Strict Efficiency in Vector Optimization with Nearly Convexlike Set-Valued Maps

Abstract

The concept of the well posedness for a special scalar problem is linked with strictly efficient solutions of vector optimization problem involving nearly convexlike set-valued maps. Two scalarization theorems and two Lagrange multiplier theorems for strict efficiency in vector optimization involving nearly convexlike set-valued maps are established. A dual is proposed and duality results are obtained in terms of strictly efficient solutions. A new type of saddle point, called strict saddle point, of an appropriate set-valued Lagrange map is introduced and is used to characterize strict efficiency.

1. Introduction

One important problem in vector optimization is to find the efficient points of a set. As observed by Kuhn, Tucker, and later by Geoffrion, some efficient points exhibit certain abnormal properties. To eliminate such abnormal efficient points, various concepts of proper efficiency have been introduced. The original concept was introduced by Kuhn and Tucker [1] and Geoffrion [2] and was later modified and formulated in a more general framework by Borwein [3], Hartley [4], Benson [5], Henig [6], and Borwein and Zhuang [7]; also see the references therein. In particular, the concept of strict efficiency was first introduced by Bednarczak and Song [8] in order to obtain upper semicontinuity of the section mapping 𝐺(𝑦)=𝑆∩(𝑦−𝐶) at an efficient point. Zaffaroni [9] used a special scalar function to characterize the strict efficiency and obtained some properties of strict efficiency, which includes well posedness.

Recently, several authors have turned their interests to vector optimization of set-valued maps. For instance, see [10–16]. Li [17] extended the concept of Benson proper efficiency to set-valued maps and presented two scalarization theorems and Lagrange multiplier theorems for set-valued vector optimization problem under cone subconvexlikeness. Mehra [18] and Xia and Qiu [19] discussed the super efficiency in vector optimization problem involving nearly cone-convexlike set-valued maps and nearly cone-subconvexlike set-valued maps, respectively. Miglierina [20] linked the properly efficient solutions of set-valued vector optimization with well-posedness hypothesis of a special scalar problem.

In this paper, inspired by [8, 17, 18], we study strict efficiency for vector optimization problem involving nearly cone-convexlike set-valued maps in the framework of real normed locally convex spaces. The paper is organized as follows. In Section 2, we recall some basic concepts and lemmas. In Section 3, the well posedness of a special scalar problems on strict efficiency involving nearly cone-convexlike set-valued maps is discussed. In Section 4, two scalarization theorems for strict efficiency in vector optimization problems involving nearly cone-convexlike set-valued maps are obtained. In Section 5, we establish two Lagrange multiplier theorems which show that strictly efficient solution of the constrained vector optimization problem is equivalent to strictly efficient solution of an appropriate unconstrained vector optimization problem. In Section 6, some results on strict duality are given. In Section 7, a new concept of strict saddle point for set-valued Lagrangian map is introduced and is then utilized to characterize strict efficiency.

2. Preliminaries

Throughout this paper, let 𝑋 be a linear space and 𝑌 and 𝑍 two real normed locally convex spaces, with topological dual spaces 𝑌^∗ and 𝑍^∗. For a set 𝐴⊂𝑌, cl𝐴, int𝐴, 𝜕𝐴, and 𝐴^𝑐 denote the closure, the interior, the boundary, and the complement of 𝐴, respectively. Moreover, we will denote by 𝐵 the closed unit ball of 𝑌. A set 𝐶⊂𝑌 is said to be a cone if 𝜆𝑐∈𝐶 for any 𝑐∈𝐶 and 𝜆≥0. A cone 𝐶 is said to be convex if 𝐶+𝐶⊂𝐶, and it is said to be pointed if 𝐶∩(−𝐶)={0}. The generated cone of 𝐶 is defined by cone𝐶∶={𝜆𝑐∣𝜆≥0,𝑐∈𝐶}. (1) The dual cone of 𝐶 is defined as 𝐶⁺∶=𝜑∈𝑌^∗∣𝜑(𝑐)≥0,∀𝑐∈𝐶. (2) The quasi-interior of 𝐶⁺ is the set 𝐶^+𝑖∶=𝜑∈𝑌^∗∣𝜑(𝑐)>0,∀𝑐∈𝐶⧵0_𝑌. (3) Recall that a base of a cone 𝐶 is a convex subset Θ of 𝐶 such that 0_𝑌∉clΘ,𝐶=coneΘ. (4) Of course, 𝐶 is pointed whenever 𝐶 has a base. Furthermore, if 𝐶 is a nonempty closed convex pointed cone in 𝑌, then 𝐶^+𝑖≠∅ if and only if 𝐶 has a base.

Definition 1. Let 𝑆 be a nonempty subset of 𝑌. The set of all strictly efficient points and the set of all efficient points with respect to the convex cone 𝐶 with nonempty interior are defined as ()∩(),𝐸()𝑆𝑡𝐸𝑆,𝐶∶=𝑦∣forevery𝜖>0,∃𝛿>0suchthat𝑆−𝑦𝛿𝐵−𝐶⊆𝜖𝐵𝑆,𝐶∶=𝑦∣𝑆−𝑦∩−𝐶=0_𝑌. (5)

It is easy to verify that 𝑆𝑡𝐸(𝑆,𝐶)⊆𝐸(𝑆,𝐶). (6) Also, in this paper, we assume that 𝐶⊂𝑌 and 𝐷⊂𝑍 are pointed closed convex cone with nonempty interior. 𝐹∶𝑋→2^𝑌 and 𝐺∶𝑋→2^𝑍 are set-valued maps with nonempty value. Let 𝐿(𝑍,𝑌) be the space of continuous linear operations from 𝑍 to 𝑌, and let 𝐿₊(𝑍,𝑌)∶={𝑇∈𝐿(𝑍,𝑌)∣𝑇(𝐷)⊂𝐶}. (7)Let (𝐹,𝐺) be the set-valued map from 𝑋 to 𝑌×𝑍, denoted by (𝐹,𝐺)(𝑥)=𝐹(𝑥)×𝐺(𝑥). (8) If 𝜑∈𝑌^∗, 𝑇∈𝐿(𝑍,𝑌), we also define 𝜑𝐹∶𝑋→2^ℝ and 𝐹+𝑇𝐺∶𝑋→2^𝑌 by (𝜑𝐹)(𝑥)=𝜑[𝐹(𝑥)] and (𝐹+𝑇𝐺)(𝑥)=𝐹(𝑥)+𝑇[𝐺(𝑥)], respectively.

