Truong Xuan Duc Ha. August 2, 2010

Transcription

1 Optimality conditions for various efficient solutions involving coderivatives: from set-valued optimization problems to set-valued equilibrium problems Truong Xuan Duc Ha August 2, 2010 Abstract In this paper, we present a new approach to the study of various efficient solutions of a set-valued equilibrium problem (for short, SEP) through the study of corresponding solutions of a set-valued optimization problem with a geometric constraint (for short, SOP). The solutions under consideration are: 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 obtain in a unified scheme scalar characterization with the help of Hiriart-Urruty s signed distance function, necessary/sufficient conditions for weakly, strongly, properly efficient solutions of SOP involving coderivatives and normal cones in the senses of Clarke, Ioffe and Mordukhovich, and recall Bao-Mordukhovich s necessary conditions for efficient solutions of SOP involving the coderivative and normal cone in the sense of Mordukhovich. Secondly, we show that there is an equivalence between solutions of SOP and solutions of SEP and based on this equivalence we derive similar results for solutions of SEP as well as for solutions 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 equilibrium problem (for short, VEP) in [2] and to a set-valued equilibrium problem (for Institute of Mathematics, 18 Hoang Quoc Viet Road, Hanoi, Viet Nam, txdha@math.ac.vn 1

2 short, SEP) in [3]. SEP and its special cases, VEP and VVI, have applications in different areas of optimization, optimal control, operations research and economics. Numerous works have been devoted to the existence, scalarization, optimality conditions or connectedness of various efficient solutions of these problems; see, for instance, [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 for efficient solutions of SEP. Since SEP is closely related to a set-valued optimization problem, it is natural to borrow techniques and results on optimality conditions for the latter to try to come up with similar results for the former. Inspired by this idea, we present a new approach to the study of various efficient solutions of SEP through the study of corresponding solutions of a set-valued optimization problem with a geometric constraint (for short, SOP). The solutions under consideration are: 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 and Benson properly efficient solutions. Firstly, we obtain in a unified scheme scalar characterization with the help of Hiriart-Urruty s signed distance function, necessary/sufficient conditions for weakly, strongly, properly efficient solutions of SOP involving coderivatives and normal cones in the senses of Clarke, Ioffe and Mordukhovich, and recall Bao- Mordukhovich s necessary conditions [25] for efficient solutions of SOP involving the coderivative and normal cone in the sense of Mordukhovich. Secondly, we show that there is an equivalence between solutions of SOP and solutions of SEP and based on this equivalence we derive similar results for solutions of SEP, VEP and VVI. 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 of normal cone, subdifferential and coderivative and establish a property of a setvalued pseudo-lipschitz map. In Section 3, we recall the unified approach to the study of weakly, strongly, properly efficient points of a set and scalarization of these points presented in [26]. Scalarization and optimality conditions for various efficient solutions 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] without proofs, thus making our exposition self-contained. We first recall concepts of normal cone, subdifferential and coderivative in the senses of Ioffe, Clarke and Mordukhovich 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 and Y, respectively. The closed unit ball and the open unit ball in any space, say X, are denoted by B X and B X ; 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 2

3 nonempty set A X, inta and cla stand for the interior and closure of A and conea := {ta : t R +, a A}, where R + = [0, [. Further, d(x; A) is the distance from x to A and χ A (x) is the indicator function associated to A, i.e., χ A (x) = 0 if x A and χ A (x) = otherwise. Recall that a Banach space is Asplund if each of its separable subspace has a separable dual. This class of spaces has been comprehensively investigated in geometric theory of Banach spaces and has been largely employed in variational analysis; see, e.g. [38, 39]. Examples of Asplund spaces are the Banach spaces R n, L p [0,1] and lp (1 < p < ). 2.1 Normal cone, subdifferential and coderivative in the senses of Ioffe, Clarke and Mordukhovich Assume that g : X R { } is a function and F : X 2 Y is a set-valued map (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 Lipschitz near x domg. The Ioffe approximate subdifferential of g at x [30-32] is the set A g(x) = lim sup ε g y+l (y), L F (ε,y) (0 +,x) where F is the collection of all finite dimensional subspaces of X, g y+l (u) = g(u) if u y + L and g y+l (u) = + otherwise, for ε 0 ε g y+l (y) = {x X x (v) ε v + + lim inf t 0 + t 1 [g y+l (y + tv) g y+l (y)], v X}. The Clarke generalized subdifferential of g at x [29] is the set C g(x) = {x X x (v) g 0 (x; v), v X}, where g 0 (x; v) is the generalized directional derivative of g at x in the direction v g 0 g(y + tv) g(y) (x; v) = lim sup. y x, t 0 + t Let Ω be a nonempty subset of X different from X and x clω. approximate normal cone to Ω at x [30-32] is given by The Ioffe N A (x; Ω) = λ>0 λ A d(x; Ω) and the Clarke normal cone to Ω at x [29] is given by ( ) N C (x; Ω) = cl λ C d(x; Ω). λ>0 3

