Optimality conditions for various efficientsolutions involving coderivatives: from

set-valued optimization problems toset-valued equilibrium problems

Truong Xuan Duc Ha∗

August 2, 2010

Abstract

In this paper, we present a new approach to the study of various efficientsolutions of a set-valued equilibrium problem (for short, SEP) through thestudy of corresponding solutions of a set-valued optimization problem witha geometric constraint (for short, SOP). The solutions under considerationare: efficient solutions, weakly efficient solutions, strongly efficient solutions,and properly efficient solutions such as positive properly efficient solutions,Henig global properly efficient solutions, Henig properly efficient solutions,super efficient solutions, Benson properly efficient solutions. Firstly, we obtainin a unified scheme scalar characterization with the help of Hiriart-Urruty’ssigned distance function, necessary/sufficient conditions for weakly, strongly,properly efficient solutions of SOP involving coderivatives and normal cones inthe senses of Clarke, Ioffe and Mordukhovich, and recall Bao-Mordukhovich’snecessary conditions for efficient solutions of SOP involving the coderivativeand normal cone in the sense of Mordukhovich. Secondly, we show that thereis an equivalence between solutions of SOP and solutions of SEP and basedon this equivalence we derive similar results for solutions of SEP as well as forsolutions of a vector equilibrium problem and a vector variational inequality.

Key Words: Set-valued optimization problem, set-valued equilibrium problem,vector equilibrium problem, vector variational inequality, normal cone, coderivative,optimality conditions, efficient solutions (proper, weakly, strongly), scalarization.

Mathematics subject classifications (MSC 2010): 49J53, 49K27, 90C29, 90C33

1 Introduction

Inspired by the pioneering work [1] by Giannessi on vector variational inequalities(for short, VVI), a scalar equilibrium problem has been extended to a vector equi-librium problem (for short, VEP) in [2] and to a set-valued equilibrium problem (for

∗Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Viet Nam, email:txdha@math.ac.vn

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short, SEP) in [3]. SEP and its special cases, VEP and VVI, have applications indifferent areas of optimization, optimal control, operations research and economics.Numerous works have been devoted to the existence, scalarization, optimality con-ditions or connectedness of various efficient solutions of these problems; see, forinstance, [4-24]. Note that most results are concerned with VEP and VVI and,to our knowledge, there are few papers which deal with optimality conditions forefficient solutions of SEP.

Since SEP is closely related to a set-valued optimization problem, it is naturalto borrow techniques and results on optimality conditions for the latter to try tocome up with similar results for the former. Inspired by this idea, we present anew approach to the study of various efficient solutions of SEP through the study ofcorresponding solutions of a set-valued optimization problem with a geometric con-straint (for short, SOP). The solutions under consideration are: efficient solutions,weakly efficient solutions, strongly efficient solutions, and properly efficient solutionssuch as positive properly efficient solutions, Henig global properly efficient solutions,Henig properly efficient solutions, super efficient solutions and Benson properly ef-ficient solutions. Firstly, we obtain in a unified scheme scalar characterization withthe help of Hiriart-Urruty’s signed distance function, necessary/sufficient conditionsfor weakly, strongly, properly efficient solutions of SOP involving coderivatives andnormal cones in the senses of Clarke, Ioffe and Mordukhovich, and recall Bao-Mordukhovich’s necessary conditions [25] for efficient solutions of SOP involvingthe coderivative and normal cone in the sense of Mordukhovich. Secondly, we showthat there is an equivalence between solutions of SOP and solutions of SEP andbased on this equivalence we derive similar results for solutions of SEP, VEP andVVI. Our arguments involve the unified approach of [26] and some techniques of[27].

The paper is organized as follows. In Section 2, we recall various concepts ofnormal cone, subdifferential and coderivative and establish a property of a set-valued pseudo-Lipschitz map. In Section 3, we recall the unified approach to thestudy of weakly, strongly, properly efficient points of a set and scalarization of thesepoints presented in [26]. Scalarization and optimality conditions for various efficientsolutions of SOP are formulated in Section 4 and similar results for solutions of SEP,VEP and VVI are formulated in Section 5.

2 Some tools from Variational Analysis

For the convenience of the reader we repeat the relevant material from [27-42] with-out proofs, thus making our exposition self-contained. We first recall concepts ofnormal cone, subdifferential and coderivative in the senses of Ioffe, Clarke and Mor-dukhovich and then discuss some properties of a set-valued pseudo-Lipschitz map.

Throughout the paper, X and Y are Banach spaces with their duals X∗ andY ∗, respectively. The closed unit ball and the open unit ball in any space, say X,are denoted by BX and BX ; we omit the subscript X when no confusion occurs.We will use the same notation ‖.‖ for the norms in X, Y , X∗ and Y ∗. For a

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nonempty set A ⊂ X, intA and clA stand for the interior and closure of A andconeA := ta : t ∈ R+, a ∈ A, where R+ = [0,∞[. Further, d(x;A) is the distancefrom x to A and χA(x) is the indicator function associated to A, i.e., χA(x) = 0if x ∈ A and χA(x) = ∞ otherwise. Recall that a Banach space is Asplund ifeach of its separable subspace has a separable dual. This class of spaces has beencomprehensively investigated in geometric theory of Banach spaces and has beenlargely employed in variational analysis; see, e.g. [38, 39]. Examples of Asplundspaces are the Banach spaces Rn, Lp[0,1] and lp (1 < p <∞).

2.1 Normal cone, subdifferential and coderivative in thesenses of Ioffe, Clarke and Mordukhovich

Assume that g : X → R ∪ ∞ is a function and F : X → 2Y is a set-valuedmap (for the sake of convenience we assume that F (x) is nonempty for all x ∈ X).The domain, epigraph of g and the graph of F are the sets domg = x ∈ X |g is finite at x, epig = (x, t) ∈ X × R | g(x) ≤ t and grF = (x, y) ∈ X × Y |y ∈ F (x), respectively.

To define the concept of subdifferential of g, suppose first that g is locally Lip-schitz near x ∈domg. The Ioffe approximate subdifferential of g at x [30-32] is theset

∂Ag(x) =⋂L∈F

lim sup(ε,y)→(0+,x)

∂−ε gy+L(y),

where F is the collection of all finite dimensional subspaces of X, gy+L(u) = g(u) ifu ∈ y + L and gy+L(u) = +∞ otherwise, for ε ≥ 0

∂−ε gy+L(y) = x∗ ∈ X∗ | x∗(v) ≤ ε‖v‖++ lim inft→0+ t−1[gy+L(y + tv)− gy+L(y)],∀v ∈ X.

The Clarke generalized subdifferential of g at x [29] is the set

∂Cg(x) = x∗ ∈ X∗ | x∗(v) ≤ g0(x; v), ∀v ∈ X,

where g0(x; v) is the generalized directional derivative of g at x in the direction v

g0(x; v) = lim supy→x, t→0+

g(y + tv)− g(y)

t.

Let Ω be a nonempty subset of X different from X and x ∈ clΩ. The Ioffeapproximate normal cone to Ω at x [30-32] is given by

NA(x; Ω) =⋃λ>0

λ∂Ad(x; Ω)

and the Clarke normal cone to Ω at x [29] is given by

NC(x; Ω) = cl

(⋃λ>0

λ∂Cd(x; Ω)

).

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Assuming that X is an Asplund space, the Mordukhovich normal cone to Ω at x[34-39] is defined by

NM(x; Ω) = lim supx′

Ω→x

N(x′; Ω),

where the limit in the right-hand side means the sequential Kuratowski-Painleveupper limit with respect to the norm topology in X and the weak-star ω∗ topology

in X∗, x′Ω→ x refers to all sequences converging to x which remain in Ω and N(x; Ω)

is the Frechet normal cone to Ω at x given by

N(x; Ω) =

x∗ ∈ X∗

∣∣∣∣ lim supx′

Ω→x

x∗(x′ − x)

‖x′ − x‖≤ 0

.

(Note that the Mordukhovich normal cone and related to it concepts are defined inany Banach space but here we restrict ourselves to the Asplund space setting, wherethey enjoy full calculus).

