannessi’s work, some authors [3–7] established a series of optimality conditions of set-valued optimization problems under different types of generalized convexity of set-valued maps in topological spaces. Since linear spaces are wider than topological spaces, generalizing some results of the above mentioned references from topological spaces to linear spaces is an interesting topic. Li [8] introduced a cone subconvexlike set-valued map involving the algebraic interior and established Kuhn-Tucker conditions. Huang and Li [9] studied Lagrangian multiplier rules of set-valued optimization problems with generalized cone subconvexlike set-valued maps in linear spaces. When the algebraic interior of the convex cone is empty, Hernández et al. [10] used the relative algebraic interior of the convex cone to introduce cone subconvexlikeness of set-valued maps and investigated Benson proper efficiency of set-valued optimization problems in linear spaces.

The aim of this paper is to study globally proper efficiency of set-valued optimization problems in linear spaces. This paper is organized as follows. In Section 2, we recalled some basic notions and gave some lemmas. In Section 3, we presented a generalized alternative theorem of the partial and generalized cone subconvexlike set-valued map and established Kuhn-Tucker conditions of set-valued optimization problems in the sense of globally proper efficiency.

2. Preliminaries

In this paper, let 𝑌 and 𝑍 be two real-ordered linear spaces, and let 0 denote the zero element of every space. Let 𝐾 be a nonempty subset in 𝑌. The cone hull of 𝐾 is defined as cone𝐾∶={𝜆𝑘∣𝑘∈𝐾,𝜆≥0}.𝐾 is called a convex cone if and only if 𝜆₁𝑘₁+𝜆₂𝑘₂∈𝐾,∀𝜆₁,𝜆₂≥0,∀𝑘₁,𝑘₂∈𝐾. (2.1) A cone 𝐾 is said to be pointed if and only if 𝐾∩(−𝐾)={0}. A cone 𝐾 is said to be nontrivial if and only if 𝐾≠{0} and 𝐾≠𝑌.

Let 𝑌^∗ and 𝑍^∗ stand for the algebraic dual spaces of 𝑌 and 𝑍, respectively. Let 𝐶 and 𝐷 be nontrivial, pointed, and convex cones in 𝑌 and 𝑍, respectively. The algebraic dual cone 𝐶⁺ of 𝐶 is defined as 𝐶⁺∶={𝑦^∗∈𝑌^∗∣⟨𝑦,𝑦^∗⟩⩾0𝑥000850,∀𝑦∈𝐶}, and the strictly algebraic dual cone 𝐶^+𝑖 of 𝐶 is defined as 𝐶^+𝑖∶={𝑦^∗∈𝑌^∗∣⟨𝑦,𝑦^∗⟩>0,∀𝑦∈𝐶⧵{0}}, where ⟨𝑦,𝑦^∗⟩ denotes the value of the linear functional 𝑦^∗ at the point 𝑦. The meaning of 𝐷⁺ is similar to that of 𝐶⁺.

Let 𝐾 be a nonempty subset of 𝑌. The linear hull span𝐾 of 𝐾 is defined as span𝐾∶={𝑘∣𝑘=∑^𝑛_𝑖=1𝜆_𝑖𝑘_𝑖,𝜆_𝑖∈ℝ,𝑘_𝑖∈𝐾,𝑖=1,…,𝑛}, and the affine hull aff𝐾 of 𝐾 is defined as aff𝐾∶={𝑘∣𝑘=∑^𝑛_𝑖=1𝜆_𝑖𝑘_𝑖,∑^𝑛_𝑖=1𝜆_𝑖=1,𝜆_𝑖∈ℝ,𝑘_𝑖∈𝐾,𝑖=1,…,𝑛}. The generated linear subspace 𝐿(𝐾) of 𝐾 is defined as 𝐿(𝐾)∶=span(𝐾−𝐾).