Definition 2 (see [12]). A set-valued map 𝐹∶𝑋→2^𝑌 is said to be nearly 𝐶-convexlike on 𝑋 if cl(𝐹(𝑥)+𝐶) is convex in 𝑌.

Lemma 3 (see [12]). Let 𝐹∶𝑋→2^𝑌 be nearly 𝐶-convexlike set-valued map on 𝑋. Then exactly one of the following statement holds:(i)∃𝑥∈𝑋 such that 𝐹(𝑥)∩(−int𝐶)≠∅,(ii)∃𝜑∈𝐶⁺⧵{0_𝑌} such that (𝜑𝐹)(𝑥)⊂ℝ₊.

Lemma 4 (see [12]). If (𝐹,𝐺) is nearly 𝐶×𝐷-convexlike on 𝑋, then(i)for each 𝜑∈𝐶⁺⧵{0_𝑌}, (𝜑𝐹,𝐺) is nearly ℝ₊×𝐷-convexlike on 𝑋;(ii)for each 𝑇∈𝐿₊(𝑍,𝑌), 𝐹+𝑇𝐺 is nearly 𝐶-convexlike on 𝑋.

Lemma 5 (see [21]). Let 𝑆 be a closed convex subset of 𝑌 and 𝐾 a closed convex pointed cone. Then 𝑆∩(−𝐾⧵{0_𝑌})=∅, if and only if there exists 𝜑∈𝐾^+𝑖 such that 𝜑(𝑠)≥0, for all 𝑠∈𝑆.

3. Strict Efficiency and Well Posedness

Consider the following vector optimization problem with set-valued maps: (𝑥)(𝑥)∩()𝐶-min𝐹s.t.𝐺−𝐷≠∅,𝑥∈𝑋. (VP) Denote the feasible solution set of (VP) by 𝐴∶={𝑥∈𝑋∶𝐺(𝑥)∩(−𝐷)≠∅}, (9) And denote the image of 𝐴 under 𝐹 by 𝐹(𝐴)=_𝑥∈𝐴𝐹(𝑥). (10)

Definition 6. A point 𝑥 is said to be a strictly efficient solution of (VP), if there exists 𝑦∈𝐹(𝑥) such that 𝑦∈𝑆𝑡𝐸[𝐹(𝐴),𝐶], and the point (𝑥,𝑦) is said to be a strictly efficient minimizer of (VP).

Definition 7. For a set 𝑆⊆𝑌, let the function Δ_𝑆∶𝑌→ℝ∪{±∞} be defined as Δ_𝑆𝑦=𝑑_𝑆𝑦−𝑑_𝑌⧵𝑆𝑦, (11) where 𝑑_𝑆(𝑦)=inf{‖𝑠−𝑦‖∶𝑠∈𝑆} with 𝑑_∅(𝑦)=+∞.

The function Δ was first introduced in [22], and its main properties are gathered together in the following proposition.

Proposition 8 (see [9]). If the set 𝑆 is nonempty and 𝑆≠𝑌, then(1)Δ_𝑆  is real valued; (2)Δ_𝑆  is 1-Lipschitzian; (3)Δ_𝑆(𝑦)<0  for every 𝑦∈int𝐴, Δ_𝑆(𝑦)=0  for every   𝑦∈𝜕𝐴, and   Δ_𝑆(𝑦)>0  for every   𝑦∈int𝑆^𝑐; (4)if  𝑆 is closed, then it holds that  𝑆={𝑦∶Δ_𝑆(𝑦)≤0};(5)if  𝑆 is a convex, then  Δ_𝑆 is convex;(6)if  𝑆 is a cone, then  Δ_𝑆 is positively homogeneous;(7)if  𝑆 is a closed convex cone, then  Δ_𝑆 is nonincreasing with respect to the ordering relation induced on  𝑌.

We consider the following parameterized scalar problem: minΔ_−𝐶𝑦−𝑦s.t.𝑦∈𝐹(𝐴). P_𝑦

The following theorem characterize the relation between strictly efficient points of (VP) and the parameterized scalar problem P_𝑦.

Theorem 9. Let 𝑥∈𝐴, 𝑦∈𝐹(𝑥). Then (𝑥,𝑦) is a strictly efficient minimizer of (VP) if and only if there exists a nondecreasing function 𝜙∶ℝ₊→ℝ₊ with 𝜙(0)=0 and 𝜙(𝑡)>0 for all 𝑡>0, such that Δ_−𝐶(𝑦−𝑦)≥𝜙(‖𝑦−𝑦‖) for all 𝑦∈𝐹(𝐴).

Proof. Since (𝑥,𝑦) is a strictly efficient minimizer of vector optimization problem (VP) can be rephrased as follows: for every 𝜖>0 there exists 𝛿>0 such that 𝑑_−𝐶(𝑦−𝑦)≥𝛿 for every 𝑦∈𝐹(𝐴) with ‖𝑦−𝑦‖>𝜖. So suppose that the point (𝑥,𝑦) is a strictly efficient minimizer of (VP) and consider the following functions: 𝜙₀(𝜖)=inf𝑑_−𝐶𝑦−𝑦∣𝑦∈𝐹(𝐴),𝑦−𝑦,𝜙(𝜖)𝜙≥𝜖=min₀(𝜖),1. (12) It is evident that 𝜙 is nondecreasing, null at the origin, and positive elsewhere; moreover, for every 𝑦∈𝐹(𝐴) it holds that Δ_−𝐶𝑦−𝑦=𝑑_−𝐶𝑦−𝑦≥𝜙𝑦−𝑦. (13) If, on the other hand, there exists a nondecreasing function 𝜙 with the above properties and such that Δ_−𝐶(𝑦−𝑦)≥𝜙(‖𝑦−𝑦‖) for all 𝑦∈𝐹(𝐴), then it holds that Δ_−𝐶(𝑦−𝑦)=𝑑_−𝐶(𝑦−𝑦)>0 for all 𝑦∈𝐹(𝐴) with 𝑦≠𝑦. To show that (𝑥,𝑦) is a strictly efficient minimizer of (VP), for every 𝜖>0, we can let 𝛿=inf{𝑑_−𝐶(𝑦−𝑦)∶‖𝑦−𝑦‖>𝜖}, and it implies that the proof is completed.