4 Assuming that X is an Asplund space, the Mordukhovich normal cone to Ω at x [34-39] is defined by N M (x; Ω) = lim sup ˆN(x ; Ω), x Ω x where the limit in the right-hand side means the sequential Kuratowski-Painlevé upper 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 ˆN(x; Ω) is the Fréchet normal cone to Ω at x given by { } ˆN(x; Ω) = x X x (x x) lim sup x x 0. (Note that the Mordukhovich normal cone and related to it concepts are defined in any Banach space but here we restrict ourselves to the Asplund space setting, where they enjoy full calculus). Now assume that the function g is lower semicontinuous and the set-valued map F is closed (i.e., its graph is a closed set). The subdifferentials of g and coderivatives of F in the senses of Ioffe, Clarke or Mordukhovich are defined through the corresponding normal cones as follows (for the sake of convenience, we make the convention that when no confusion occurs the same notations N, g and D F are used for the normal cone, subdifferentials and coderivatives in the above senses or in the sense of convex analysis and that the spaces under consideration are Asplund whenever the subdifferential, the normal cone and the coderivative are understood in the sense of Mordukhovich) and x Ω x g(x) = {x X (x, 1) N((x, g(x)); epig)} D F (x, y)(y ) = {x X (x, y ) N((x, y); grf )}. Along with the Mordukhovich coderivative, we also consider (in the Asplund space setting) the mixed coderivative [38] given by D F (x, y)(y ) := {x X sequences (x k, y k ) grf ( x, ȳ), x w k x, yk y with (x k, y k ) ˆN((x k, y k ); grf ), k N}. When F is single-valued, we write D F (x) and D F (x) instead of D F (x, y) and D 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 given a map h : X Y. Recall that h is said to admit a strict derivative at x, an element of L(X, Y ) denoted h (x), provided that for each v, the following holds: h(x + tv) h(x ) lim x x, t 0 t = h (x)(v), and provided the convergence is uniform for v in compact sets (this condition automatically holds if h is Lipschitz near x). 4

5 Proposition 2.1. Assume that g : X R { } is lower semicontinuous and Ω is 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 the subdifferential 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 one function is strictly differentiable near x. (v) χ Ω (x) = N(x; Ω) and if Ω is convex then the above normal cones reduce to the 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) = B X. Proposition 2.2. Assume that a map g : X Y is strictly differentiable near x then D g( x) = D g( x) = {[g ( x)] }. Here, [g ( x)] : Y X is the adjoint map to g ( x) defined by [g ( x)] (y )(x) = y (g ( x)(x)), (x, y ) X Y. Remark 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 approach to differentiability of maps, we may consider different specific coderivatives for setvalued maps generated by different normal cones to their graphs. The Mordukhovich coderivative 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 has been further developed to its full and comprehensive calculus and has been used in the study of optimal control, differential inclusions, scalar and vector optimization, economics..., see [38, 39]. It turns out that coderivative for single- or set-valued maps is 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 lower semicontinuous function presented above is termed as the G-nucleus of the G-normal cone and the G-nucleus of the G-subdifferential in [31]. For the calculus of the approximate coderivative see [33]. The approximate normal cone is contained in the Clarke normal cone. The Mordukhovich normal cone is smaller than the approximate normal cone and they coincide in finite dimensional spaces. 3. We mention that Clarke never introduced nor used any coderivative concept for either set-valued or single-valued maps, but the coderivative generated by the Clarke normal cone in the scheme of [36] as above has been used under the name Clarke s coderivative in [37]. 5