Now assume that the function g is lower semicontinuous and the set-valuedmap F is closed (i.e., its graph is a closed set). The subdifferentials of g andcoderivatives of F in the senses of Ioffe, Clarke or Mordukhovich are defined throughthe corresponding normal cones as follows (for the sake of convenience, we make theconvention that when no confusion occurs the same notations N , ∂g and D∗F areused for the normal cone, subdifferentials and coderivatives in the above senses orin the sense of convex analysis and that the spaces under consideration are Asplundwhenever the subdifferential, the normal cone and the coderivative are understoodin the sense of Mordukhovich)

∂g(x) = x∗ ∈ X∗ | (x∗,−1) ∈ N((x, g(x)); epig)

andD∗F (x, y)(y∗) = x∗ ∈ X∗ | (x∗,−y∗) ∈ N((x, y); grF ).

Along with the Mordukhovich coderivative, we also consider (in the Asplundspace setting) the mixed coderivative [38] given by

D∗F (x, y)(y∗) := x∗ ∈ X∗| ∃ sequences (xk, yk)grF→ (x, y), x∗k

w∗→ x∗, ‖y∗k‖ → y∗

with (x∗k,−y∗k) ∈ N((xk, yk); grF ), k ∈ N.

When F is single-valued, we write D∗F (x) and D∗F (x) instead of D∗F (x, y) andD∗F (x, y).

Next, we recall some results that will be used in the sequel, see [29-38, 40].Denote by L(X, Y ) the space of continuous linear maps from X to Y . Let be givena map h : X → Y . Recall that h is said to admit a strict derivative at x, an elementof L(X, Y ) denoted h′(x), provided that for each v, the following holds:

limx′→x, t↓0

h(x′ + tv)− h(x′)

t= h′(x)(v),

and provided the convergence is uniform for v in compact sets (this condition auto-matically holds if h is Lipschitz near x).

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Proposition 2.1. Assume that g : X → R ∪ ∞ is lower semicontinuous and Ωis a nonempty closed subset of X.

(i) If the function g is strictly differentiable near x then ∂g(x) = g′(x).

(ii) If g is convex and Lipschitz near x, the above subdifferentials reduce to thesubdifferential of convex analysis, i.e.,

∂g(x) = x∗ ∈ X∗ | x∗(x′ − x) ≤ g(x′)− g(x), ∀ x′ ∈ domg.

(iii) If g(x′) ≥ g(x) for all x′ in a neighborhood of x ∈ domg, then 0 ∈ ∂g(x).

(iv) (sum rule) Assume that h : X → R ∪ +∞ is Lipschitz near x ∈ domg∩domh, then ∂(g + h)(x) ⊂ ∂g(x) + ∂h(x) and the equality holds if at least onefunction is strictly differentiable near x.

(v) ∂χΩ(x) = N(x; Ω) and if Ω is convex then the above normal cones reduce tothe normal cone of convex analysis, i.e. to the set

x∗ ∈ X∗ | x∗(x′ − x) ≤ 0, ∀ x′ ∈ Ω.

(vi) For the norm ‖.‖ in X one has ∂‖.‖(0) = BX∗.

Proposition 2.2. Assume that a map g : X → Y is strictly differentiable near xthen D∗g(x) = D∗g(x) = [g′(x)]∗. Here, [g′(x)]∗ : Y ∗ → X∗ is the adjoint map tog′(x) defined by

[g′(x)]∗(y∗)(x) = y∗(g′(x)(x)), ∀(x, y∗) ∈ X × Y ∗.

Remark 2.1.1. Note that in [35] Mordukhovich introduced the notion of coderivatives of a

set-valued map regardless of the normal cone used. After he suggested this approachto differentiability of maps, we may consider different specific coderivatives for set-valued maps generated by different normal cones to their graphs. The Mordukhovichcoderivative related to a normal cone in a finite dimensional space was introduced in[35] and was extended to Banach spaces in [34]. The Mordukhovich coderivative hasbeen further developed to its full and comprehensive calculus and has been used inthe study of optimal control, differential inclusions, scalar and vector optimization,economics..., see [38, 39]. It turns out that coderivative for single- or set-valued mapsis the right tool for formulation of optimality conditions, see [25-27, 39, 43-48].

2. The approximate normal cone and the approximate subdifferential for a lowersemicontinuous function presented above is termed as the G-nucleus of the G-normalcone and the G-nucleus of the G-subdifferential in [31]. For the calculus of the ap-proximate coderivative see [33]. The approximate normal cone is contained in theClarke normal cone. The Mordukhovich normal cone is smaller than the approxi-mate normal cone and they coincide in finite dimensional spaces.

3. We mention that Clarke never introduced nor used any coderivative conceptfor either set-valued or single-valued maps, but the coderivative generated by theClarke normal cone in the scheme of [36] as above has been used under the name“Clarke’s coderivative” in [37].

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2.2 Properties of a set-valued pseudo-Lipschitz map

Let Ω ⊂ X be a nonempty set, F be a set-valued map from Ω to Y , x ∈ Ω and(x, y) ∈grF . Recall that F is pseudo-Lipschitz [28] around (x, y) with some modulusγ > 0 if there are neighborhoods U of x and V of y such that

F (x) ∩ V ⊂ F (x′) + γ‖x− x′‖BY , ∀x, x′ ∈ U. (1)

Mordukhovich proposed the term “Lipschitz-like” or “Aubin property” for thisproperty, see [38]. Note that pseudo-Lipschitz property is fundamental in nonlinearand variational analysis; it is in fact equivalent to the two other underlying propertiesfor the inverse map F−1 known as linear openness/covering and metric regularityaround (x, y). Thibault established [42] that if F is pseudo-Lipschitz, then

d(y;F (x)) ≤ (γ + 1)d((x, y); grG) (2)

for (x, y) near (x, y). Rockafellar proved [41] that F is pseudo-Lipschitz around(x, y) with the modulus γ iff there exists r > 0 such that for all x, x′ ∈ x+ rBX andy, y′ ∈ y + rBY

|d(y;F (x))− d(y′;F (x′))| ≤ γ‖x− x′‖+ ‖y − y′‖. (3)

Below we establish a property of a set-valued pseudo-Lipschitz map which playsan important role in our proof of necessary conditions.

Proposition 2.3. Suppose that

(i) The set Ω is closed.

(ii) F is closed and is pseudo-Lipschitz around (x, y) with the modulus γ.

Then for (x, y) near (x, y) we have

d((x, y); Λ) ≤ (γ + 1)[d((x, y); grF ) + d(x; Ω)],

where Λ := (x, y) ∈ X × Y |x ∈ Ω, y ∈ F (x).

Proof. The proof is motivated by that of Theorem 2.5 in [27]. Fix r > 0 given by(1)-(3), x ∈ x+ (r/3)BX and y ∈ y+ (r/3)BY . Fix an arbitrary a ∈ (x+ rBX)∩Ω.It is clear that (a, b) ∈ Λ for any b ∈ F (a). Therefore,

d((x, y); Λ) ≤ ‖x− a‖+ ‖y − b‖

and since b ∈ F (a) is arbitrary,

d((x, y); Λ) ≤ ‖x− a‖+ d(y;F (a)).

Putting x′ = a and y = y′ in (3), we get d(y;F (a)) ≤ d(y;F (x)) + γ‖x − a‖ andhence,

d((x, y); Λ) ≤ (γ + 1)‖x− a‖+ d(y;F (x)),

which together with (2) yield

d((x, y); Λ) ≤ (γ + 1)[‖x− a‖+ d((x, y); grF )]. (4)

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Observe that for u ∈ Ω and u /∈ x+ rBX we have

‖x− u‖ = ‖x− x+ x− u‖ ≥ ‖u− x‖ − ‖x− x‖ > r − r/3 = 2r/3

and since d(x; Ω) ≤ ‖x− x‖ ≤ r/3, we obtain

d(x; Ω) = inf‖x− u‖ | u ∈ (x+ rBX) ∩ Ω. (5)

Recalling that (4) holds for all a ∈ (x+ rBX) ∩ Ω, we deduce from (5) that

d((x, y); Λ) ≤ (γ + 1)[d(x; Ω) + d((x, y); grF )].