The scalar problem P_𝑦 is Tikhonov well posed if Δ_−𝐶(𝑦−𝑦)>0 for all 𝑦∈𝐹(𝐴) with 𝑦≠𝑦 and 𝑦_𝑛∈𝐹(𝐴),𝑑_−𝐶𝑦_𝑛−𝑦⟶0⟹𝑦_𝑛⟶𝑦. (14)

Theorem 10. Let 𝑥∈𝐴, 𝑦∈𝐹(𝑥). The (𝑥,𝑦) is a strictly efficient minimizer of (VP) if and only if 𝑦 is a solution of P_𝑦 and the scalar problem P_𝑦 is Tikhonov well posed.

Proof. If (𝑥,𝑦) is a strictly efficient minimizer of (VP), then, by Theorem 9, 𝑦 is the unique solution of P_𝑦 and there exists a forcing function 𝜙 such that Δ_−𝐶(𝑦−𝑦)≥𝜙(‖𝑦−𝑦‖), for all 𝑦∈𝐹(𝐴). Since 𝜙(𝑡_𝑛)→0 implies 𝑡_𝑛→0, hence for any sequence 𝑦_𝑛∈𝐹(𝐴) such that Δ_−𝐶(𝑦_𝑛−𝑦)→0, then it must converge to 𝑦.
Conversely, if the scalar problem P_𝑦 is Tikhonov well posed, then 𝑑_−𝐶(𝑦−𝑦)=Δ_−𝐶(𝑦−𝑦) holds for every 𝑦∈𝐹(𝐴). Thus, we consider the function 𝜙(𝜖)=inf{𝑑_−𝐶(𝑦−𝑦)∶‖𝑦−𝑦‖≥𝜖}; it holds by the construction that 𝑑_−𝐶(𝑦−𝑦≥𝜙(𝑦−𝑦), and it is to see that 𝜙 is nondecreasing on [0,∞) with 𝜙(0)=0 and 𝜙(𝑡)>0 for all 𝑡>0. Hence, again, by the Theorem 9, we get that (𝑥,𝑦) is a strictly efficient minimizer of (VP).

4. Strict Efficiency and Linear Scalarization

In association with the vector optimization problem (VP) involving set-valued maps, we consider the following linearly scalar optimization problem with a set-valued map: min𝜑𝐹(𝑥)s.t.𝑥∈𝐴, LSP_𝜑 where 𝜑∈𝑌^∗⧵{0_𝑌^∗}.

Definition 11. If 𝑥∈𝐴, 𝑦∈𝐹(𝐴) and 𝜑𝑦≤𝜑𝑦,∀𝑦∈𝐹(𝐴), (15) then 𝑥 and (𝑥,𝑦) are called a minimal solution and a minimizer of LSP_𝜑, respectively.

Lemma 12. Let 𝑦∈𝑆⊂𝑌, and 𝐶 is a closed convex cone with having a compact base Θ. Then 𝑦 is a strictly efficient point of 𝑆 if and only if cl(𝑆+𝐶−𝑦)∩−𝐶={0_𝑌}.

Proof. The definition of strict efficiency of 𝑆 can be rephrased as follows: for every 𝜖>0, there exists 𝛿>0 such that 𝑑_−𝐶(𝑦−𝑦)>𝛿 for every 𝑦∈𝑆 with ‖𝑦−𝑦‖>𝜖. Hence, if there exists sequence 𝑦_𝑛∈𝑆, 𝑐_𝑛∈𝐶, and some 𝑐∈int𝐶 such that 𝑦_𝑛+𝑐_𝑛−𝑦→−𝑐, then 𝑦_𝑛+𝑐_𝑛+𝑐−𝑦→0; hence, 𝑑_−𝐶(𝑦−𝑦)→0. Since 𝑦_𝑛−𝑦=−(𝑐_𝑛+𝑐), 𝑐≠0, and 𝐶 is pointed, 𝑦_𝑛−𝑦 is outside some small ball around the origin. This shows that 𝑦 is not the strictly efficient point of 𝑆, this contraction shows that cl(𝑆+𝐶−𝑦)∩−𝐶={0_𝑌}.
Conversely, if 𝑦 is not a strictly efficient point of 𝑆, then there exists 𝜖>0 and a sequence 𝑦_𝑛∈𝑆, and 𝑐_𝑛∈𝐶 such that 𝑦_𝑛−𝑦≥𝜖,𝑦_𝑛+𝑐_𝑛−𝑦⟶0. (16) We write 𝑐_𝑛=𝜆_𝑛𝜃_𝑛 with 𝜆_𝑛>0 and 𝜃_𝑛∈Θ, then, by (16) and as Θ is compact, there exists 𝛼>0 and 𝑁∈𝑅⁺ such that 𝛼<𝜆_𝑛. Indeed, by (16), we have 𝑐_𝑛∉(𝜖/2)𝐵; furthermore, since Θ is compact, thus 𝜆_𝑛 does not converse to 0, and it implies that there exists a real number 𝛼>0 and 𝑁∈ℝ⁺ such that 𝛼<𝜆_𝑛 for all 𝑛≥𝑁. Now, we define 𝑐^′_𝑛=(𝜆_𝑛−𝛼)𝜃_𝑛, 𝑛≥𝑁.
Thus, we obtain 𝑦_𝑛+𝑐^′_𝑛−𝑦=𝑦_𝑛+𝑐_𝑛−𝑦−𝛼𝜃_𝑛⟶−𝛼𝜃≠0. (17) Hence, cl𝑆+𝐶−𝑦∩−𝐶⧵0_𝑌≠∅. (18) This contradiction shows that 𝑦 is a strictly efficient point of 𝑆.

Theorem 13. Let 𝑥∈𝐴, 𝑦∈𝐹(𝑥), let 𝐶 have a compact base, and let 𝜑∈𝐶^+𝑖 be fixed. If (𝑥,𝑦) is a minimizer of LSP_𝜑, then (𝑥,𝑦) is a strictly efficient minimizer of (VP).