6 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, ȳ) grf. Recall that F is pseudo-lipschitz [28] around ( x, ȳ) with some modulus γ > 0 if there are neighborhoods U of x and V of ȳ such that F (x) V F (x ) + γ x x B Y, x, x U. (1) Mordukhovich proposed the term Lipschitz-like or Aubin property for this property, see [38]. Note that pseudo-lipschitz property is fundamental in nonlinear and variational analysis; it is in fact equivalent to the two other underlying properties for the inverse map F 1 known as linear openness/covering and metric regularity around ( x, ȳ). 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, ȳ). Rockafellar proved [41] that F is pseudo-lipschitz around ( x, ȳ) with the modulus γ iff there exists r > 0 such that for all x, x x + rb X and y, y ȳ + rb Y 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 plays an 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, ȳ) with the modulus γ. Then for (x, y) near ( x, ȳ) 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)b X and y ȳ + (r/3)b Y. Fix an arbitrary a ( x + rb X ) Ω. It is clear that (a, b) Λ for any b F (a). Therefore, and since b F (a) is arbitrary, d((x, y); Λ) x a + y b 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 and hence, 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) 6

7 Observe that for u Ω and u / x + rb X 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 + rb X ) Ω}. (5) Recalling that (4) holds for all a ( x + rb X ) 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 exploited certain normal compactness properties of sets and maps, which are automatic in finite dimensions while are unavoidably needed in infinite-dimensional spaces due to the natural lack of compactness therein. In what follows these properties are used in the framework of Asplund spaces, and so the given definitions are specified to this setting, see [38]. Recall that a set Ω X is sequentially normally compact (SNC) at x Ω if for any sequences x Ω k x and x w k 0 with x k ˆN(x k ; Ω), k N, we have x k 0 as k. A set-valued map F between Asplund spaces X and Y is SNC at ( x, ȳ) grf if its graph is SNC at this point. Further, we say that F is partially SNC (PSNC) at ( x, ȳ) if, sequentially, [(x k, y k ) grf ( x, ȳ), x k w 0, y k 0, (x k, y k) ˆN((x k, y k ); grf )] = [ x k 0] as k. The PSNC property is automatically implied by robust Lipschitzian behavior of set-valued and single-valued maps; in particular, when F is pseudo- Lipschitz around ( x, ȳ). 3 Efficient points of a set This section is devoted to various concepts of efficient points of a set. We recall some results from [26], namely, the unified approach to studying these points and their 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 with apex at zero (pointedness means K ( K) = {0}). A convex set Θ Y is called a base for K if 0 / clθ and K = {tθ : t R +, θ Θ}. When Θ is bounded, we say that K has a bounded base. Denote K + = {y Y y (k) 0, k K} and K +i = {y Y y (k) > 0, k K \ {0}}. 7

8 It is known that K has a base iff K +i and K has a bounded base iff intk + [50]. Nonnegative orthants in the Banach spaces R n, C [0,1], L p [0,1] and lp (1 p < ) have bases and the nonnegative orthants in L 1 [0,1], l1 have bounded bases [50]. Throughout this section, let A be a nonempty subset of Y and ā A. In this paper we consider the following concepts of efficiency and proper efficiency. Definition 3.1. We say that (i) ā is an efficient (or Pareto minimal) point of A if (A ā) ( K \ {0}) =. (ii) supposing that int K, ā is a weakly efficient point of A if (A ā) ( intk) =. (iii) ā is a strongly (or ideal) efficient point of A if A ā K. (iv) supposing that K +i, ā is a positive properly efficient point of A if there exists ϕ K +i such that ϕ(a) ϕ(ā), a A. (v) ā is a Henig global properly efficient point of A if there exists a convex cone C with apex at zero with K \ {0} intc such that (A ā) ( intc) =. (vi) supposing that K has a base Θ, ā is a Henig properly efficient point of A if there is a scalar ɛ > 0 such that clcone(a ā) ( clcone(θ + ɛb)) = {0}. (vii) ā is a super efficient point of A if there is a scalar ρ > 0 such that clcone(a ā) (B K) ρb. (viii) ā is a Benson properly efficient point of A if clcone[(a ā) + K] ( K) = {0}. The sets of efficient points defined in Definition 3.1 are denoted by Min(A), W M in(a), StrM in(a), P os(a), GHe(A), He(A), SE(A) and Be(A), respectively. We refer to [51-53] for the concepts of efficiency, weak efficiency and strong efficiency. Note that positive proper efficiency has been introduced by Hurwicz in [54], Benson proper efficiency has been presented in [55], Henig proper efficiency 8

