Scalarization approach for approximation of weakly efficient solutions of set-valued optimization problems

Abstract: In this paper, by using the scalarization method and normal subdifferential for set-valued maps, we consider an extension of Minty variational-like inequalities and obtain some relations between their solutions and set-valued optimization problems. An existence result for generalized Minty variational-like inequalities and set-valued optimization problems is also given. Moreover, the concept of approximate efficient solutions due to Kutateladze is investigated and by the Tammer–Weidner nonlinear functional, we characterize them for cone constrained set-valued optimization problems.


Introduction
Variational inequalities are identified either in the form presented by the Minty (1967) or in the form by Stampacchia (1960). Giannessi (1980) was the first author who obtained the equivalence between solutions of a Minty variational inequality and efficient solution of differentiable, convex optimization problem. Afterward, some authors focused their works to nonsmooth functions (see, e.g. Al-Homidan & Ansari, 2010;Alshahrani, Ansari, & Al-Homidan, 2014;Chen & Huang, 2012;Yang & Yang, 2006). Al-Homidan and Ansari (2010) obtained these results for invex functions with Clarke's generalized directional derivative. By using the scalarization method, Santos, Rojas-Medar,

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The concept of variational inequalities has wide applications in many problems such as economics, transportation, optimization, and engineering sciences. In this paper, by using the scalarization method, we establish some relations between them and optimization problems. Also, we obtain a kind of penalization for approximate efficient solutions of a cone-constrained optimization problem.
Ruiz-Garzáon, and Rufiáan-Lizana (2008) studied scalarized variational-like inequalities presented in terms of Clarke's generalized directional derivative and showed that the set of their solutions is equal to weak efficient solutions set. In Alshahrani et al. (2014) further extended results in Santos et al. (2008) to establish some existence results for solutions of nonsmooth variational-like inequalities under a dense pseudomonotonicity assumption. Moreover, Oveisiha and Zafarani (2014) considered generalized Stampacchia variational-like inequalities and set-valued optimization problems and also obtained some characterizations of the solution sets of pseudoinvex extremum problems. Very recently, Ruiz-Garzón, Osuna-Gómez, Rufián-Lizana, and Hernández-Jiménez (2015) introduced a new concept of generalized invexity for continuous-time programming problems and show the equivalence of efficient and weak efficient solutions for the multiobjective continuous-time programming problem, and solutions of Stampacchia and Minty variational-like type inequality problems.
The notion of approximate solutions has been defined in several ways (see, e.g. Helbig, 1992;Kutateladze, 1979;Tanaka, 1994;White, 1986). The first concept was introduced by Kutateladze (1979) and has been used to construct approximate Kuhn-Tucker type conditions, approximate duality theorems and so forth, see, for instance (Bednarczuk & Przybyla, 2007;Bolintinéanu, 2001;Chen, Huang, & Yang, 2005;Göpfert, Riahi, Tammer, & Zălinescu, 2003;Gutiérrez, Jiménez, & Novo, 2008;Vályi, 1987). Rong and Wu (2000) considered the notion of -weak efficient solution and studied Lagrangian multiplier and duality properties for set-valued optimization problems with cone subconvexlike mappings based on the separation theorems of convex sets. In Gutiérrez, Jiménez, and Novo (2011) introduced a new concept of -efficient point based on set-valued mapping. They obtained some existence results and properties on the behavior of these approximate efficient and weak efficient solutions. Very recently, Huerga, Gutiérrez, Jiménez, and Novo (2015), studied the concept of -efficiency, defined by Kutateladze, and proved that the limit of them, when the precision tends to zero is the set of weak efficient solutions for single vector-valued optimization problems and also obtained Kuhn-Tucker optimality conditions for -efficient solutions of nondifferentiable convex Pareto multiobjective problems with inequality constraints.
In this paper, we consider generalized variational-like inequalities in terms of normal subdifferential for set-valued maps. By introducing a scalarized Minty variational-like inequalities (MVLI), we show that any solution of a scalarized set-valued optimization problems (SOP) is also a solution of Minty variational-like inequalities under standard assumptions and that the inverse implications hold under the additional generalized K-convexity assumption, where K is an ordering cone of the considered image space. Also, by using the Tammer-Weidner nonlinear scalarization functional a characterization of -efficient solutions of cone constrained set-valued optimization problems is given. The paper is organized as follows: in Section 2, some basic definitions and preliminary results are presented. Section 3 is devoted to study several relationships between scalarized Minty variational-like inequalities and set-valued optimization problems. Also, we obtain an existence result for (MVLI) and set-valued optimization problems. In Section 4, we follow an approach presented in Gutiérrez et al. (2011), Huerga et al. (2015. We state a kind of penalization scheme for approximate solutions of a cone constrained set-valued optimization problems. Finally, in Section 5, some conclusions are presented, which summarize this work.