For the study of vector optimization problems, Mordukhovich and Bao exploitedcertain normal compactness properties of sets and maps, which are automatic infinite dimensions while are unavoidably needed in infinite-dimensional spaces dueto the natural lack of compactness therein. In what follows these properties areused in the framework of Asplund spaces, and so the given definitions are specifiedto this setting, see [38]. Recall that a set Ω ⊂ X is sequentially normally compact

(SNC) at x ∈ Ω if for any sequences xkΩ→ x and x∗k

w∗→ 0 with x∗k ∈ N(xk; Ω), k ∈ N,we have ‖x∗k‖ → 0 as k →∞. A set-valued map F between Asplund spaces X andY is SNC at (x, y) ∈grF if its graph is SNC at this point. Further, we say that Fis partially SNC (PSNC) at (x, y) if, sequentially,

[(xk, yk)grF→ (x, y), x∗k

w∗→ 0, ‖y∗k‖ → 0, (x∗k, y∗k) ∈ N((xk, yk); grF )] =⇒ [‖x∗k‖ → 0]

as k → ∞. The PSNC property is automatically implied by robust Lipschitzianbehavior of set-valued and single-valued maps; in particular, when F is pseudo-Lipschitz around (x, y).

3 Efficient points of a set

This section is devoted to various concepts of efficient points of a set. We recallsome results from [26], namely, the unified approach to studying these points andtheir scalarization by Hiriart-Urruty’s signed distance function introduced in[49].

3.1 Definitions

Throughout the paper, let K ⊂ Y be a nonempty closed pointed convex cone withapex at zero (pointedness means K ∩ (−K) = 0). A convex set Θ ⊂ Y is calleda base for K if 0 /∈ clΘ and K = tθ : t ∈ R+, θ ∈ Θ. When Θ is bounded, we saythat K has a bounded base. Denote

K+ = y∗ ∈ Y ∗ | y∗(k) ≥ 0, ∀k ∈ K

andK+i = y∗ ∈ Y ∗ | y∗(k) > 0, ∀k ∈ K \ 0.

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It is known that K has a base iff K+i 6= ∅ and K has a bounded base iff intK+ 6= ∅[50]. Nonnegative orthants in the Banach spaces Rn, C[0,1], L

p[0,1] and lp (1 ≤ p <∞)

have bases and the nonnegative orthants in L1[0,1], l

1 have bounded bases [50].

Throughout this section, let A be a nonempty subset of Y and a ∈ A. In thispaper we consider the following concepts of efficiency and proper efficiency.

Definition 3.1. We say that

(i) a is an efficient (or Pareto minimal) point of A if

(A− a) ∩ (−K \ 0) = ∅.

(ii) supposing that int K 6= ∅, a is a weakly efficient point of A if

(A− a) ∩ (−intK) = ∅.

(iii) a is a strongly (or ideal) efficient point of A if

A− a ⊂ K.

(iv) supposing that K+i 6= ∅, a is a positive properly efficient point of A if thereexists ϕ ∈ K+i such that

ϕ(a) ≥ ϕ(a), ∀ a ∈ A.

(v) a is a Henig global properly efficient point of A if there exists a convex cone Cwith apex at zero with K \ 0 ⊂ intC such that

(A− a) ∩ (−intC) = ∅.

(vi) supposing that K has a base Θ, a is a Henig properly efficient point of A ifthere is a scalar ε > 0 such that

clcone(A− a) ∩ (−clcone(Θ + εB)) = 0.

(vii) a is a super efficient point of A if there is a scalar ρ > 0 such that

clcone(A− a) ∩ (B −K) ⊂ ρB.

(viii) a is a Benson properly efficient point of A if

clcone[(A− a) +K] ∩ (−K) = 0.

The sets of efficient points defined in Definition 3.1 are denoted by Min(A),WMin(A), StrMin(A), Pos(A), GHe(A), He(A), SE(A) and Be(A), respectively.

We refer to [51-53] for the concepts of efficiency, weak efficiency and strongefficiency. Note that positive proper efficiency has been introduced by Hurwicz in[54], Benson proper efficiency has been presented in [55], Henig proper efficiency

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and Henig global proper efficiency have been presented in [56] and super efficiencyhas been introduced by Borwein and Zhuang in [57]. The above definition of Henigproperly efficient points can be found in [57, 58] , see also [59, 60] and the notationHe(A,K) is taken from [59, 60] while a slightly notation He(A,Θ) emphasizing thebase Θ is used in other references. Let us recall an equivalent definition of a Henigproperly efficient point. Let Θ be as before a base of K. Setting

δ := inf‖θ‖ : θ ∈ Θ > 0,

for each 0 < η < δ, we can associate to K with another convex, pointed and opencone Vη, defined by

Vη = cone(Θ + ηBY ) \ 0.It is known that a is a Henig properly efficient point of A w.r.t. to Θ iff there is ascalar η ∈ [0, δ[ such that

(A− a) ∩ (−Vη) = ∅.We refer the reader to [61] for another equivalent definition of a Henig properly

efficient point by means of a functional from K+i and to [62] for a survey andmaterials on proper efficiency.

In the sequel, when speaking of weakly efficient points (resp. positive properlyefficient points) we mean that intK (resp. K+i) is nonempty, when speaking of Henigproperly efficient points or of K+i we mean that K has a base Θ, when speakingthat K has a bounded base we mean that Θ is bounded.

We recall known relations among the above efficient points in the propositionbelow, see [51-53, 62, 63].

Proposition 3.2. (i) StrMin(A) ⊂Min(A) ⊂ WMin(A).

(ii) Pos(A) ⊂ GHe(A).

(iii) SE(A) ⊂ He(A) ⊂ GHe(A) ⊂ Be(A) ⊂Min(A).

(iv) If K has a bounded base then SE(A) = He(A).

Proposition 3.3. (Corollaries 4.2 and 4.5 in [63]) Suppose that

(i) Y is a separable Banach space, or Y is a reflexive Banach space and K has abase.

(ii) A− a is nearly K-subconvexlike.

If a is a Benson proper efficien point of A then it is a positive properly efficientpoint of A.

Recall [64] that a nonempty set A ⊂ Y is nearly K-subconvexlike if the setclcone(A+K) is convex.

From now on, for the sake of convenience we make a convention that by “prop-erly efficient” (point/solution) we mean positive properly efficient, Henig properlyefficient, Henig global properly efficient, super efficient and Benson properly efficient(point/solution).

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3.2 A unified approach to weakly, strongly, properly effi-cient points

A brief inquiry into the matter reveals that weakly, strongly, properly efficient pointsin Definition 3.1 can be described through disjointness between some set and somenonempty open (not necessarily convex) cone Q. Inspired by this fact, we presentedin [26] the notion of Q-minimal point of a set, which contains as special cases weakly,strongly and properly efficient points. Therefore, the latter can be studied in a uni-fied scheme: one first studies Q-minimal points and then derives similar results forthem. Moreover, it turns out that some arguments developed for weakly efficientpoints are still valid for Q-minimal points without assuming the ordering cone tohave nonempty interior and Q-minimal points can be characterized by a special func-tion introduced in optimization by Hiriart-Urruty which has a nice subdifferentialityproperty due to the openness of Q.

From now on unless otherwise specified let Q ⊂ Y be an arbitrary nonemptyopen cone with apex at zero and different from Y.

Definition 3.4. We say that a is a Q-minimal point of A and write a ∈ Qmin(A)if

A ∩ (a−Q) = ∅or, equivalently,

(A− a) ∩ (−Q) = ∅.

Remark 3.5. Makarov and Rachkovski studied in more details some concepts ofproper efficiency and introduced the notion of D-efficiency i.e. efficiency w.r.t. afamily of dilating cones, see [61]. Recall that an open cone in Y is said to be adilating cone (or a dilation) of K, or dilating K if it contains K \ 0. GivenD ∈ F(K), where F(K) is the class of families of cones dilating K, they called a aD-minimal point of A (a ∈ DMin(A)) if there exists C ∈ D such that

(A− a) ∩ (−C) = ∅.

It has been established [61] that Henig global proper efficiency, Henig proper effi-ciency, super efficiency and some other concepts of proper efficiency are D-efficiencywith D being appropriately chosen family of dilating cones. In contrast with D-efficiency, our concept includes also the concepts of strong efficiency and weak effi-ciency.

We recall a result from Theorem 21.7 in [26] stating that weakly, strongly, prop-erly efficient points of Definition 3.1 are in fact Q-minimal points with Q beingappropriately chosen cones.