9 and Henig global proper efficiency have been presented in [56] and super efficiency has been introduced by Borwein and Zhuang in [57]. The above definition of Henig properly efficient points can be found in [57, 58], see also [59, 60] and the notation He(A, K) is taken from [59, 60] while a slightly notation He(A, Θ) emphasizing the base Θ is used in other references. Let us recall an equivalent definition of a Henig properly 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 open cone V η, defined by V η = cone(θ + η B Y ) \ {0}. It is known that ā is a Henig properly efficient point of A w.r.t. to Θ iff there is a scalar η [0, δ[ such that (A ā) ( V η ) =. 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 and materials on proper efficiency. In the sequel, when speaking of weakly efficient points (resp. positive properly efficient points) we mean that intk (resp. K +i ) is nonempty, when speaking of Henig properly efficient points or of K +i we mean that K has a base Θ, when speaking that K has a bounded base we mean that Θ is bounded. We recall known relations among the above efficient points in the proposition below, see [51-53, 62, 63]. Proposition 3.2. (i) StrMin(A) Min(A) W Min(A). (ii) P os(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 a base. (ii) A ā is nearly K-subconvexlike. If ā is a Benson proper efficien point of A then it is a positive properly efficient point of A. Recall [64] that a nonempty set A Y is nearly K-subconvexlike if the set clcone(a + K) is convex. From now on, for the sake of convenience we make a convention that by properly efficient (point/solution) we mean positive properly efficient, Henig properly efficient, Henig global properly efficient, super efficient and Benson properly efficient (point/solution). 9

10 3.2 A unified approach to weakly, strongly, properly efficient points A brief inquiry into the matter reveals that weakly, strongly, properly efficient points in Definition 3.1 can be described through disjointness between some set and some nonempty open (not necessarily convex) cone Q. Inspired by this fact, we presented in [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 unified scheme: one first studies Q-minimal points and then derives similar results for them. Moreover, it turns out that some arguments developed for weakly efficient points are still valid for Q-minimal points without assuming the ordering cone to have nonempty interior and Q-minimal points can be characterized by a special function introduced in optimization by Hiriart-Urruty which has a nice subdifferentiality property due to the openness of Q. From now on unless otherwise specified let Q Y be an arbitrary nonempty open cone with apex at zero and different from Y. Definition 3.4. We say that ā is a Q-minimal point of A and write ā Qmin(A) if A (ā Q) = or, equivalently, (A ā) ( Q) =. Remark 3.5. Makarov and Rachkovski studied in more details some concepts of proper efficiency and introduced the notion of D-efficiency i.e. efficiency w.r.t. a family of dilating cones, see [61]. Recall that an open cone in Y is said to be a dilating cone (or a dilation) of K, or dilating K if it contains K \ {0}. Given D F(K), where F(K) is the class of families of cones dilating K, they called ā a D-minimal point of A (ā DMin(A)) if there exists C D such that (A ā) ( C) =. It has been established [61] that Henig global proper efficiency, Henig proper efficiency, super efficiency and some other concepts of proper efficiency are D-efficiency with D being appropriately chosen family of dilating cones. In contrast with D- efficiency, our concept includes also the concepts of strong efficiency and weak efficiency. We recall a result from Theorem 21.7 in [26] stating that weakly, strongly, properly efficient points of Definition 3.1 are in fact Q-minimal points with Q being appropriately chosen cones. Theorem 3.6. (i) ā StrMin(A) iff ā Qmin(A) with Q = Y \ ( K). (ii) ā W Min(A) iff ā Qmin(A) with Q = intk. (iii) ā P os(a) iff ā Qmin(A) with Q = {y Y ϕ(y) > 0} for some ϕ K +i. (iv) ā GHe(A) iff ā Qmin(A) with Q = C with C being some open pointed convex cone dilating K. 10

11 (v) ā He(A) iff ā Qmin(A) with Q = V η for some η ]0, δ[. (vi) supposing that K has a bounded base, ā SE(A) iff ā Qmin(A) with Q = V η for some η ]0, δ[. (vii) ā Be(A) iff ā Qmin(A) with Q = Y \ clcone[(a ā) + K]. 3.3 Scalarization of weakly, strongly, properly efficient points of a set First, we recall scalarization of a Q-minimal point obtained in [26]. Our scalarizing function is the signed distance function introduced by Hiriart-Urruty [49]. Recall that for a subset U of Y, this function is defined by U (y) = d(y; U) d(y; Y \ U). Theorem 3.7. ā Qmin(A) iff the function Q (. ā) attains its minimum over A at ā, that is Q (a ā) Q (ā ā) = 0, a A. (6) As a consequence of Theorems 3.6 and 3.7 we have the following scalarization for weakly, strongly, properly efficient points. Theorem 3.8. (i) ā W Min(A) iff (6) holds with Q = intk. (ii) ā StrMin(A) iff (6) holds with Q = Y \ ( K). (iii) ā P os(a) iff (6) holds with Q = {y Y ϕ(y) > 0} for some ϕ K +i. (iv) ā GHe(A) iff (6) holds with Q = C, C being an open pointed convex cone dilating K. (v) ā He(A) iff (6) holds with Q = V η for some scalar η ]0, δ[. (vi) supposing that K has a bounded base, ā SE(A) iff (6) holds with Q = V η for some scalar η ]0, δ[. (vii) ā Be(A) iff (6) holds with Q = Y \ clcone[(a ā) + K]. Remark 3.9. Hiriart-Urruty s signed distance function has been used for scalarization in vector optimization in [47, 65]. We collect some known properties of the function U from [49] in the proposition below. Proposition (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. 11