Proof. By Lemma 12, we need only to prove that cl𝐹(𝐴)+𝐶−𝑦∩−𝐶=0_𝑌. (19) Indeed, let 𝑐∈cl(𝑆+𝐶−𝑦)∩−𝐶. Then there exists {𝑦_𝑛}⊂𝐹(𝐴), {𝑐_𝑛}⊂𝐶 such that 𝑐=lim_𝑛→+∞𝑦_𝑛+𝑐_𝑛−𝑦, (20) hence, 𝜑(𝑐)=lim_𝑛→+∞𝜑𝑦_𝑛+𝜑𝑐_𝑛−𝜑𝑦. (21) Since (𝑥,𝑦) is a minimizer of LSP_𝜑 and 𝑦_𝑛∈𝐹(𝐴), we have 𝜑(𝑦_𝑛)≥𝜑(𝑦), while 𝑐_𝑛∈𝐶 and 𝜑∈𝐶^+𝑖 imply that 𝜑(𝑐_𝑛)≥0. Hence, 𝜑(𝑐)≥0. On the other hand, since 𝑐∈−𝐶, we have 𝜑(𝑐)≤0. Thus 𝜑(𝑐)=0. Again, by 𝜑∈𝐶⁺ and 𝑐∈−𝐶, we must have 𝑐=0_𝑌.
Hence, we have shown that cl(𝑆+𝐶−𝑦)∩−𝐶={0_𝑌}. Therefore, this proof is completed.

Theorem 14. Let 𝐹 be nearly 𝐶-convexlike on 𝐴, 𝑥∈𝐴, and 𝑦∈𝐹(𝑥), and let 𝐶 have a compact base. If (𝑥,𝑦) is a strictly efficient minimizer of (VP), then there exists 𝜑∈𝐶⁺ such that (𝑥,𝑦) is a minimizer of LSP_𝜑.

Proof. Since (𝑥,𝑦) is a strictly efficient minimizer of (VP), thus by Lemma 12, we have −𝐶∩cl𝐹(𝐴)+𝐶−𝑦=0_𝑌. (22)
By the definition of nearly 𝐶-convexlike set-valued map 𝐹, we have that cl[𝐹(𝐴)+𝐶−𝑦] is closed convex set in 𝑌, since 𝐶 is a closed, convex, pointed, compact cone. Thus, by Lemma 5, there exists 𝜑∈𝐶^+𝑖 such that 𝜑cl𝐹(𝐴)+𝐶−𝑦≥0. (23) Since 𝐹(𝐴)−𝑦⊂𝐹(𝐴)+𝐶−𝑦⊂cl𝐹(𝐴)+𝐶−𝑦, (24) we obtain 𝜑𝑦−𝜑𝑦≥0,∀𝑦∈𝐹(𝐴). (25) Therefore, (𝑥,𝑦) is a minimizer of LSP_𝜑.

If we denote by 𝑆𝑡𝐸(VP) the set of strictly efficient minimizer of (VP) and by 𝑀(LSP_𝜑) the set of minimizer of LSP_𝜑, then from Theorems 13 and 14, we get immediately the following corollary.

Corollary 15. Let 𝐹 be nearly 𝐶-convexlike on 𝑋. Then, 𝑆𝑡𝐸(𝑉𝑃)=_{𝜑∈𝐶+𝑖}𝑀𝐿𝑆𝑃_𝜑. (26)

5. Strict Efficiency and Lagrange Multipliers

In this section, we establish two Lagrange multiplier theorems which show that the set of strictly efficient minimizer of the constrained set-valued vector optimization problem (VP), it is equivalent to the set of an appropriate unconstrained vector optimization problem.

The following concept is a generalization of Slater constraint qualification in mathematical programming and in vector optimization.

Definition 16. We say that (VP) satisfies the generalized Slater constraint qualification if there exists 𝑥∈𝑋 such that 𝐺(𝑥)∩(−int𝐷)≠∅.

Theorem 17. Let 𝐹 be nearly 𝐶-convexlike on 𝑋. Let (𝐹,𝐺) be nearly 𝐶×𝐷-convexlike on 𝑋 and let 𝐶 have a compact base. Furthermore, let (VP) satisfy the generalized Slater constraint qualification. If (𝑥,𝑦) is a strictly efficient minimizer of (VP), then there exists 𝑇∈𝐿₊(𝑍,𝑌) such that 𝑇[𝐺(𝑥)∩(−𝐷)]=0_𝑌 and (𝑥,𝑦) is a strictly efficient minimizer of the following unconstrained vector optimization problem: 𝐶-min𝐹(𝑥)+𝑇[𝐺(𝑥)]𝑠.𝑡.𝑥∈𝑋. (UVP)