Definition 2.1 (see [11]). Let 𝐾 be a nonempty subset of 𝑌. The algebraic interior of 𝐾 is the set cor𝐾∶=𝑘∈𝐾∣∀𝑘^′∈𝑌,∃𝜆^′>0,∀𝜆∈0,𝜆^′,𝑘+𝜆𝑘^′∈𝐾. (2.2)

Definition 2.2 (see [12]). Let 𝐾 be a nonempty subset of 𝑌. The relative algebraic interior of 𝐾 is the set icr𝐾=𝑘∈𝐾∣∀𝑣∈aff𝐾−𝑘,∃𝜆₀>0,∀𝜆∈0,𝜆₀,𝑘+𝜆𝑣∈𝐾. (2.3)
Clearly, aff𝐾−𝑘=𝐿(𝐾),forall𝑘∈𝐾. Therefore, Definition 2.2 is consistent with the definition of the relative algebraic interior of 𝐾 in [13, 14]. However, Definition 2.2 seems to be more convenient than the ones in [13, 14].

It is worth noting that if 𝐾 is a nontrivial and pointed cone in 𝑌, then 0∉icr𝐾, and if 𝐾 is a convex cone, then icr𝐾 is a convex set, and icr𝐾∪{0} is a convex cone.

Lemma 2.3 (see [13]). If 𝐾 is a convex cone in 𝑌, then 𝐾+icr𝐾=icr𝐾.

Lemma 2.4 (see [10, 12, 14]). If 𝐾 is a nonempty subset in 𝑌, then(a) aff𝐾−𝑘=aff𝐾−𝐾,forall𝑘∈𝐾;
if 𝐾 is convex in 𝑌 and icr𝐾≠∅, then(b) icr(icr𝐾)=icr𝐾; (c) aff(icr𝐾)=aff𝐾.

Lemma 2.5 (see [12]). Let 𝐾 be a convex set with icr(𝐾)≠∅ in 𝑌. If 0∉icr𝐾, then there exists 𝑦^∗∈𝑌^∗⧵{0} such that 𝑘,𝑦^∗≥0,∀𝑘∈𝐾. (2.4)

3. Main Results

Let 𝐴 be a nonempty set, and let 𝐹∶𝐴⇉𝑌 and 𝐺∶𝐴⇉𝑍 be two set-valued maps on 𝐴. Write 𝐹(𝐴)∶=⋃_𝑥∈𝐴𝐹(𝑥) and ⟨𝐹(𝑥),𝑦^∗⟩∶={⟨𝑦,𝑦^∗⟩∣𝑦∈𝐹(𝑥)}. The meanings of 𝐺(𝐴) and ⟨𝐺(𝑥),𝑧^∗⟩ are similar to those of 𝐹(𝐴) and ⟨𝐹(𝑥),𝑦^∗⟩.

Now, we introduce a new notion of the partial and generalized cone subconvexlike set-valued map.

Definition 3.1. A set-valued map 𝐽=(𝐹,𝐺)∶𝐴⇉𝑌×𝑍 is called partial and generalized 𝐶×𝐷-subconvexlike on 𝐴 if and only if cone(𝐽(𝐴))+icr𝐶×𝐷 is a convex set in 𝑌×𝑍.

The following theorem will give some equivalent characterizations of the partial and generalized 𝐶×𝐷-subconvexlike set-valued map in linear spaces.

Theorem 3.2. Let icr𝐶≠∅. Then the following statements are equivalent:(a)the set-valued map 𝐽∶𝐴⇉𝑌×𝑍 is partial and generalized 𝐶×𝐷-subconvexlike on 𝐴,(b)Forall(𝑐,𝑑)∈icr𝐶×𝐷,∀𝑥₁,𝑥₂∈𝐴,∀𝜆∈]0,1[, (𝑐,𝑑)+𝜆𝐽𝑥₁+(1−𝜆)𝐽𝑥₂⊆cone(𝐽(𝐴))+icr𝐶×𝐷, (3.1)(c)∃𝑐^′∈icr𝐶,∀𝑥₁,𝑥₂∈𝐴,∀𝜆∈]0,1[,∀𝜀>0, 𝜀𝑐^′,0+𝜆𝐽𝑥₁+(1−𝜆)𝐽𝑥₂⊆cone(𝐽(𝐴))+𝐶×𝐷, (3.2)(d)∃𝑐^′′∈𝐶,∀𝑥₁,𝑥₂∈𝐴,∀𝜆∈]0,1[,∀𝜀>0, 𝜀𝑐^′′,0+𝜆𝐽𝑥₁+(1−𝜆)𝐽𝑥₂⊆cone(𝐽(𝐴))+𝐶×𝐷. (3.3)