12 (iv) U (y) < 0 iff y int U, U (y) = 0 iff y bd U and U (y) > 0 iff y Y \int U. (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 cases established in [26] play an important role in formulating optimality conditions. Proposition (i) intk (0) K + \ {0}. (ii) Let C be an open convex cone dilating K. Then (iii) For η ]0, δ[, we have C (0) K +i. 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 constraint 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 and Ω X is a nonempty set. Throughout the section, let x Ω and ( x, ȳ) grf. We establish necessary and sufficient conditions for ( x, ȳ) to be a weakly, strongly or properly efficient solution of SOP and recall Bao-Mordukhovich s necessary conditions for ( x, ȳ) 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 corresponding concepts of solutions of SOP. In this paper, we restrict ourselves to global solutions. Denote F (Ω) := x Ω F (x). Definition 4.1. We say that ( x, ȳ) is an (Pareto) efficient (resp., weakly efficient, strongly efficient, positive properly efficient, Henig global properly efficient, Henig properly efficient, super efficient, Benson properly efficient and Q-minimal) solution of SOP if ȳ is an efficient (resp., weakly efficient, strongly efficient, positive properly efficient, 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 between the above efficient solutions of SOP. 12

13 4.2 Scalarization of weakly, strongly, properly efficient solutions of SOP It is easy to derive from Definition 4.1 and Theorem 3.8 the following scalarization for weakly, strongly, properly efficient solutions of SOP through the following relation Q (y ȳ) Q (ȳ ȳ) = 0, y F (Ω). (7) Theorem 4.3. Q = intk. (i) ( x, ȳ) is a weakly efficient solution of SOP iff (7) holds with (ii) ( x, ȳ) is a strongly efficient solution of SOP iff (7) holds with Q = Y \ ( K). (iii) ( x, ȳ) is a Henig global properly efficient solution of SOP iff (7) holds with Q = C, C being an open pointed convex cone dilating K. (iv) ( x, ȳ) is a Henig properly efficient solution of SOP iff (7) holds with Q = V η for some scalar η ]0, δ[. (v) supposing that K has a bounded base, ( x, ȳ) is a super efficient solution of SOP iff (7) holds with Q = V η for some scalar η ]0, δ[. (vi) ( x, ȳ) is a Benson properly efficient solution of SOP iff (7) holds with Q = Y \ clcone[(f (Ω) ȳ) + 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 normal cones in the senses of Clarke, Ioffe and Mordukhovich in the form Theorem 4.4. Suppose that 0 D F ( x, ȳ)(y ) + N( x; Ω). (8) (a) (b) The set Ω is closed. F is closed and pseudo-lipschitz around ( x, ȳ). Then (i) For ( x, ȳ) to be a Q-minimal solution of SOP, it is necessary that (8) holds with some nonzero y Q (0). (ii) For ( x, ȳ) to be a weakly efficient solution of SOP, it is necessary that (8) holds with some y K + \ {0}. (iii) For ( x, ȳ) to be a strongly efficient solution of SOP, it is necessary that (8) holds for all y K +. 13