Proof. Since (𝑥,𝑦) is a strictly efficient minimizer of (VP), by Theorem 14, there exists 𝜑∈𝐶^+𝑖 such that 𝜑𝐹(𝑥)−𝑦≥0,∀𝑥∈𝐴. (27) Define 𝐻∶𝑋→2^ℝ×𝑍 by 𝐻(𝑥)=𝜑𝐹(𝑥)−𝑦×𝐺(𝑥)=𝜑𝐹,𝐺(𝑥)−𝜑𝑦,0_𝑍. (28) Since (𝐹,𝐺) is nearly 𝐶×𝐷-convexlike on 𝑋, by Lemma 4, we have that 𝐻 is nearly ℝ₊×𝐷-convexlike on 𝑋, while (27) implies that 𝐻(𝑥)∩−intℝ₊×𝐷≠∅,∀𝑥∈𝑋, (29) has no solution, and hence, by Lemma 3, there exists (𝜆,𝜓)∈ℝ₊×𝐷⧵{0_𝑌^∗} such that 𝜆𝜑𝐹(𝑥)−𝑦+𝜓[𝐺(𝑥)]≥0,∀𝑥∈𝑋. (30) Since 𝑥∈𝐴, that is, 𝐺(𝑥)∩(−𝐷)≠∅, this implies that there exists 𝑧∈𝐺(𝑥) such that −𝑧∈𝐷. Then, since 𝜓∈𝐷⁺, we get 𝜓(𝑧)≤0. (31) Also, let 𝑥=𝑥 in (30) and noting that 𝑦∈𝐹(𝑥), and 𝑧∈𝐺(𝑥), we get 𝜓(𝑧)≥0.
Hence, 𝜓[𝐺(𝑥)∩−𝐷]=0_𝑌. (32) We now claim that 𝜆≠0. If this is not the case, then 𝜓∈𝐷⁺⧵0_𝑌^∗. (33) By the generalized Slater constraint qualification, there exists 𝑥∈𝑋 such that 𝐺(𝑥)∩−int𝐷≠∅, (34) and so there exists 𝑧∈𝐺(𝑥) such that −𝑧∈int𝐷. (35) Hence, 𝜓(𝑧)<0. But substituting 𝜆=0 into (30), and by taking 𝑥=𝑥 and 𝑧∈𝐺(𝑥) in (30), we have 𝜓(𝑧)≥0. (36) This contradiction shows that 𝜆>0. From this and 𝜑∈𝐶^+𝑖, we can choose 𝑐∈𝐶⧵{0_𝑌} such that 𝜆𝜑(𝑐)=1 and define the operator 𝑇∶𝑍→𝑌 by 𝑇(𝑧)=𝜓(𝑧)𝑐. (37) Obviously, 𝑇∈𝐿₊(𝑍,𝑌),𝑇[𝐺(𝑥)∩−𝐷]=0_𝑌. (38) Thus, 0_𝑌∈𝑇[𝐺(𝑥)],𝑦∈𝐹(𝑥)⊂𝐹(𝑥)+𝑇[𝐺(𝑥)]. (39) From (30) and (37), we obtain [𝐹(𝑥)(𝑥[𝐹(𝑥[𝐺(𝑥(𝑐)[𝐹(𝑥[𝐺(𝑥𝜆𝜑+𝑇𝐺)]=𝜆𝜑)]+𝜓)]𝜆𝜑=𝜆𝜑)]+𝜓)]≥𝜆𝜑𝑦,∀𝑥∈𝑋. (40) Dividing the above inequality by 𝜆>0, we obtain 𝜑[𝐹(𝑥)+𝑇𝐺(𝑥)]≥𝜑𝑦,∀𝑥∈𝑋. (41) Since, (𝐹,𝐺) is nearly 𝐶×𝐷-convexlike on 𝑋, by Lemma 4, 𝐹+𝑇𝐺 is nearly 𝐶-convexlike on 𝑋. Therefore, by Theorem 13 and 𝜑∈𝐶^+𝑖, we have that (𝑥⋅𝑦) is a strictly efficient minimizer of (UVP).

Theorem 18. Let 𝑥, 𝑦∈𝐹(𝑥) and let 𝐶 have a compact base. If there exists 𝑇∈𝐿₊(𝑍,𝑌) such that 0_𝑌∈𝑇[𝐺(𝑥)] and (𝑥,𝑦) is a strictly efficient minimizer of (UVP), then (𝑥,𝑦) is a strictly efficient minimizer of (VP).

Proof. From the assumption, we have 𝑦∈𝐹(𝑥)⊂𝐹(𝑥)+𝑇[𝐺(𝑥)],(−𝐶)∩cl_𝑥∈𝑋(𝐹(𝑥)+𝑇[𝐺(𝑥)])+𝐶−𝑦=0_𝑌. (42)(43) Since 𝑥∈𝐴, we have 𝐺(𝑥)∩(−𝐷)≠∅. Thus there exists 𝑧_𝑥∈𝐺(𝑥)∩(−𝐷). Then, −𝑇𝑧_𝑥∈𝐶, (44) which implies that 𝐶−𝑇(𝑧_𝑥)⊂𝐶; that is, 𝐶⊂𝐶+𝑇(𝑧_𝑥), ∀𝑥∈𝐴. So, we have 𝐹(𝐴)+𝐶−𝑦⊂_𝑥∈𝐴𝐹(𝑥)+𝐶−𝑦⊂_𝑥∈𝐴𝐹(𝑥)+𝑇[𝐺(𝑥)]+𝐶−𝑦⊂_𝑥∈𝐴(𝐹(𝑥)+𝑇[𝐺(𝑥)])+𝐶−𝑦. (45) This together with (43) implies (−𝐶)∩cl𝐹(𝐴)+𝐶−𝑦=0_𝑌. (46) Noting that 𝑥∈𝐴, 𝑦∈𝐹(𝑥)⊂𝐹(𝐴) and by Lemma 12, we have that (𝑥,𝑦) is a strictly efficient minimizer of (VP).

6. Strict Efficiency and Duality

Definition 19. The Lagrange map for (VP) is the set-valued map ℒ∶𝑋×𝐿₊(𝑍,𝑌)→2^𝑌 defined by ℒ(𝑥,𝑇)=𝐹(𝑥)+𝑇[𝐺(𝑥)]. (47)

We denote ℒ(𝑋,𝑇)=_𝑥∈𝑋ℒ(𝑥,𝑇)=_𝑥∈𝑋(𝑥),ℒ𝐹+𝑇𝐺)(𝑥,𝐿₊(𝑍,𝑌)=_{𝑇∈𝐿+(𝑍,𝑌)}ℒ()=𝑥,𝑇_{𝑇∈𝐿+(𝑍,𝑌)}(𝐹(𝑥)+𝑇[𝐺(𝑥)]). (48)

Definition 20. The set-valued map Φ∶𝐿₊(𝑍,𝑌)→2^𝑌 is defined as Φ(𝑇)=𝑆𝑡𝐸[ℒ(𝑋,𝑇),𝐶],∀𝑇∈𝐿₊(𝑍,𝑌), (49) which is called a strict dual map of (VP).

Using this definition, we define the Lagrange dual problem associated with the primal problem (VP) as follows: 𝐶-maxΦ(𝑇)s.t.𝑇∈𝐿₊(𝑍,𝑌). (VD)

Definition 21. 𝑦∈⋃_{𝑇∈𝐿+(𝑍,𝑌)}Φ(𝑇) is called an efficient point of (VD) if 𝑦−𝑦∉𝐶⧵0_𝑌,𝑦∈_{𝑇∈𝐿+(𝑍,𝑌)}Φ(𝑇). (50)

We can now establish the following dual theorems.