Theorem 3.6. (i) a ∈ StrMin(A) iff a ∈ Qmin(A) with Q = Y \ (−K).

(ii) a ∈ WMin(A) iff a ∈ Qmin(A) with Q = intK.

(iii) a ∈ Pos(A) iff a ∈ Qmin(A) with Q = y ∈ Y | ϕ(y) > 0 for some ϕ ∈ K+i.

(iv) a ∈ GHe(A) iff a ∈ Qmin(A) with Q = C with C being some open pointedconvex cone dilating K.

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(v) a ∈ He(A) iff a ∈ Qmin(A) with Q = Vη for some η ∈]0, δ[.

(vi) supposing that K has a bounded base, a ∈ SE(A) iff a ∈ Qmin(A) with Q = Vηfor some η ∈]0, δ[.

(vii) a ∈ Be(A) iff a ∈ Qmin(A) with Q = Y \ −clcone[(A− a) +K].

3.3 Scalarization of weakly, strongly, properly efficient pointsof a set

First, we recall scalarization of a Q-minimal point obtained in [26]. Our scalarizingfunction is the signed distance function introduced by Hiriart-Urruty [49]. Recallthat for a subset U of Y , this function is defined by

∆U(y) = d(y;U)− d(y;Y \ U).

Theorem 3.7. a ∈ Qmin(A) iff the function ∆−Q(.− a) attains its minimum overA at a, that is

∆−Q(a− a) ≥ ∆−Q(a− a) = 0, ∀a ∈ A. (6)

As a consequence of Theorems 3.6 and 3.7 we have the following scalarizationfor weakly, strongly, properly efficient points.

Theorem 3.8. (i) a ∈ WMin(A) iff (6) holds with Q = intK.

(ii) a ∈ StrMin(A) iff (6) holds with Q = Y \ (−K).

(iii) a ∈ Pos(A) iff (6) holds with Q = y ∈ Y | ϕ(y) > 0 for some ϕ ∈ K+i.

(iv) a ∈ GHe(A) iff (6) holds with Q = C, C being an open pointed convex conedilating K.

(v) a ∈ He(A) iff (6) holds with Q = Vη for some scalar η ∈]0, δ[.

(vi) supposing that K has a bounded base, a ∈ SE(A) iff (6) holds with Q = Vη forsome scalar η ∈]0, δ[.

(vii) a ∈ Be(A) iff (6) holds with Q = Y \ −clcone[(A− a) +K].

Remark 3.9. Hiriart-Urruty’s signed distance function has been used for scalariza-tion in vector optimization in [47, 65].

We collect some known properties of the function ∆U from [49] in the propositionbelow.

Proposition 3.10. (i) ∆U is Lipschitz of rank 1.

(ii) ∆Y \U = −∆U .

(iii) ∆U is convex if U is convex and ∆U is concave if U is reverse convex, i.e.U = Y \ V with V being convex.

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(iv) ∆U(y) < 0 iff y ∈int U , ∆U(y) = 0 iff y ∈ bd U and ∆U(y) > 0 iff y ∈ Y \intU .

(v) Suppose that U is convex and has a nonempty interior, and 0 ∈bdU . Then

∂∆U(0) ⊂ N(0;U) \ 0.

The following properties of the subdifferential of ∆U(0) in some special casesestablished in [26] play an important role in formulating optimality conditions.

Proposition 3.11. (i) ∂∆−intK(0) ⊂ K+ \ 0.

(ii) Let C be an open convex cone dilating K. Then

∂∆−C(0) ⊂ K+i.

(iii) For η ∈]0, δ[, we have

∂∆−Vη(0) ⊂ y∗ ∈ K+i | y∗(θ) ≥ η, ∀ θ ∈ Θ.

Proof. For the assertions (i)-(ii) see Proposition 3.4 in [26] and for the assertion (iii)see Proposition 3.3 in [26] and its proof.

4 Optimality conditions for solutions of SOP

In this section we consider a set-valued optimization problem with geometric con-straint SOP

Minimize F (x) subject to x ∈ Ω,

where F is a set-valued maps from a Banach space X into a Banach space Y andΩ ⊂ X is a nonempty set. Throughout the section, let x ∈ Ω and (x, y) ∈grF . Weestablish necessary and sufficient conditions for (x, y) to be a weakly, strongly orproperly efficient solution of SOP and recall Bao-Mordukhovich’s necessary condi-tions for (x, y) to be an efficient solution of SOP.

4.1 Concepts of solutions of SOP

Various concepts of efficient points of a set in Definition 3.1 naturally induce corre-sponding concepts of solutions of SOP. In this paper, we restrict ourselves to globalsolutions. Denote

F (Ω) := ∪x∈ΩF (x).

Definition 4.1. We say that (x, y) is an (Pareto) efficient (resp., weakly efficient,strongly efficient, positive properly efficient, Henig global properly efficient, Henigproperly efficient, super efficient, Benson properly efficient and Q-minimal) solutionof SOP if y is an efficient (resp., weakly efficient, strongly efficient, positive properlyefficient, Henig global properly efficient, Henig properly efficient, super efficient,Benson properly efficient and Q-minimal) point of F (Ω).

Remark 4.2. It is easy to derive from Propositions 3.2 and 3.3 relations betweenthe above efficient solutions of SOP.

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4.2 Scalarization of weakly, strongly, properly efficient so-lutions of SOP

It is easy to derive from Definition 4.1 and Theorem 3.8 the following scalarization forweakly, strongly, properly efficient solutions of SOP through the following relation

∆−Q(y − y) ≥ ∆−Q(y − y) = 0, ∀y ∈ F (Ω). (7)

Theorem 4.3. (i) (x, y) is a weakly efficient solution of SOP iff (7) holds withQ = intK.

(ii) (x, y) is a strongly efficient solution of SOP iff (7) holds with Q = Y \ (−K).

(iii) (x, y) is a Henig global properly efficient solution of SOP iff (7) holds withQ = C, C being an open pointed convex cone dilating K.

(iv) (x, y) is a Henig properly efficient solution of SOP iff (7) holds with Q = Vηfor some scalar η ∈]0, δ[.

(v) supposing that K has a bounded base, (x, y) is a super efficient solution of SOPiff (7) holds with Q = Vη for some scalar η ∈]0, δ[.

(vi) (x, y) is a Benson properly efficient solution of SOP iff (7) holds with

Q = Y \ −clcone[(F (Ω)− y) +K].

4.3 Necessary and sufficient conditions for weakly, strongly,properly efficient solutions of SOP

In this subsection, we prove necessary and sufficient conditions for weakly, strongly,properly efficient solutions of SOP expressed in terms of coderivatives and normalcones in the senses of Clarke, Ioffe and Mordukhovich in the form

0 ∈ D∗F (x, y)(y∗) +N(x; Ω). (8)

Theorem 4.4. Suppose that

(a) The set Ω is closed.

(b) F is closed and pseudo-Lipschitz around (x, y).

Then

(i) For (x, y) to be a Q-minimal solution of SOP, it is necessary that (8) holdswith some nonzero y∗ ∈ ∂∆−Q(0).

(ii) For (x, y) to be a weakly efficient solution of SOP, it is necessary that (8) holdswith some y∗ ∈ K+ \ 0.

(iii) For (x, y) to be a strongly efficient solution of SOP, it is necessary that (8)holds for all y∗ ∈ K+.

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(iv) For (x, y) to be a positive properly efficient solution of SOP, it is necessarythat (8) holds with some y∗ ∈ K+i.

(v) For (x, y) to be a Henig global properly efficient solution of SOP when K+i 6= ∅,it is necessary that (8) holds with some y∗ ∈ K+i.

(vi) For (x, y) to be a Henig properly efficient solution of SOP , it is necessary that(8) holds with some

y∗ ∈ v∗ ∈ K+i | infθ∈Θ

v∗(θ) > 0.

(vii) For (x, y) to be a super efficient solution of SOP when K has a bounded base,it is necessary that (8) holds with some

y∗ ∈ v∗ ∈ K+i | infθ∈Θ

v∗(θ) > 0.

(viii) For (x, y) to be a Benson properly efficient solution of SOP, it is necessarythat (8) holds with some y∗ ∈ K+i provided that (c) Y is a separable Banachspace, or Y is a reflexive Banach space and K has a base and (d) F − y isnearly K-subconvexlike on Ω i.e. clcone (F (Ω)− y +K) is convex.