14 (iv) For ( x, ȳ) to be a positive properly efficient solution of SOP, it is necessary that (8) holds with some y K +i. (v) For ( x, ȳ) to be a Henig global properly efficient solution of SOP when K +i, it is necessary that (8) holds with some y K +i. (vi) For ( x, ȳ) to be a Henig properly efficient solution of SOP, it is necessary that (8) holds with some y {v K +i inf θ Θ v (θ) > 0}. (vii) For ( x, ȳ) 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 inf θ Θ v (θ) > 0}. (viii) For ( x, ȳ) to be a Benson properly efficient solution of SOP, it is necessary that (8) holds with some y K +i provided that (c) Y is a separable Banach space, or Y is a reflexive Banach space and K has a base and (d) F ȳ is nearly K-subconvexlike on Ω i.e. clcone (F (Ω) ȳ + K) is convex. Proof. Our proof is motivated by that of Theorem 3.7 in [27]. (i) By Theorem 4.3, ( x, ȳ) is a Q-minimal solution of SOP iff the function Q (. ȳ) attains its minimum over F (Ω) at ȳ, that is Q (y ȳ) Q (0) = 0, y F (Ω). This means that ( x, ȳ) is a minimizer of the problem where q(x, y) = Q (y ȳ) and Minimize q(x, y) subject to (x, y) Λ, (9) Λ := {(x, y) X Y x Ω, y F (x)}. Then by the Clarke penalization, see Proposition in [29], for some scalar l > 0 large enough, ( x, ȳ) 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, ȳ) is an unconstrained local minimizer 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 subdifferentials, that is there exist y 1 q( x, ȳ) = Q (0), (x 2, y 2) l d(( x, ȳ); gr F ) N(( x, ȳ); gr F ) 14

15 and such that x 3 l d( x; Ω) N( x; Ω) (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 have y Q (0) and x 2 D F ( x, ȳ)(y ). The latter together with 0 = x 2 + x 3 and x 3 N( x; Ω) yields 0 D F ( x, ȳ)(y ) + N( x; Ω). (ii) The assertion follows from Theorem 3.6, the assertion (i) applied to Q =-intk and Proposition (iii) Let ȳ K + be arbitrarily chosen. From Definitions 3.1 and 4.1, we have y ȳ K, y F (Ω) and since ȳ K +, it follows that ȳ (y ȳ) 0, y F (Ω). Therefore, ( x, ȳ) is an unconstrained local minimizer of (9), where q(x, y) = ȳ (y ȳ). By the same arguments as in the proof of the assertion (i), we can prove that (8) holds with y = ȳ. Since ȳ K + is arbitrary, the assertion follows. (iv) Let ϕ K +i be such that ϕ(y ȳ) 0, y F (Ω). It suffices to apply the arguments of the proof of the assertion (iii). (v) Let C be an open convex cone dilating K associated to ȳ as in the definition of the Henig global properly efficient point ȳ of F (Ω). The assertions follows from Theorem 3.6, the assertion (i) applied to Q = C and Proposition (vi) Suppose that ( x, ȳ) is a Henig properly efficient solution of SOP. By Proposition 3.11, ȳ Qmin(F (Ω)) with Q = V η for some η ]0, δ[. By the assertion (i), we can find y Vȳ(0) such that (8) holds. From Proposition 3.11 we have Vη (0) {y K +i y (θ) η, θ Θ}. Since inf θ 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, ȳ) being a Benson properly efficient solution of SOP also is a positive properly efficient solution. The assertion then follows from the assertion (iv). Remark One can also prove the assertions (ii)-(viii) by using the arguments of the proof of the assertion (i) and Theorem Similar necessary conditions for a weakly efficient solution of SOP in the case F is a single-valued map have been obtained in [46]. 3. Necessary conditions in the form of Lagrange multiplier rule involving different coderivatives for a general constrained set-valued optimization problem (in short, CSOP) which includes not only a geometric constraint (the case SOP) but also a constraint G(x) K (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 15

16 (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 being additionally 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 have y (y ȳ) 0, y F (Ω). (10) Proof. By (8), we can find x 1, x 2 X such that x 1 D F ( x, ȳ)(y ), x 2 N( x; Ω) and x 1 + x 2 = 0. Since (x 1, y ) N(( x, ȳ); grf ), according to the definition of normal cone of convex analysis, we have and x 1(x x) y (y ȳ) 0, (x, y) grf x 2(x x) 0, x Ω. Summarizing two latter relations and taking account of x 1 + x 2 = 0, we obtain the desired relation (10). Lemma 4.8. Suppose that holds for some y Y and y Y. Then (i) If y K + \ {0} and intk, then y / intk. y (y) 0 (11) (ii) For a fixed y Y, if (11) holds for all y K + then y K. (iii) If y K +i and inf θ Θ y (θ) = ζ, then y / V η, where η = min{ζ/(2 y ), δ/2}. (12) Proof. (i) Suppose to the contrary that y intk or y intk. As y 0, there exists v Y such that y (v) < 0. Since y intk, there exists a scalar ρ > 0 such that y + ρv K. As y K + \ {0}, we have y ( y + ρv) 0. Hence, we obtain y (y) ρy (v) < 0, a contradiction to (11). (ii) Suppose to the contrary that y / K. Since K is a closed convex cone, a separation theorem implies the existence of a scalar t and ỹ Y \ {0} such that ỹ (y) < t ỹ (k), k K. A standard argument yields that ỹ K +, t 0 and ỹ (y) < t 0, contradicting the assumption that (11) holds for all y K +. (iii) Suppose to the contrary that y V η. By the definition of V η, there exist a scalar ρ > 0, θ Θ and b B Y such that y = ρ( θ + ηb). Taking (12) and b 1 into account we get y (y) This is a contradiction to (11). = ρy ( θ + ηb) = ρy ( θ) + η(y (b) ρ( ζ + η y b ) ρ( ζ + ζ/2) = ρζ/2 < 0. 16