Proof. (a)⇒(b). Let (𝑐,𝑑)∈icr𝐶×𝐷,𝑥₁,𝑥₂∈𝐴,𝜆∈]0,1[,(𝑦₁,𝑧₁)∈𝐽(𝑥₁), and (𝑦₂,𝑧₂)∈𝐽(𝑥₂). Clearly, 𝑦₁,𝑧₁+()(𝐽(𝐴𝑦𝑐,𝑑∈cone))+icr𝐶×𝐷,₂,𝑧₂+(𝑐,𝑑)∈cone(𝐽(𝐴))+icr𝐶×𝐷. (3.4) Since 𝐽 is partial and generalized 𝐶×𝐷-subconvexlike on 𝐴, it follows from (3.4) that (𝑐,𝑑)+𝜆𝑦₁,𝑧₁+(1−𝜆)𝑦₂,𝑧₂=𝜆𝑦₁,𝑧₁+(𝑐,𝑑)+(1−𝜆)𝑦₂,𝑧₂+(𝑐,𝑑)∈cone(𝐽(𝐴))+icr𝐶×𝐷, (3.5) which implies that (3.1) holds.
The implications (b)⇒(c)⇒(d) are clear.
(d)⇒(a). Let (𝑚_𝑖,𝑛_𝑖)∈cone(𝐽(𝐴))+icr𝐶×𝐷(𝑖=1,2),𝜆∈]0,1[. Then there exist 𝜌_𝑖≥0,𝑥_𝑖∈𝐴,(𝑦_𝑖,𝑧_𝑖)∈𝐽(𝑥_𝑖), and (𝑐_𝑖,𝑑_𝑖)∈icr𝐶×𝐷(𝑖=1,2) such that (𝑚_𝑖,𝑛_𝑖)=𝜌_𝑖(𝑦_𝑖,𝑧_𝑖)+(𝑐_𝑖,𝑑_𝑖).
Case one: if 𝜌₁=0 or 𝜌₂=0, we have 𝜆(𝑚₁,𝑛₁)+(1−𝜆)(𝑚₂,𝑛₂)∈cone(𝐽(𝐴))+icr𝐶×𝐷.
Case two: if 𝜌₁>0 and 𝜌₂>0, we have 𝜆𝑚₁,𝑛₁+(1−𝜆)𝑚₂,𝑛₂=𝜆𝜌₁𝑦₁,𝑧₁+𝑐₁,𝑑₁+(1−𝜆)𝜌₂𝑦₂,𝑧₂+𝑐₂,𝑑₂=𝜆𝑐₁,𝑑₁+(1−𝜆)𝑐₂,𝑑₂+𝜆𝜌₁𝑦₁,𝑧₁+(1−𝜆)𝜌₂𝑦₂,𝑧₂1=𝛽𝛽𝜆𝑐₁,𝑑₁+(1−𝜆)𝑐₂,𝑑₂+𝜆𝜌₁𝛽𝑦₁,𝑧₁+(1−𝜆)𝜌₂𝛽𝑦₂,𝑧₂, (3.6) where 𝛽=𝜆𝜌₁+(1−𝜆)𝜌₂.
By Lemma 2.4, we obtain −𝑐^′′1∈𝐶−𝐶⊆aff𝐶−𝐶=aff𝐶−𝛽𝜆𝑐₁+(1−𝜆)𝑐₂()−1=afficr𝐶𝛽𝜆𝑐₁+(1−𝜆)𝑐₂. (3.7) Since (1/𝛽)[𝜆𝑐₁+(1−𝜆)𝑐₂]∈icr𝐶=icr(icr𝐶), there exists 𝜆₀>0 such that 1𝛽𝜆𝑐₁+(1−𝜆)𝑐₂+𝜆₀−𝑐^′′∈icr𝐶. (3.8) By (3.3), (3.6), (3.8), and Lemma 2.3, we have 𝜆𝑚₁,𝑛₁+(1−𝜆)𝑚₂,𝑛₂1=𝛽𝛽𝜆𝑐₁,𝑑₁+(1−𝜆)𝑐₂,𝑑₂+𝜆₀−𝑐^′′,0+𝜆₀𝑐^′′,0+𝜆𝜌₁𝛽𝑦₁,𝑧₁+(1−𝜆)𝜌₂𝛽𝑦₂,𝑧₂1=𝛽𝛽𝜆𝑐₁+(1−𝜆)𝑐₂+𝜆₀−𝑐^′′,1𝛽𝜆𝑑₁+(1−𝜆)𝑑₂+𝜆₀𝑐^′′,0+𝜆𝜌₁𝛽𝑦₁,𝑧₁+(1−𝜆)𝜌₂𝛽𝑦₂,𝑧₂∈𝛽(icr𝐶×𝐷)+cone(𝐽(𝐴))+𝐶×𝐷⊆cone(𝐽(𝐴))+icr𝐶×𝐷. (3.9)
Cases one and two imply that cone(𝐽(𝐴))+icr𝐶×𝐷 is a convex set in 𝑌×𝑍. Therefore, (a) holds.