Theorem 22 (weak duality). If 𝑥∈𝐴 and 𝑦₀∈⋃_{𝑇∈𝐿+(𝑍,𝑌)}Φ(𝑇), then 𝑦₀−𝐹(𝑥)∩𝐶⧵0_𝑌=∅. (51)

Proof. From 𝑦₀∈⋃_{𝑇∈𝐿+(𝑍,𝑌)}Φ(𝑇), there exists 𝑇∈𝐿₊(𝑍,𝑌) such that 𝑦₀∈Φ𝑇∈𝑆𝑡𝐸_𝑥∈𝑋𝐹(𝑥)+𝑇𝐺(𝑥),𝐶⊂𝐸_𝑥∈𝑋𝐹(𝑥)+𝑇𝐺(𝑥),𝐶. (52) Hence, 𝑦₀−𝐹(𝑥)−𝑇[𝐺(𝑥)]∩𝐶⧵0_𝑌=∅,∀𝑥∈𝑋. (53) In particular, 𝑦₀−𝑦−𝑇(𝑧)∉𝐶⧵0_𝑌,∀𝑦∈𝐹(𝑥),∀𝑧∈𝐺(𝑥). (54) Noting that 𝑥∈𝐴, we choose 𝑧∈𝐺(𝑥)∩(−𝐷). Then, −𝑇(𝑧)∈𝐶, and taking 𝑧=𝑧 in (54), we have 𝑦₀−𝑦−𝑇(𝑧)∉𝐶⧵0_𝑌,∀𝑦∈𝐹(𝑥). (55) Hence, from −𝑇(𝑧)∈𝐶 and 𝐶+𝐶⧵{0_𝑌}⊆𝐶⧵{0_𝑌}, we obtain 𝑦₀−𝑦∉𝐶⧵0_𝑌,∀𝑦∈𝐹(𝑥). (56) This completes the proof.

Theorem 23 (strong duality). Let 𝐹 be nearly C-convexlike on 𝑋. Let (𝐹,𝐺) be nearly (𝐶×𝐷)-convexlike on 𝑋 and let 𝐶 have a compact base. Furthermore, let (VP) satisfy the generalized Slater constraint qualification. If 𝑥 is a strictly efficient solution of (VP), then there exists that 𝑦∈𝐹(𝑥) is an efficient point of (VD).

Proof. Since 𝑥 is a strictly efficient solution of (VP), then there exists 𝑦∈𝐹(𝑥) such that (𝑥,𝑦) is a strictly efficient minimizer of (VP). According to Theorem 17, there exists 𝑇∈𝐿₊(𝑍,𝑌) such that 𝑇[𝐺(𝑥)∩(−𝐷)]={0_𝑌}, and (𝑥,𝑦) is a strictly efficient minimizer of (UVP). Hence, 𝑦∈𝑆𝑡𝐸[⋃_𝑥∈𝑋(𝐹(𝑥)+𝑇[𝐺(𝑥)]),𝐶]=Φ(𝑇)⊂⋃_{𝑇∈𝐿+(𝑍,𝑌)}Φ(𝑇). By Theorem 22, we have 𝑦−𝑦∉𝐶⧵0_𝑌,∀𝑦∈_{𝑇∈𝐿+(𝑍,𝑌)}Φ(𝑇). (57) Therefore, by Definition 20, we know that 𝑦 is an efficient point of (VD).

7. Strict Efficient and Strict Saddle Point

We will now introduce a new concept of strict saddle point for a set-valued Lagrange map ℒ and use it to characterize strict efficiency.

For a nonempty subset 𝑆 of 𝑌, we define a set 𝑆𝑡𝑀(𝑆,𝐶)=𝑦∈𝑆∣∀𝜖>0,∃𝛿>0suchthat𝑆−𝑦∩(−𝛿𝐵+𝐶)⊆𝜖𝐵. (58) It is easy to find that 𝑦∈𝑆𝑡𝑀(𝑆,𝐶) if and only if −𝑦∈𝑆𝑡𝐸(−𝑆,𝐶), and if 𝑌 is normed space and 𝐶 has a compact base. Then by Lemma 12, we have 𝑦∈𝑆𝑡𝑀(𝑆,𝐶) if and only if cl(𝑆−𝐶−𝑦)∩𝐶={0_𝑌}.

Definition 24. A pair (𝑥,𝑇)∈𝑋×𝐿₊(𝑍,𝑌) is said to be a strict saddle point of Lagrange map ℒ if ℒ𝑥,𝑇∩𝑆𝑡𝐸_𝑥∈𝑋ℒ𝑥,𝑇,𝐶∩𝑆𝑡𝑀_{𝑇∈𝐿+(𝑍,𝑌)}ℒ(𝑥,𝑇),𝐶≠∅. (59)

We first present an important equivalent characterization for a strict saddle point of the Lagrange map ℒ.

Lemma 25. Let 𝐶 have a compact base. (𝑥,𝑇)∈𝑋×𝐿₊(𝑍,𝑌) is said to be a strict saddle point of Lagrange map ℒ if only if there exist 𝑦∈𝐹(𝑥) and 𝑧∈𝐺(𝑥) such that(i)𝑦∈𝑆𝑡𝐸[⋃_𝑥∈𝑋ℒ(𝑥,𝑇),𝐶]∩𝑆𝑡𝑀[⋃_{𝑇∈𝐿+(𝑍,𝑌)}ℒ(𝑥,𝑇),𝐶], (ii)𝑇(𝑧)=0_𝑌.