Proof. Our proof is motivated by that of Theorem 3.7 in [27].(i) By Theorem 4.3, (x, y) is a Q-minimal solution of SOP iff the function ∆−Q(.−

y) attains its minimum over F (Ω) at y, that is

∆−Q(y − y) ≥ ∆−Q(0) = 0, ∀y ∈ F (Ω).

This means that (x, y) is a minimizer of the problem

Minimize q(x, y) subject to (x, y) ∈ Λ, (9)

where q(x, y) = ∆−Q(y − y) and

Λ := (x, y) ∈ X × Y | x ∈ Ω, y ∈ F (x).

Then by the Clarke penalization, see Proposition 2.4.3 in [29], for some scalar l′ > 0large enough, (x, y) is an unconstrained local minimizer of

(x, y) −→ q(x, y) + l′d((x, y); Λ).

Proposition 2.3 implies that for l = (γ + 1)l′, (x, y) is an unconstrained local mini-mizer of

(x, y) −→ q(x, y) + ld((x, y); gr F ) + ld(x; Ω).

By the assertions (iii) and (iv) of Proposition 2.1, 0 is in the sum of the subdiffer-entials, that is there exist

y∗1 ∈ ∂q(x, y) = ∂∆−Q(0),

(x∗2, y∗2) ∈ l∂d((x, y); gr F ) ⊂ N((x, y); gr F )

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andx∗3 ∈ l∂d(x; Ω) ⊂ N(x; Ω)

such that(0, 0) = (0, y∗1) + (x∗2, y

∗2) + (x∗3, 0).

Hence, we have 0 = x∗2 + x∗3 and 0 = y∗1 + y∗2. Putting y∗ = y∗1 = −y∗2, we havey∗ ∈ ∂∆−Q(0) and x∗2 ∈ D∗F (x, y)(y∗). The latter together with 0 = x∗2 + x∗3 andx∗3 ∈ N(x; Ω) yields

0 ∈ D∗F (x, y)(y∗) +N(x; Ω).

(ii) The assertion follows from Theorem 3.6, the assertion (i) applied to Q =-intKand Proposition 3.11.

(iii) Let y∗ ∈ K+ be arbitrarily chosen. From Definitions 3.1 and 4.1, we havey − y ∈ K, ∀ y ∈ F (Ω) and since y∗ ∈ K+, it follows that

y∗(y − y) ≥ 0, ∀ y ∈ F (Ω).

Therefore, (x, y) is an unconstrained local minimizer of (9), where q(x, y) = y∗(y−y).By the same arguments as in the proof of the assertion (i), we can prove that (8)holds with y∗ = y∗. Since y∗ ∈ K+ is arbitrary, the assertion follows.

(iv) Let ϕ ∈ K+i be such that ϕ(y− y) ≥ 0, ∀ y ∈ F (Ω). It suffices to apply thearguments of the proof of the assertion (iii).

(v) Let C be an open convex cone dilating K associated to y as in the definitionof the Henig global properly efficient point y of F (Ω). The assertions follows fromTheorem 3.6, the assertion (i) applied to Q = −C and Proposition 3.11.

(vi) Suppose that (x, y) is a Henig properly efficient solution of SOP. By Propo-sition 3.11, y ∈ Qmin(F (Ω)) with Q = Vη for some η ∈]0, δ[. By the assertion (i),we can find y∗ ∈ ∂∆Vy(0) such that (8) holds. From Proposition 3.11 we have

∂∆−Vη(0) ⊂ y∗ ∈ K+i | y∗(θ) ≥ η, ∀ θ ∈ Θ.

Since infθ∈Θ y∗(θ) ≥ η > 0, the assertion follows.

(vii) The assertion is immediate from Proposition 3.2 and the assertion (vi).(viii) Under the assumptions (c) and (d), Proposition 3.3 implies that (x, y)

being a Benson properly efficient solution of SOP also is a positive properly efficientsolution. The assertion then follows from the assertion (iv).

Remark 4.5. 1. One can also prove the assertions (ii)-(viii) by using the argumentsof the proof of the assertion (i) and Theorem 4.3.

2. Similar necessary conditions for a weakly efficient solution of SOP in the caseF is a single-valued map have been obtained in [46].

3. Necessary conditions in the form of Lagrange multiplier rule involving differentcoderivatives for a general constrained set-valued optimization problem (in short,CSOP) which includes not only a geometric constraint (the case SOP) but also aconstraint G(x) ∩ K 6= ∅ (with G and K being another set-valued map and cone)have been established in [26, 27] under some metrict regularity assumptions on G.Here, we provide a direct proof for SOP.

Theorem 4.6. Suppose that

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(a) The set Ω is closed and convex.

(b) F is closed and convex, i.e. the graph of F is closed and convex.

Then the necessary conditions in Theorem 4.4 also are sufficient (with Q beingadditionally convex in the case (i)).

To prove this theorem we need the following lemmas.

Lemma 4.7. Under the assumptions (a)-(b) of Theorem 4.6, for any y∗ ∈ Y ∗

satisfying (8) we havey∗(y − y) ≥ 0, ∀ y ∈ F (Ω). (10)

Proof. By (8), we can find x∗1, x∗2 ∈ X∗ such that x∗1 ∈ D∗F (x, y)(y∗), x∗2 ∈ N(x; Ω)

and x∗1 + x∗2 = 0. Since (x∗1,−y∗) ∈ N((x, y); grF ), according to the definition ofnormal cone of convex analysis, we have

x∗1(x− x)− y∗(y − y) ≤ 0, ∀ (x, y) ∈ grF

andx∗2(x− x) ≤ 0, ∀ x ∈ Ω.

Summarizing two latter relations and taking account of x∗1 + x∗2 = 0, we obtain thedesired relation (10).

Lemma 4.8. Suppose thaty∗(y) ≥ 0 (11)

holds for some y∗ ∈ Y ∗ and y ∈ Y . Then

(i) If y∗ ∈ K+ \ 0 and intK 6= ∅, then y /∈ −intK.

(ii) For a fixed y ∈ Y , if (11) holds for all y∗ ∈ K+ then y ∈ K.

(iii) If y∗ ∈ K+i and infθ∈Θ y∗(θ) = ζ, then y /∈ −Vη, where

η = minζ/(2‖y∗‖), δ/2. (12)

Proof. (i) Suppose to the contrary that y ∈ −intK or −y ∈ intK. As y∗ 6= 0, thereexists v ∈ Y such that y∗(v) < 0. Since −y ∈ intK, there exists a scalar ρ > 0 suchthat −y + ρv ∈ K. As y∗ ∈ K+ \ 0, we have y∗(−y + ρv) ≥ 0. Hence, we obtainy∗(y) ≤ ρy∗(v) < 0, a contradiction to (11).

(ii) Suppose to the contrary that y /∈ K. Since K is a closed convex cone, aseparation theorem implies the existence of a scalar t and y∗ ∈ Y ∗ \ 0 such that

y∗(y) < t ≤ y∗(k), ∀ k ∈ K.

A standard argument yields that y∗ ∈ K+, t ≤ 0 and y∗(y) < t ≤ 0, contradictingthe assumption that (11) holds for all y∗ ∈ K+.

(iii) Suppose to the contrary that y ∈ −Vη. By the definition of Vη, there exist ascalar ρ > 0, θ ∈ Θ and b ∈ BY such that y = ρ(−θ + ηb). Taking (12) and ‖b‖ ≤ 1into account we get

y∗(y) = ρy∗(−θ + ηb) = ρy∗(−θ) + η(y∗(b)≤ ρ(−ζ + η‖y∗‖‖b‖) ≤ ρ(−ζ + ζ/2) = −ρζ/2 < 0.

This is a contradiction to (11).

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Now we can prove Theorem 4.6.

Proof. Suppose that (8) holds. By Lemma 4.7, (10) holds with some y∗ ∈ Y ∗ chosenin correspondence with each cases (i)-(viii). We will apply Lemma 4.8 to each caseto obtain desired assertions.

(i) As Q is convex and y∗ ∈ ∂∆−Q(0), the definition of subdifferential of convexanalysis and (10) yield

∆−Q(y − y)−∆−Q(0) ≥ y∗(y − y) ≥ 0, ∀ y ∈ F (Ω).