Remark 3.3. Theorem 3.2 generalizes the sixth item of Proposition  2.4 in [14], Lemma  2.1 in [15], and Lemma  2 in [16].
Now, we will give a generalized alternative theorem of the partial and generalized 𝐶×𝐷-subconvexlike map. We consider the following two systems.

System 1. There exists 𝑥₀∈𝐴 such that −𝐽(𝑥₀)∩(icr𝐶×𝐷)≠∅.

System 2. There exists (𝑦^∗,𝑧^∗)∈(𝐶⁺×𝐷⁺)⧵{(0,0)} such that 𝑦,𝑦^∗+𝑧,𝑧^∗≥0,∀𝑦,𝑧∈𝐽(𝐴). (3.10)

Theorem 3.4 (generalized alternative theorem). Let icr(cone(𝐽(𝐴))+icr𝐶×𝐷)≠∅, and let the set-valued map 𝐽∶𝐴⇉𝑌×𝑍 be partial and generalized 𝐶×𝐷-subconvexlike on 𝐴. Then, (i)if System 1 has no solutions, then System 2 has a solution; (ii)if (𝑦^∗,𝑧^∗)∈𝐶^+𝑖×𝐷⁺ is a solution of System 2, then System 1 has no solutions.

Proof. (i) Firstly, we assert that (0,0)∉cone(𝐽(𝐴))+icr𝐶×𝐷. Otherwise, there exist 𝑥₀∈𝐴 and 𝛼≥0 such that (0,0)∈𝛼𝐽(𝑥₀)+icr𝐶×𝐷.
Case one: if 𝛼=0, then 0∈icr𝐶. Since 𝐶 is a nontrivial, pointed, and convex cone, 0∉icr𝐶. Thus, we obtain a contradiction.
Case two: if 𝛼>0, then there exists (𝑦₀,𝑧₀)∈𝐽(𝑥₀) such that −𝑦₀,𝑧₀∈1𝛼(icr𝐶×𝐷)⊆icr𝐶×𝐷, (3.11) which contradicts that System 1 has no solutions.
Cases one and two show that our assertion is true. Since the set-valued map 𝐽 is partial and generalized 𝐶×𝐷-subconvexlike on 𝐴,cone(𝐽(𝐴))+icr𝐶×𝐷 is a convex set in 𝑌×𝑍. Note that icr(cone(𝐽(𝐴))+icr𝐶×𝐷)≠∅. Thus, all conditions of Lemma 2.5 are satisfied. Therefore, there exists (𝑦^∗,𝑧^∗)∈(𝑌^∗×𝑍^∗)⧵{(0,0)} such that 𝑟𝑦+𝑐,𝑦^∗+𝑟𝑧+𝑑,𝑧^∗≥0,∀𝑟≥0,𝑥∈𝐴,𝑦∈𝐹(𝑥),𝑧∈𝐺(𝑥),𝑐∈icr𝐶,𝑑∈𝐷. (3.12) Letting 𝑟=1 in (3.12), we have 𝑦+𝑐,𝑦^∗+𝑧+𝑑,𝑧^∗≥0,∀𝑥∈𝐴,𝑦∈𝐹(𝑥),𝑧∈𝐺(𝑥),𝑐∈icr𝐶,𝑑∈𝐷. (3.13)
We again assert that 𝑦^∗∈𝐶⁺. Otherwise, there exists 𝑦^′∈𝐶 such that ⟨𝑦^′,𝑦^∗⟩<0. Let 𝑥∈𝐴,𝑦∈𝐹(𝑥),𝑧∈𝐺(𝑥),𝑐∈icr𝐶, and 𝑑∈𝐷 be fixed. Then there exists sufficiently large positive number 𝜆 such that 𝜆⟨𝑦^′,𝑦^∗⟩+⟨𝑦+𝑐,𝑦^∗⟩+⟨𝑧+𝑑,𝑧^∗⟩<0, that is, 𝑦+𝑐+𝜆𝑦^′,𝑦^∗+𝑧+𝑑,𝑧^∗<0. (3.14) By Lemma 2.3, 𝑐+𝜆𝑦^′∈icr𝐶. Thus, (3.14) contradicts (3.13). Therefore, 𝑦^∗∈𝐶⁺. Similarly, we can prove that 𝑧^∗∈𝐷⁺.