Proof (necessity). Since (𝑥,𝑇) is a strict saddle point of the map ℒ, by Definition 24 there exists 𝑦∈𝐹(𝑥), 𝑧∈𝐺(𝑥) such that 𝑦+𝑇(𝑧)∈𝑆𝑡𝐸_𝑥∈𝑋ℒ𝑥,𝑇,𝐶,𝑦+𝑇(𝑧)∈𝑆𝑡𝑀_{𝑇∈𝐿+(𝑍,𝑌)}ℒ(𝑥,𝑇),𝐶. (60)(61) From (61), we have 𝐶∩cl_{𝑇∈𝐿+(𝑍,𝑌)}ℒ(𝑥,𝑇)−𝐶−𝑦+𝑇(𝑧)=0_𝑌. (62) Since every 𝑇∈𝐿₊(𝑍,𝑌), we have 𝑇(𝑧)−𝑇(𝑧)=𝑦+𝑇(𝑧)−𝑦+𝑇(𝑧()∈𝐹𝑥)+𝑇[𝐺(𝑥)]−𝑦+𝑇(𝑧()=ℒ𝑥,𝑇)−𝑦+𝑇(𝑧). (63) We have 𝑇(𝑧)∶𝑇∈𝐿₊(𝑍,𝑌)−𝐶−𝑇(𝑧)⊂_{𝑇∈𝐿+(𝑍,𝑌)}ℒ(𝑥,𝑇)−𝐶−𝑦+𝑇(𝑧). (64) Hence, cl_{𝑇∈𝐿+(𝑍,𝑌)}{𝑇(𝑧)}−𝐶−𝑇(𝑧)⊂cl_{𝑇∈𝐿+(𝑍,𝑌)}ℒ(𝑥,𝑇)−𝐶−𝑦+𝑇(𝑧). (65) Thus, from (62), we have 𝐶∩cl_{𝑇∈𝐿+(𝑍,𝑌)}{𝑇(𝑧)}−𝐶−𝑇(𝑧)=0_𝑌. (66) Let 𝑓∶𝐿(𝑍,𝑌)→𝑌 be defined by 𝑓(𝑇)=−𝑇(𝑧). (67) Then, (66) can be written as (−𝐶)∩cl𝑓𝐿₊(𝑍,𝑌)+𝐶−𝑓𝑇=0_𝑌. (68) This together with Lemma 12 shows that 𝑇∈𝐿₊(𝑍,𝑌) is a strictly efficient point of the vector optimization problem 𝐶-min𝑓(𝑇)s.t.𝑇∈𝐿₊(𝑍,𝑌). (69)
Since 𝑓 is a linear map, then of course −𝑓 is nearly 𝐶-convexlike on 𝐿₊(𝑍,𝑌). Hence, by Theorem 14, there exists 𝜑∈𝐶^+𝑖 such that 𝜑−𝑇(𝑧)=𝜑𝑓𝑇𝑓(𝑇[(≤𝜑)=𝜑−𝑇𝑧)],∀𝑇∈𝐿₊(𝑍,𝑌). (70) Now, we claim that −𝑧∈𝐷. (71) If this is not true, then since 𝐷 is a closed convex cone set, by the strong separation theorem in topological vector space [21], there exists 𝜇∈𝑍^∗⧵{0_𝑍^∗} such that 𝜇(−𝑧)<𝜇(𝜆𝑑),∀𝑑∈𝐷,∀𝜆>0. (72) In the above expression, taking 𝑑=0_𝑧∈𝐷 gets 𝜇(𝑧)>0, (73) while letting 𝜆→+∞ leads to 𝜇(𝑑)≥0,∀𝑑∈𝐷. (74) Hence, 𝜇∈𝐷⁺⧵0_𝑍^∗. (75) Let 𝑐^∗∈int𝐶 be fixed, and define 𝑇^∗∶𝑍→𝑌 as 𝑇^∗(𝑧)=𝜇(𝑧)𝜇(𝑧)𝑐^∗+𝑇(𝑧). (76) It is evident that 𝑇^∗∈𝐿(𝑍,𝑌) and that 𝑇^∗(𝑑)=𝜇(𝑑)𝜇(𝑧𝑐^∗)+𝑇(𝑑)∈𝐶+𝐶⊂𝐶,∀𝑑∈𝐷. (77) Hence, 𝑇^∗∈𝐿₊(𝑍,𝑌). Taking 𝑧=𝑧 in (66), we obtain 𝑇^∗(𝑧)−𝑇(𝑧)=𝑐^∗. (78) Hence, 𝜑𝑇^∗(𝑧)−𝜑𝑇(𝑧)=𝜑𝑐^∗>0, (79) which contradicts (70). Therefore, −𝑧∈𝐷. (80) Thus, −𝑇(𝑧)∈𝐶, since 𝑇∈𝐿₊(𝑍,𝑌). If 𝑇(𝑧)≠0_𝑌, then −𝑇(𝑧)∈𝐶⧵0_𝑌, (81) hence, 𝜑[𝑇(𝑧)]<0, by 𝜑∈𝐶^+𝑖, while taking 𝑇=0∈𝐿₊(𝑍,𝑌) leads to 𝜑𝑇(𝑧)≥0. (82) This contradiction shows that 𝑇(𝑧)=0_𝑌, that is, condition (ii) holds.
Therefore, by (60) and (61), we know 𝑦∈𝑆𝑡𝐸_𝑥∈𝑋ℒ𝑥,𝑇,𝐶∩𝑆𝑡𝑀_{𝑇∈𝐿+(𝑍,𝑌)}ℒ(𝑥,𝑇),𝐶; (83) that is, condition (i) holds.
Sufficiency. From 𝑦∈𝐹(𝑥), 𝑧∈𝐺(𝑥), and condition (ii), we get 𝑦=𝑦+𝑇(𝑧)∈𝐹(𝑥)+𝑇[𝐺(𝑥)]=ℒ𝑥,𝑇. (84) And by condition (i), we obtain 𝑦∈ℒ𝑥,𝑇∩𝑆𝑡𝐸_𝑥∈𝑋ℒ𝑥,𝑇,𝐶∩𝑆𝑡𝑀_{𝑇∈𝐿+(𝑍,𝑌)}ℒ(𝑥,𝑇),𝐶. (85) Therefore, (𝑥,𝑇) is a strict saddle point of the set-valued Lagrange map ℒ, and the proof is complete.

The following saddle-point theorems allow us to express a strictly efficient solution of (VP) as a strict saddle of the set-valued Lagrange map ℒ.

Theorem 26. Let 𝐹 be nearly 𝐶-convexlike on 𝐴, let (𝐹,𝐺) be nearly (𝐶×𝐷)-convexlike on 𝐴, and let 𝐶 have a compact base. Moreover, (VP) satisfies generalized Slater constrained qualification.(i)If (𝑥,𝑇) is a strict saddle point of the map ℒ, then 𝑥 is strictly efficient solution of (VP).(ii)If (𝑥,𝑦) is a strictly efficient minimizer of (VP), and 𝑦∈𝑆𝑡𝑀[⋃_{𝑇∈𝐿+(𝑍,𝑌)}ℒ(𝑥,𝑇),𝐶], then there exists 𝑇∈𝐿₊(𝑍,𝑌) such that (𝑥,𝑇) is a strict saddle point of the map ℒ.