By Theorem 3.7, y ∈ Qmin(F (Ω)) and hence, (x, y) is a Q-minimal solution of SOP.(ii) Applying Lemma 4.8(i) we get y − y /∈ −intK, ∀ y ∈ F (Ω), which means

that (x, y) is a weakly efficient solution of SOP.(iii) Applying Lemma 4.8(ii) we get y − y ∈ K, ∀ y ∈ F (Ω), which means that

(x, y) is a strongly efficient solution of SOP.(iv) It is obvious.(v) Let C := u ∈ Y |ϕ(u) > 0∪0. It is clear that intC = u ∈ Y |ϕ(u) > 0.

Since ϕ ∈ K+i, we have K \ 0 ⊂ intC. Further, as ϕ(y − y) ≥ 0, ∀ y ∈ F (Ω),we get y − y /∈ −intC, ∀ y ∈ F (Ω). Thus, (x, y) is a Henig global properly efficientsolution of SOP.

(vi) Suppose that infθ∈Θ y∗(θ) = ζ, then Lemma 4.8(iii) implies that y− y /∈ −Vη,

where η is the scalar defined by (12). It is clear that η ∈]0, δ[. Theorem 3.6 (vi)implies that y is a Henig properly efficient point of F (Ω). Hence, (x, y) is a Henigglobal properly efficient solution of SOP.

(vii) It is a consequence of (vi) and Proposition 3.2(iv).(viii) It is a consequence of (iv) and Proposition 3.3.

Remark 4.9. A similar sufficient conditions for weakly efficient solutions of CSOPhave been established in [27].

4.4 Necessary conditions for efficient solutions of SOP

In this subsection, we recall Bao-Mordukhovich’s result [25], which has been obtainedby using the extremal principle [38] and other advanced tools of variational analysis.Unfortunately, in the case of efficient solutions we have only necessary conditions.

Theorem 4.10. Suppose that

(i) X and Y are Asplund spaces.

(ii) The set Ω is closed.

(iii) F is closed and the qualification condition

D∗F (x, y)(0) ∩NM(x; Ω) = 0 (13)

is satisfied, and that either F is PSNC at (x, y) or Ω is SNC at x; both thequalification condition (13) and the PSNC property of F are automaticallysatisfied if F is pseudo-Lipschitz around (x, y).

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(iv) either K is SNC at the origin or F−1Ω is PSNC at (y, x), where FΩ(x) = F (x)

if x ∈ Ω and FΩ(x) = ∅, otherwise.

If (x, y) is an (Pareto) efficient solution of SOP then there exists y∗ ∈ K∗ with‖y∗‖ = 1 such that

0 ∈ D∗MF (x, y)(y∗) +NM(x; Ω).

Here, D∗M and NM denote the coderivative and normal cone in the sense ofMordukhovich.

5 Optimality conditions for solutions of SEP, VEP

and VVI

In this section, we recall concepts of solutions of SEP, VEP and VVI. Establishingan equivalence between solution of SEP and these of SOP, we derive scalaizationand optimality conditions for solutions of SEP, VEP and VVI from the ones forsolutions of SOP obtained in Section 4 (Theorems 4.3, 4.4, 4.6 and 4.10).

Throughout this section, we make the following assumptions:1. A ⊂ X is a nonempty set and x ∈ A.2. Φ : A× A→ 2Y is a set-valued map.3. φ : A× A→ Y is a single-valued map.4. f : X → L(X, Y ) is a single-valued map, where L(X, Y ) is, as before, the set

of all continuous linear maps from X to Y .

We will use the following notations

Φ(x, A) := ∪x∈AΦ(x, x),

φ(x, A) := φ(x, x) | x ∈ A

andf(x)(A− x) := f(x)(x− x) | x ∈ A.

By Φx and φx we mean the maps from X to Y induced by Φ and φ, respectively,as follows

Φx(x) = Φ(x, x), φx(x) = φ(x, x).

5.1 Concepts of solutions of SEP, VEP, VVI

Definition 5.1. We say that

(i) x is an (Pareto) efficient solution of SEP if

Φ(x, x) ∩ (−K \ 0) = ∅, ∀ x ∈ A.

(ii) supposing that int K 6= ∅, x is a weakly efficient solution of SEP if

Φ(x, x) ∩ (−intK) = ∅, ∀ x ∈ A.

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(iii) x is a strongly (or ideal) efficient solution of SEP if

Φ(x, x) ⊂ K, ∀ x ∈ A.

(iv) supposing that K+i 6= ∅, x is a positive properly efficient solution of SEP ifthere exists ϕ ∈ K+i such that

ϕ(Φ(x, x)) ⊂ R+, ∀ x ∈ A.

(v) x is a Henig global properly efficient solution of SEP if there exists a convexcone C with apex at zero with K \ 0 ⊂ intC such that

Φ(x, x) ∩ (−intC) = ∅, ∀ x ∈ A.

(vi) supposing that K has a base Θ, x is a Henig properly efficient solution of SEPif there is a scalar ε > 0 such that

clconeΦ(x, A) ∩ (−clcone(Θ + εB)) = 0.

(vii) x is a super efficient solution of SEP if there is a scalar ρ > 0 such that

clcone(Φ(x, A)) ∩ (B −K) ⊆ ρB.

(viii) x is a Benson properly efficient solution of SEP if

clcone[Φ(x, A) +K] ∩ (−K) = 0.

Replacing Φ in Definition 5.1 by φ, we obtain concepts of solutions of VEP andreplacing Φ(x, x), Φ(x, A) in Definition 5.1 by f(x)(x− x), f(x)(A− x), respectively,we obtain concepts of solutions of VVI.

Remark 5.2. Ansari et al. in [3] introduced the concepts of efficient, weakly efficientand strongly efficient solutions for VEP and SEP. Note that the definition of weaklyefficient solutions for SEP presented in [3] is in a slightly different way: supposingthat int K 6= ∅, x is called a weakly efficient solution to SEP if

Φ(x, x) 6⊂ −intK ∀x ∈ A.

For the concepts of efficient, weakly efficient, Henig properly efficient, Henig globalproperly efficient, super efficient solutions to VEP as above see [12, 15] and forthe concept of Benson efficient solutions to VEP see [13]. The concepts of superefficient solutions to SEP was introduced in [16]. Note that it has been defined in[16] also the concept of a ψ-efficient solution of VEP, which is slightly different fromthe concept of a positive properly efficient solution. Namely, given ψ ∈ K∗ \ 0, xis a ψ-efficient solution of SEP

ψ(φ(x, x)) ≥ 0, ∀x ∈ A.

From now on, we make the convention that the condition

0 ∈ Φx(x) or, equivalently, 0 ∈ Φ(x, x) (14)

holds whenever we consider SEP and the condition

φ(x, x) = 0

holds whenever we consider VEP.

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5.2 Equivalence between solutions of SOP and solutions ofSEP

It is immediate from Definitions 4.1, 5.1 and (14) the following equivalence.

Theorem 5.3. (Equivalence of SOP and SEP) The following assertions are equiv-alent:

(i) x is a solution of SEP in some sense of Definition 5.1.

(ii) (x, 0) is a solution of SOP with F = Φx, Ω = A in corresponding sense ofDefinition 4.1.

5.3 Scalarization of weakly, strongly, properly efficient so-lutions of SEP, VEP and VVI

In this subsection, P is any of SEP, VEP and VVI. We derive from Theorem 3.7scalarization for weakly, strongly, properly efficient solutions of P in the form

∆−Q(y) ≥ ∆−Q(0) = 0, ∀y ∈ SP , (15)

where SP is a set associated to P given by

SP :=

Φ(x, A) if P is SEPφ(x, A) if P is VEPf(x)(A− x) if P is VVI

Theorem 5.4. (i) x is a weakly efficient solution of P iff (15) holds with Q =intK.

(ii) x is a strongly efficient solution of P iff (15) holds with Q = Y \ (−K).

(iii) x is a Henig global properly efficient solution of P iff (15) holds with Q = C,C being an open pointed convex cone dilating K.

(iv) x is a Henig properly efficient solution of P iff (15) holds with Q = Vη forsome scalar η ∈]0, δ[.