Let 𝑐∈icr𝐶 be fixed in (3.13). Then, 𝛽𝑐∈icr𝐶,∀𝛽>0. Letting 𝑑=0 in (3.13), we have 𝑦,𝑦^∗+𝛽𝑐,𝑦^∗+𝑧,𝑧^∗≥0,∀𝑥∈𝐴,𝑦∈𝐹(𝑥),𝑧∈𝐺(𝑥). (3.15) Letting 𝛽→0 in (3.15), we obtain 𝑦,𝑦^∗+𝑧,𝑧^∗≥0,∀𝑥∈𝐴,𝑦∈𝐹(𝑥),𝑧∈𝐺(𝑥), (3.16) which implies that System 2 has a solution.
(ii) If (𝑦^∗,𝑧^∗)∈𝐶^+𝑖×𝐷⁺ is a solution of System 2, then 𝑦,𝑦^∗+𝑧,𝑧^∗≥0,∀𝑥∈𝐴,𝑦∈𝐹(𝑥),𝑧∈𝐺(𝑥). (3.17) We assert that System 1 has no solutions. Otherwise, there exist 𝑝∈𝐹(𝑥₀) and 𝑞∈𝐺(𝑥₀) such that −𝑝∈icr𝐶⊆𝐶⧵{0} and −𝑞∈𝐷. Therefore, we have ⟨𝑝,𝑦^∗⟩+⟨𝑞,𝑧^∗⟩<0, which contradicts (3.17). Therefore, our assertion is true.

Remark 3.5. If 𝑌×𝑍 is a finite-dimensional space, then the partial and generalized 𝐶×𝐷-subconvexlikeness of 𝐽∶𝐴⇉𝑌×𝑍 implies that cone(𝐽(𝐴))+icr𝐶×𝐷 is a nonempty convex in 𝑌×𝑍, which in turn implies that the condition icr(cone(𝐽(𝐴))+icr𝐶×𝐷)≠∅ holds trivially.

Remark 3.6. Theorem 3.4 generalizes Theorem  3.7 in [14], Theorem  2.1 in [15], and Theorem  1 in [16].
From now on, we suppose that icr𝐶≠∅.

Definition 3.7 (see [17]). Let 𝐵⊆𝑌.𝑦∈𝐵 be called a global properly efficient point with respect to 𝐶 (denoted by 𝑦∈GPE(𝐵,𝐶)) if and only if there exists a nontrivial, pointed, and convex cone 𝐶^′ with 𝐶⧵{0}⊆icr𝐶^′ such that (𝐵−𝑦)∩(−𝐶^′⧵{0})=∅.
Now, we consider the following set-valued optimization problem: (𝑥)(𝑥)Min𝐹subjectto−𝐺∩𝐷≠∅. (3.18) The feasible set of (3.18) is defined by 𝑆∶={𝑥∈𝐴∣−𝐺(𝑥)∩𝐷≠∅}.

Definition 3.8. Let 𝑥∈𝑆 be called a global properly efficient solution of (3.18) if and only if there exists 𝑦∈𝐹(𝑥) such that 𝑦∈GPE(𝐹(𝑆),𝐶). The pair (𝑥,𝑦) is called a global properly efficient element of (3.18).
Now, we will establish Kuhn-Tucker conditions of set-valued optimization problem (3.18) in the sense of globally proper efficiency.