17 Now we can prove Theorem 4.6. Proof. Suppose that (8) holds. By Lemma 4.7, (10) holds with some y Y chosen in correspondence with each cases (i)-(viii). We will apply Lemma 4.8 to each case to obtain desired assertions. (i) As Q is convex and y Q (0), the definition of subdifferential of convex analysis and (10) yield Q (y ȳ) Q (0) y (y ȳ) 0, y F (Ω). By Theorem 3.7, ȳ Qmin(F (Ω)) and hence, ( x, ȳ) is a Q-minimal solution of SOP. (ii) Applying Lemma 4.8(i) we get y ȳ / intk, y F (Ω), which means that ( x, ȳ) is a weakly efficient solution of SOP. (iii) Applying Lemma 4.8(ii) we get y ȳ K, y F (Ω), which means that ( x, ȳ) 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 ȳ) 0, y F (Ω), we get y ȳ / intc, y F (Ω). Thus, ( x, ȳ) is a Henig global properly efficient solution of SOP. (vi) Suppose that inf θ Θ y (θ) = ζ, then Lemma 4.8(iii) implies that y ȳ / V η, where η is the scalar defined by (12). It is clear that η ]0, δ[. Theorem 3.6 (vi) implies that ȳ is a Henig properly efficient point of F (Ω). Hence, ( x, ȳ) is a Henig global 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 CSOP have 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 obtained by 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 Suppose that (i) X and Y are Asplund spaces. (ii) (iii) The set Ω is closed. F is closed and the qualification condition D F ( x, ȳ)(0) N M ( x; Ω) = 0 (13) is satisfied, and that either F is PSNC at ( x, ȳ) or Ω is SNC at x; both the qualification condition (13) and the PSNC property of F are automatically satisfied if F is pseudo-lipschitz around ( x, ȳ). 17

18 (iv) either K is SNC at the origin or F 1 Ω if x Ω and F Ω (x) =, otherwise. is PSNC at (ȳ, x), where F Ω(x) = F (x) If ( x, ȳ) is an (Pareto) efficient solution of SOP then there exists y y = 1 such that 0 DMF ( x, ȳ)(y ) + N M ( x; Ω). K with Here, D M and N M denote the coderivative and normal cone in the sense of Mordukhovich. 5 Optimality conditions for solutions of SEP, VEP and VVI In this section, we recall concepts of solutions of SEP, VEP and VVI. Establishing an equivalence between solution of SEP and these of SOP, we derive scalaization and optimality conditions for solutions of SEP, VEP and VVI from the ones for solutions 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 2 Y 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. and We will use the following notations Φ( x, A) := x A Φ( x, x), φ( x, A) := {φ( x, x) x A} f( x)(a x) := {f( x)(x x) x A}. By Φ x and φ x we mean the maps from X to Y induced by Φ and φ, respectively, as follows Φ x (x) = Φ( x, x), φ x (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, x is a weakly efficient solution of SEP if Φ( x, x) ( intk) =, x A. 18

19 (iii) x is a strongly (or ideal) efficient solution of SEP if Φ( x, x) K, x A. (iv) supposing that K +i, x is a positive properly efficient solution of SEP if there 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 convex cone 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 SEP if there is a scalar ɛ > 0 such that clconeφ( x, A) ( clcone(θ + ɛb)) = {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 and replacing 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 efficient and strongly efficient solutions for VEP and SEP. Note that the definition of weakly efficient solutions for SEP presented in [3] is in a slightly different way: supposing that int K, x is called a weakly efficient solution to SEP if Φ( x, x) intk x A. For the concepts of efficient, weakly efficient, Henig properly efficient, Henig global properly efficient, super efficient solutions to VEP as above see [12, 15] and for the concept of Benson efficient solutions to VEP see [13]. The concepts of super efficient 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 from the concept of a positive properly efficient solution. Namely, given ψ K \ {0}, x is a ψ-efficient solution of SEP ψ(φ( x, x)) 0, x A. From now on, we make the convention that the condition 0 Φ x ( x) or, equivalently, 0 Φ( x, x) (14) holds whenever we consider SEP and the condition holds whenever we consider VEP. φ( x, x) = 0 19