Proof. (i) By the necessity of Lemma 25, we have 0_𝑌∈𝑇[𝐺(𝑥)], (86) and there exists 𝑦∈𝐹(𝑥) such that (𝑥,𝑦) is a strictly efficient minimizer of the problem 𝐶-min𝐹(𝑥)+𝑇[𝐺(𝑥)]s.t.𝑥∈𝑋. (UVP)
According to Theorem 18, (𝑥,𝑦) is a strictly efficient minimizer of (VP). Therefore, 𝑥 is strictly efficient solution of (VP).
(ii) From the assumption, and by the Theorem 17, there exists 𝑇∈𝐿₊(𝑍,𝑌) such that 𝑦∈𝑆𝑡𝐸_𝑥∈𝑋ℒ𝑥,𝑇,𝑇[𝐺(,𝐶𝑥)∩(−𝐷)]=0_𝑌. (87) Therefore, there exists 𝑧∈𝐺(𝑥) such that 𝑇(𝑧)=0_𝑌. Hence, from Lemma 25, it follows that (𝑥,𝑇) is a strict saddle point of the map ℒ.

Lemma 27. Let (𝑥,𝑇)∈𝑋×𝐿₊(𝑍,𝑌), 𝑦∈𝐹(𝑥), and 𝑧∈𝐺(𝑥). Then the following conditions:(a)𝑦∈𝑆𝑡𝐸[⋃_𝑥∈𝑋ℒ(𝑥,𝑇),𝐶]∩𝑆𝑡𝑀[⋃_{𝑇∈𝐿+(𝑍,𝑌)}ℒ(𝑥,𝑇),𝐶],(b)𝑇(𝑧)=0_𝑌,are equivalent to the following conditions:(i)𝑦∈𝑆𝑡𝐸[⋃_𝑥∈𝑋ℒ(𝑥,𝑇),𝐶]∩𝑆𝑡𝑀[𝐹(𝑥),𝐶], (ii)𝐺(𝑥)⊂−𝐷,(iii)𝑇(𝑧)=0_𝑌.

Proof. By conditions (a) and (b), it is easy to verify that conditions (i) and (iii) hold. Now, we show that 𝐺(𝑥)⊂−𝐷. If this is not true, then there would exist 𝑧₀∈𝐺(𝑥) such that −𝑧₀∉𝐷. (88) Then, since 𝐷 is a closed convex set, by the strong separation theorem in topological vector space (see [21]), there exists 𝜓₀∈𝑍^∗⧵{0_𝑍^∗} such that 𝜓₀−𝑧₀<𝜓(𝜆𝑑),∀𝑑∈𝐷,∀𝜆>0. (89) In the above expression, taking 𝑑=0_𝑧∈𝐷 gives 𝜓₀𝑧₀>0, (90) and taking 𝜆→+∞ leads to 𝜓₀(𝑑)≥0,∀𝑑∈𝐷. (91) Hence, 𝜓₀∈𝐷⁺⧵0_𝑧∗. (92) Take 𝑐₀∈int𝐶 and define 𝑇₀∶𝑍→𝑌 as 𝑇₀(𝑧)=𝜓₀(𝑧)𝑐₀. (93) Then, 𝑇₀∈Ł₊(𝑍,𝑌) and 𝑇₀𝑧₀=𝜓₀𝑧₀𝑐₀∈int𝐶⊂𝐶⧵0_𝑌. (94) And, by condition (a), we have 𝑦−𝑦∉𝐶⧵0_𝑌,∀𝑦∈_{𝑇∈𝐿+(𝑍,𝑌)}ℒ(𝑥,𝑇). (95) From 𝑦+𝑇₀𝑧₀∈𝐹(𝑥)+𝑇₀[𝐺(𝑥)]=𝐿𝑥,𝑇₀⊂_{𝑇∈𝐿+(𝑍,𝑌)}ℒ(𝑥,𝑇), (96) and by (95), we have 𝑇₀𝑧₀=𝑦+𝑇₀𝑧₀−𝑦∉𝐶⧵0_𝑌. (97) This conflicts with (99). Therefore, 𝐺(𝑥)⊂−𝐷.
Conversely, by (i), we have 𝑦∈𝑆𝑡𝑀[𝐹(𝑥),𝐶], (98) that is, cl𝐹(𝑥)−𝐶−𝑦∩𝐶=0_𝑌. (99) By condition (ii), for every 𝑇∈𝐿₊(𝑍,𝑌), we have 𝑇[𝐺(𝑥)]⊂𝑇(−𝐷)⊂−𝐶. (100) Thus, _𝑥∈𝑋ℒ(𝑥,𝑇)−𝐶−𝑦=_𝑥∈𝑋[𝐹(𝑥)+𝑇[𝐺(𝑥)]]−𝐶−𝑦=_𝑥∈𝑋𝑇[𝐺(𝑥)]+𝐹(𝑥)−𝐶−𝑦⊂𝐹(𝑥)−𝐶−𝐶−𝑦⊂𝐹(𝑥)−𝐶−𝑦. (101) Therefore, by (99), we have cl_𝑥∈𝑋ℒ(𝑥,𝑇)−𝐶−𝑦∩𝐶=0_𝑌, (102) that is, 𝑦∈𝑆𝑡𝑀[⋃_𝑥∈𝑋ℒ(𝑥,𝑇)]. Therefore, this proof is completed.

By Lemmas 25 and 27 and Theorem 26, we can obtain immediately the following corollary.

Corollary 28. Let 𝐹 be nearly 𝐶-convexlike on 𝐴, let (𝐹,𝐺) be nearly (𝐶×𝐷)-convexlike on 𝐴, and let (VP) satisfy generalized Slater constrained qualification.(i)If (𝑥,𝑇) is a strict saddle point of the map ℒ, then 𝑥 is strictly efficient solution of (VP). (ii)If (𝑥,𝑦) is a strictly efficient minimizer of (VP) and 𝐺(𝑥)⊂−𝐷, 𝑦∈𝑆𝑡𝑀[𝐹(𝑥),𝐶], then there exists 𝑇∈𝐿₊(𝑍,𝑌) such that (𝑥,𝑇) is a strict saddle point of the map ℒ.

Acknowledgments

Zhimiao Fang’s research was supported by the Natural Science Foundation Project of CQ CSTC (Grant no. cstc2012jjA00033). The authors would like to thank two anonymous referees for their valuable comments and suggestions, which helped to improve the paper.