(v) supposing that K has a bounded base, x is a super efficient solution of P iff(15) holds with Q = Vη for some scalar η ∈]0, δ[.

(vi) x is a Benson properly efficient solution of P iff (15) holds with

Q = Y \ −clcone[SP +K].

Remark 5.5. Note that scalarization for weakly efficient solution, Henig properlyefficient solution, Henig global properly efficient solution and super efficient solutionof VEP has been obtained in [15, 16].

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5.4 Necessary and sufficient conditions for weakly, strongly,proper efficient solutions of SEP, VEP and VVI

Our optimality conditions for weakly, strongly, proper efficient solutions of SEP areexpressed in terms of coderivative and normal cone of the forms

0 ∈ D∗Φx(x, 0)(y∗) +N(x;A). (16)

Theorem 5.6. Suppose that the set A is closed, Φx is closed and pseudo-Lipschitzaround (x, 0). Then

(i) For x to be a weakly efficient solution of SEP, it is necessary that (16) holdswith some y∗ ∈ K+ \ 0.

(ii) For x to be a strongly efficient solution of SEP, it is necessary that (16) holdsfor all y∗ ∈ K+.

(iii) For x to be a positive properly efficient solution of SEP, it is necessary that(16) holds with some y∗ ∈ K+i.

(iv) For x to be a Henig global properly efficient solution of SEP when K+i 6= ∅, itis necessary that (16) holds with some y∗ ∈ K+i.

(v) For x to be a Henig properly efficient solution of SEP , it is necessary that(16) holds with some

y∗ ∈ v∗ ∈ K+i | infθ∈Θ

v∗(θ) > 0.

(vi) For x to be a super efficient solution of SEP when K has a bounded base, it isnecessary that (16) holds with some

y∗ ∈ v∗ ∈ K+i | infθ∈Θ

v∗(θ) > 0.

(vii) For x to be a Benson efficient solution of SEP, it is necessary that (16) holdswith some y∗ ∈ K+i provided that (c) Y is a separable Banach space, or Y is areflexive Banach space and K has a base and (d) Φx is nearly K-subconvexlikeon A i.e. clcone (Φx(A) +K) is convex.

Theorem 5.7. Suppose that the set A is closed, convex and Φx is closed, convex.Then the necessary conditions in Theorem 5.6 also are sufficient.

These theorems are immediate consequences of Theorems 5.3, 4.4 and 4.6.

Now let us consider special cases VEP and VVI .

1. Case VEP. Replacing Φ in Theorems 5.6 and 5.7 by φ with φx beingcontinuous on A and Lipschitz around x and replacing (16) by

0 ∈ D∗φx(x)(y∗) +N(x;A), (17)

we obtain corresponding necessary and (under the convexity assumption on thegraph of φx) sufficient conditions for weakly, strongly, properly efficient solutions

21

of VEP. If, in addition, φx is strictly differentiable around x, then Proposition 2.2implies that

D∗φx(x)(y∗) = [φ′x]∗(x)(y∗)

and (17) becomes0 ∈ [φ′x]

∗(x)(y∗) +N(x;A).

2. Case VVI. Replacing Φ(x, x) in Theorems 5.6 and 5.7 by f(x)(x − x),Φ(x, A) by f(x)(A− x) and (16) by

0 ∈ f(x)∗(y∗) +N(x;A),

we obtain corresponding necessary and (under the convexity assumption on A) suf-ficient conditions for weakly, strongly, properly efficient solutions of VVI. To seethis, it suffices to note that in this case, Φx(x) = f(x)(x− x) is an affine map so allconditions of Theorems 5.6 and 5.7 imposed on Φx are automatically satisfied and,moreover, we have

D∗Φx(x, y)(y∗) = f(x)∗(y∗).

5.5 Necessary conditions for efficient solutions of SEP, VEPand VVI

Theorem 5.8. Suppose that

(i) X and Y are Asplund spaces.

(ii) The set A is closed.

(iii) Φx is closed and the qualification condition

D∗Φx(x, 0)(0) ∩NM(x; Ω) = 0 (18)

is satisfied, and that either Φx is PSNC at (x, 0) or A is SNC at x; both thequalification condition (18) and the PSNC property of Φx are automaticallysatisfied if Φx is Lipschitz-like around (x, 0).

(iv) either K is SNC at the origin or (Φx)−1A is PSNC at (0, x), where (Φx)A(x) =

Φx(x) if x ∈ A and (Φx)A(x) = ∅, otherwise.

If x is an (Pareto) efficient solution of SEP, then there exists y∗ ∈ K∗ with ‖y∗‖ = 1such that

0 ∈ D∗MΦx(x, 0)(y∗) +NM(x;A).

Proof. An immediate consequence of Theorems 5.3 and 4.10.

One can easily derive necessary conditions for efficient solutions of VEP andVVI. As illustration, we state necessary conditions for efficient solutions of VVI.

Theorem 5.9. Suppose that

(i) X and Y are Asplund spaces.

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(ii) The set A is closed.

If x is a Pareto efficient solution of VVI, then there exists y∗ ∈ K∗ with ‖y∗‖ = 1such that

0 ∈ f(x)∗(y∗) +NM(x;A).

Proof. Since all conditions of Theorem 5.8 imposed on Φx(.) = f(x)(. − x) areautomatically satisfied, the assertion follows from this theorem.

Acknowledgment. This work was supported in part by a grant of the NationalFoundation for Science and Technology Development, Vietnam.

References

[1] F.Giannessi, Theorems of alternatives, quadratic problems, and complemen-tarity problems, in: R.W. Cottle, F. Giannessi, J.-L. Lions (Eds.), VariationalInequalities and Complementary Problems, J. Wiley, New York, 1980, pp. 151-186.

[2] M. Bianchi, N. Hadjisavvas, S. Schaible, Vector equilibrium problems with gen-eralized monotone bifunctions, J. Optim. Theor. Appl. 92 (1997), 527-542.

[3] Q.H. Ansari, W. Oetli, D. Schlager, A generalization of vector equilibria, Math.Meth. Oper. Res. 46 (1997), No. 2, 147-152.

[4] E. Allevi, A. Gnudi, S. Schaible, M.T. Vespucci, Equilibrium and least elementproblems for multivalued functions, J. Glob. Optim. 46 (2010), 561-569.

[5] Q. H. Ansari, I. V. Konnov, J.-C. Yao, On generalized vector equilibrium prob-lems, Nonlinear Analysis 47 (2001), no. 1, 543-554.

[6] Q. H. Ansari, J.-C. Yao, An existence result for the generalized vector equilib-rium problem, Applied Mathematics Letters 12 (1999), no. 8, 53-56.

[7] R. I. Bot, G.Wanka, S.-M. Grad, Duality in Vector Optimization, Springer,Berlin, 2009.

[8] G.-Y.Chen, X.Huang, X.Yang, Vector Optimization, Set-Valued and Varia-tional Analysis, Springer, Berlin, 2005.

[9] J.Y. Fu, A.H. Wan, Generalized vector quasi-equilibria problems , Math. Meth.Oper. Res. 52 (2000), 57-64.

[10] J.Y. Fu, A.H. Wan, Generalized vector equilibria problems with set-valuedmappings, Math. Meth. Oper. Res. 56 (2002), 259-268.

[11] X.H.Gong, Efficiency and Henig efficiency for vector equilibrium problems, J.Optim. Theory Appl. 108 (2001), 139-154.

[12] X.H.Gong, Strong vector equilibrium problems, J. Glob. Optim. 36 (2006),339-349.

[13] X.H.Gong, Connectedness of the solution sets and scalarization for vector equi-librium problems, J. Optim. Theory Appl. 133 (2007), 151-161.

[14] X.H.Gong, Optimality conditions for vector equilibrium problems, J. Math.Anal.Appl. 342 (2008), 1455-1466.

23

[15] X.H.Gong, Chebyshev scalarization of solution to the vector equilibrium prob-lems, J. Glob. Optim. (2010), DOI 10.1007/s10898-010-9553-5 (online first).

[16] X.H.Gong, W.T.Fu, W.Liu, Super efficiency for a vector equilibrium in locallyconvex topological vector spaces, in : F. Giannessi (Ed.), Vector VariationalInequalities and Vector equilibria”, Kluwer Academic Publishers, 2000, pp..233-252.