20 5.2 Equivalence between solutions of SOP and solutions of SEP 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 equivalent: (i) x is a solution of SEP in some sense of Definition 5.1. (ii) ( x, 0) is a solution of SOP with F = Φ x, Ω = A in corresponding sense of Definition Scalarization of weakly, strongly, properly efficient solutions of SEP, VEP and VVI In this subsection, P is any of SEP, VEP and VVI. We derive from Theorem 3.7 scalarization for weakly, strongly, properly efficient solutions of P in the form Q (y) Q (0) = 0, y S P, (15) where S P is a set associated to P given by Φ( x, A) S P := φ( x, A) f( x)(a x) if P is SEP if P is VEP 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 = V η for some 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 = V η for some scalar η ]0, δ[. (vi) x is a Benson properly efficient solution of P iff (15) holds with Q = Y \ clcone[s P + K]. Remark 5.5. Note that scalarization for weakly efficient solution, Henig properly efficient solution, Henig global properly efficient solution and super efficient solution of VEP has been obtained in [15, 16]. 20

21 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 are expressed in terms of coderivative and normal cone of the forms 0 D Φ x ( x, 0)(y ) + N( x; A). (16) Theorem 5.6. Suppose that the set A is closed, Φ x is closed and pseudo-lipschitz around ( x, 0). Then (i) For x to be a weakly efficient solution of SEP, it is necessary that (16) holds with some y K + \ {0}. (ii) For x to be a strongly efficient solution of SEP, it is necessary that (16) holds for 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, it is 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 inf θ Θ v (θ) > 0}. (vi) For x to be a super efficient solution of SEP when K has a bounded base, it is necessary that (16) holds with some y {v K +i inf θ Θ v (θ) > 0}. (vii) For x to be a Benson efficient solution of SEP, it is necessary that (16) holds with some y K +i provided that (c) Y is a separable Banach space, or Y is a reflexive Banach space and K has a base and (d) Φ x is nearly K-subconvexlike on A i.e. clcone (Φ x (A) + K) is convex. Theorem 5.7. Suppose that the set A is closed, convex and Φ x 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 φ x being continuous on A and Lipschitz around x and replacing (16) by 0 D φ x ( x)(y ) + N( x; A), (17) we obtain corresponding necessary and (under the convexity assumption on the graph of φ x ) sufficient conditions for weakly, strongly, properly efficient solutions 21

22 of VEP. If, in addition, φ x is strictly differentiable around x, then Proposition 2.2 implies that D φ x ( x)(y ) = {[φ x] ( x)(y )} and (17) becomes 0 [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) sufficient conditions for weakly, strongly, properly efficient solutions of VVI. To see this, it suffices to note that in this case, Φ x (x) = f( x)(x x) is an affine map so all conditions of Theorems 5.6 and 5.7 imposed on Φ x are automatically satisfied and, moreover, we have D Φ x ( x, ȳ)(y ) = {f( x) (y )}. 5.5 Necessary conditions for efficient solutions of SEP, VEP and VVI Theorem 5.8. Suppose that (i) X and Y are Asplund spaces. (ii) The set A is closed. (iii) Φ x is closed and the qualification condition D Φ x ( x, 0)(0) N M ( x; Ω) = 0 (18) is satisfied, and that either Φ x is PSNC at ( x, 0) or A is SNC at x; both the qualification condition (18) and the PSNC property of Φ x are automatically satisfied if Φ x is Lipschitz-like around ( x, 0). (iv) either K is SNC at the origin or (Φ x ) 1 A is PSNC at (0, x), where (Φ x) A (x) = Φ x (x) if x A and (Φ x ) A (x) =, otherwise. If x is an (Pareto) efficient solution of SEP, then there exists y K with y = 1 such that 0 D MΦ x ( x, 0)(y ) + N M ( x; A). Proof. An immediate consequence of Theorems 5.3 and One can easily derive necessary conditions for efficient solutions of VEP and VVI. As illustration, we state necessary conditions for efficient solutions of VVI. Theorem 5.9. Suppose that (i) X and Y are Asplund spaces. 22

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