Theorem 3.9. Suppose that the following conditions hold: (i)(𝑥₀,𝑦₀) is a global properly efficient element of (3.18); (ii)the set-valued map 𝐼∶𝐴⇉𝑌×𝑍 is partial and generalized 𝐶×𝐷-subconvexlike on 𝐴, where 𝐼(𝑥)=(𝐹(𝑥)−𝑦₀,𝐺(𝑥)),forall𝑥∈𝐴.

Then, there exists (𝑦^∗,𝑧^∗)∈(𝐶⁺×𝐷⁺)⧵{(0,0)} such that inf_𝑥∈𝐴𝐹(𝑥),𝑦^∗+𝐺(𝑥),𝑧^∗=𝑦₀,𝑦^∗,inf𝐺𝑥₀,𝑧^∗=0. (3.19)

Proof. Since (𝑥₀,𝑦₀) is a global properly efficient element of (3.18), there exists a nontrivial, pointed, and convex cone 𝐶^′ with 𝐶⧵{0}⊆icr𝐶^′ such that −𝐹(𝑥)−𝑦₀∩𝐶^′⧵{0}=∅,∀𝑥∈𝐴. (3.20) It follows from (3.20) that −𝐹(𝑥)−𝑦₀∩icr𝐶=∅,∀𝑥∈𝐴. (3.21) By (3.21), we obtain −𝐼(𝑥)∩(icr𝐶×𝐷)=∅,∀𝑥∈𝐴. (3.22) Since 𝐼 is partial and generalized 𝐶×𝐷-subconvexlike on 𝐴, it follows from (3.22) and Theorem 3.4 that there exists (𝑦^∗,𝑧^∗)∈(𝐶⁺×𝐷⁺)⧵{(0,0)} such that 𝐹(𝑥)−𝑦₀,𝑦^∗+𝐺(𝑥),𝑧^∗≥0,∀𝑥∈𝐴, (3.23) that is 𝐹(𝑥),𝑦^∗+𝐺(𝑥),𝑧^∗≥𝑦₀,𝑦^∗,∀𝑥∈𝐴. (3.24) Because 𝑥₀∈𝑆, there exists 𝑝∈𝐺(𝑥₀) such that −𝑝∈𝐷. Since 𝑧^∗∈𝐷⁺, we have 𝑝,𝑧^∗≤0. (3.25) Letting 𝑥=𝑥₀ in (3.24), we obtain 𝑝,𝑧^∗≥0. (3.26) It follows from (3.25) and (3.26) that 𝑝,𝑧^∗=0. (3.27) Therefore, we have 𝑦₀,𝑦^∗∈𝐹𝑥₀,𝑦^∗+𝐺𝑥₀,𝑧^∗. (3.28) By (3.24) and (3.28), we have inf_𝑥∈𝐴(⟨𝐹(𝑥),𝑦^∗⟩+⟨𝐺(𝑥),𝑧^∗⟩)=⟨𝑦₀,𝑦^∗⟩. Letting 𝑥=𝑥₀ in (3.24), we have 𝐺𝑥₀,𝑧^∗≥0. (3.29) It follows from (3.27) and (3.29) that inf⟨𝐺(𝑥₀),𝑧^∗⟩=0.

The following theorem, which can be found in [17], is a sufficient condition of global properly efficient elements of (3.18).

Theorem 3.10. Suppose that the following conditions hold: (i)𝑥₀∈𝑆,(ii)there exist 𝑦₀∈𝐹(𝑥₀) and (𝑦^∗,𝑧^∗)∈𝐶^+𝑖×𝐷⁺ such that inf_𝑥∈𝐴𝐹(𝑥),𝑦^∗+𝐺(𝑥),𝑧^∗≥𝑦₀,𝑦^∗. (3.30)Then, (𝑥₀,𝑦₀) is a global properly efficient element of (3.18).

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (Grants 11171363 and 11271391), the Natural Science Foundation of Chongqing (CSTC 2011jjA00022 and CSTC 2011BA0030), the Special Fund of Chongqing Key Laboratory (CSTC 2011KLORSE01), and the project of the Third Batch Support Program for Excellent Talents of Chongqing City High Colleges.