[17] N. Hadjisavvas, S. Schaible, From scalar to vector equilibrium problems in thequasimonotone case, J. Optim. Theory Appl. 96 (1998), no. 2, 297-309.

[18] S. H. Hou, H. Yu, G. Y. Chen, On vector quasi-equilibrium problems withset-valued maps, J. Optim. Theory Appl. 119 (2003), 485-498.

[19] I. V. Konnov, J. C. Yao, Existence of solutions for generalized vector equilibriumproblems, J. Math. Anal.Appl. 233 (1999), no. 1, 328-335.

[20] G.M. Lee, D.S. Kim, B.S. Lee, N.D. Yen, Vector variational inequality as atool for studying vector optimization problems, in : F. Giannessi (Ed.), VectorVariational Inequalities and Vector equilibria”, Kluwer Academic Publishers,2000, pp. 233-252.

[21] L. J. Lin, Z. T. Yu, and G. Kassay, Existence of equilibria for multivaluedmappings and its application to vectorial equilibria, J. Optim. Theory Appl.114 (2002), no. 1, 189-208.

[22] W. Oettli, D. Schlager, Existence of equilibria for monotone multivalued map-pings,Math. Meth. Oper. Res. 48 (1998), no. 2, 219-228.

[23] J.-W. Peng, The system of generalized set-valued equilibrium problems, J. In-equal. Appl. 2006, Art ID 16764, 9pp.

[24] W. Song, Vector equilibrium problems with set-valued mappings, in : F. Gi-annessi (Ed.), Vector Variational Inequalities and Vector equilibria”, KluwerAcademic Publishers, 2000, pp. 233-252.

[25] T.Q. Bao, B.S. Mordukhovich, Relative Pareto minimizers for multiobjectiveproblems: existence and optimality conditions. Math. Program. 122 (2010), no.2, Ser. A, 301–347.

[26] T. X. D. Ha, Optimality conditions for several types of efficient solutions of set-valued optimization problems, in : P. Pardalos, Th. M. Rassias, A. A. Khan(Eds.), Nonlinear Analysis and Variational Problems, Springer (2010), pp. 305-324.

[27] B. El Abdouni, L. Thibault, Optimality conditions for problems with set-valuedobjectives, J. Applied Analysis 2 (1996), 183–201.

[28] J.P. Aubin, Lipschitz behavior of solutions to nonconvex problems, Math. Oper.Res. 9 (1984), 87-111.

[29] F.H. Clarke, Optimization and nonsmooth analysis, Wiley, New York, 1983.

[30] A.D. Ioffe, Approximate subdifferentials and applications 2: Metric theory,Mathematica 33 (1986), 111-128.

[31] A.D. Ioffe, Approximate subdifferentials and applications 3: Metric theory,Mathematica 36 (1989), 1-38.

[32] A.D. Ioffe, Metric regularity and subdifferential calculus, Russ. Math. Surveys,55 (2000), 501-558.

24

[33] A.D. Ioffe , J.-P. Penot, Subdifferentials of performance functions and calculusof coderivatives of set-valued mappings. Well-posedness and stability of varia-tional problems. Serdica Math. J. 22 (1996), no. 3, 257–282.

[34] A.Y. Kruger, B.S. Mordukhovich, Extremal points and the Euler equations innonsmooth optimization, Dokl. Akad Nauk BSSR 24 (1980), 684-687.

[35] B.S. Mordukhovich, Maximum principle in problems of time optimal controlwith nonsmooth contraints, J. Appl. Math. Mech. 40 (1976), 960-969.

[36] B. S. Mordukhovich, Metric approximations and necessary optimality condi-tions for general classes of nonsmooth extremal problems, Soviet Math. Dokl.22 (1980), 526-530.

[37] B. S. Mordukhovich, Generalized differential calculus for nonsmooth and set-valued mappings, J. Math. Anal. Appl. 183 (1994), 250-288.

[38] B. S. Mordukhovich, Variational Analysis and Generalized differentiation, I:Basic Theory, Grundlehren Series (Fundamental Principles of MathematicalSciences) 330, Springer, Berlin, 2006.

[39] B. S. Mordukhovich, Variational Analysis and Generalized differentiation, II:Applications, Grundlehren Series (Fundamental Principles of Mathematical Sci-ences) 331, Springer, Berlin, 2006.

[40] R. T. Rockafellar, The Theory of Subgradients and its Applications to Problemsof Optimization: Convex and Nonconvex Functions, Hedelman Verlag, Berlin,1981.

[41] R. T. Rockafellar, Lipschitz property of multifunctions, Nonlinear Anal.Th.Meth. Appl. 9 (1985), 867-885.

[42] L. Thibault, On subdifferentials of optimal value functions, SIAM J. ControlOptim. 29 (1991), 1019-1036.

[43] J.P. Aubin, I.Ekeland, Applied Nonlinear Analysis, Wiley, New York, 1984.

[44] T. Q. Bao, B. S. Mordukhovich, Variational principles for set-valued mappingswith applications to multiobjective optimization, Control and Cybernetics 36(2007), 531-562.

[45] T.Q. Bao, B.S. Mordukhovich, Necessary conditions for super minimizers inconstrained multiobjective optimization. J. Global Optim. 43 (2009), no. 4,533–552.

[46] J. Dutta, C. Tammer, Lagrangian conditions for vector optimization in Banachspaces, Math. Meth. Oper. Res. 64 (2006), 521-540.

[47] T. X. D. Ha, Lagrange multipliers for set-valued optimization problems associ-ated with coderivatives, J. Math. Anal. Appl. 311 (2005), 647-663.

[48] X. Y. Zheng and K. F. Ng, The Lagrange multiplier rule for multifunctions inBanach spaces, SIAM J. Optim. 17 (2006), 1154-1175.

[49] J. B. Hiriart-Urruty, New concepts in nondifferentiable programming, Bull. Soc.Math. France 60 (1979) ,57-85.

[50] M.A. Krasnoselski, Positive Solutions of Operator Equations, Nordhoff, Gronin-gen, 1964.

[51] J. Jahn, Mathematical Vector Optimization in Partially Ordered Linear Spaces,Peter Lang, Frankfurt, 1986.

25

[52] J. Jahn, Vector Optimization - Theory, Applications, and Extensions, Springer,Berlin, 2004.

[53] D.T. Luc, Theory of Vector Optimization, Springer-Verlag, Berlin, 1989.

[54] L. Hurwicz, Programming in linear spaces, in: K.J. Arrow, L. Hurwicz, H.Uzawa (Eds.), Standford University Press, Standford, California, 1958.

[55] H.P. Benson, An improved definition of proper efficiency for vector minimizationwith respect to cones, J. Math. Anal. Appl. 71 (1978), 232-241.

[56] M.I. Henig, Proper efficiency with respect to the cones, J. Optim. Theory Appl.36 (1982), 387-407.

[57] J.M.Borwein, D. Zhuang, Super efficiency in vector optimization, Trans. Amer.Math. Society 338 (1993), 105-122.

[58] D. Zhuang, Density results for proper efficiency, SIAM J. Control Optim. 32(1994),51-58.

[59] X.Y. Zheng, Proper efficiency in local convex topological vector spaces, J. Op-tim. Theory Appl. 94 (1997), 469-486.

[60] X.Y. Zheng, Scalarization of Henig properly efficient points in a normed space,J. Optim. Theory Appl. 105 (2000), 233-247.

[61] E.K. Makarov, N.N. Rachkovski, Unified representation of proper efficiency bymeans of dilating cones, J. Optim. Theory Appl. 101 (1999),141-165.

[62] A. Guerraggio. E. Molno, A. Zaffaroni, On the notion of proper efficiency invector optimization, J. Optim. Theory Appl. 82(1994), 1-21.

[63] J.-H. Qiu, Dual characterization and scalarization for Benson proper efficiency,SIAM J. Optim. 19 (2008), 144-162.

[64] X.M. Yang, D. Li, S.Y. Wang, Near subconvexlikeness in vector optimizationwith set- valued functions, J. Optim. Theory Appl. 110 (2001), 413-427.

[65] V.V. Gorokhovich, Convex and nonsmooth problems of vector optimization,Nauka i Tekhnika, Minsk, 1990.

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