Preliminaries
Let X be a Banach space and X * be its topological dual space. The norm in X and X * will be denoted by ‖ ⋅ ‖. We denote by ⟨., .⟩, [x, y] and ]x, y[ the dual pair between X and X * , the line segment for x, y ∈ X, and the interior of [x, y], respectively. Now, we recall some concepts of subdifferentials and coderivatives that we need in next sections.
Definition 2.1 Mordukhovich (2006) Let X be a Banach space, Ω be a nonempty subset of X, x ∈ Ω and ≥ 0. The set of -normals to Ω at x is If = 0, the above set is denoted by N (x; Ω) and called regular normal cone to Ω at x. Let x ∈ Ω, the basic normal cone to Ω at x is Definition 2.2 Mordukhovich (2006) Let X be a Banach space and :X →R be finite at x ∈ X. The basic (limiting, Mordukhovich) subdifferential due to Mordukhovich (2006) of at x is defined by Given a set-valued mapping F:X ⇉ Y between Banach spaces with the range space Y partially ordered by a nonempty, closed and convex cone K. Denoting the ordering relation on Y by "≤", we have Now, we present some definitions and results about coderivatives and subdifferentials of set-valued mappings.
Definition 2.3 Mordukhovich (2006) Let F:X ⇉ Y be a set-valued mapping between Banach spaces and (x,ȳ) ∈ grF. Then the Fréchet coderivative of F at (x,ȳ) is the set-valued mapping � D * F(x,ȳ):Y * ⇉ X * given by and furthermore, the normal coderivative of F at (x,ȳ) is the set-valued mapping D * N F(x,ȳ):Y * ⇉ X * given by Definition 2.4 Bao and Mordukhovich (2007) Let F:X ⇉ Y be a set-valued mapping. Then, the epigraphical multifunction  F :X ⇉ Y is defined by The Fréchet and normal subdifferentials of F at the point (x,ȳ) ∈ epiF in the direction y * ∈ Y * are, respectively, defined by Definition 2.5 Mordukhovich (2006) Let F:Ω ⊂ X ⇉ Y with domF ≠ ∅ and B Y be the closed unit ball of Y.
(i) F is said to be Lipschitz around x ∈ domF iff there are a neighborhood U of x and ≥ 0 such that (ii) F is said to be epi-Lipschitz around x ∈ domF iff  F is Lipschitz around this point.
Let K be a closed, convex and pointed cone in Y and denote the positive polar cone of K bŷ The next object is the marginal function associated with a set-valued mapping. Given F:X ⇉ Y and y * ∈ Y * . We associate to F and y * a marginal function f y * :X → ℝ ∪ {±∞} and the minimum set Throughout this paper, we suppose that grF is closed, and for all x ∈ domF and y * ∈ K + , M y * (x) is nonempty.

Lemma 2.6 Oveisiha and Zafarani (2013) Suppose that F:Ω ⊂ X ⇉ Y is a set-valued map and
x ∈ domF. If F is epi-Lipschitz around x and y * ∈ K + , then the scalar-function f y * is locally Lipschitz at x.
The next theorem gives some relations between normal subdifferential and normal coderivative of F and limiting subdifferential of its marginal functions. (see also Theorem 3.4 and Corollary 3.5 in Oveisiha & Zafarani, 2013) Theorem 2.7 Oveisiha and Zafarani (2013) Let X, Y be Asplund spaces, F:X ⇉ Y and y * ∈ K + . Suppose that x ∈ domF and ȳ ∈ M y * (x).
Definition 2.9 (see Oveisiha & Zafarani, 2013) Let Ω ⊂ X be an invex set with respect to and F:Ω ⊂ X ⇉ Y be a set-valued mapping. Then: (1) F is said to be K-preinvex with respect to on Ω if for any x 1 , x 2 ∈ Ω and ∈ [0, 1], one has Remark 1 If F = f :X → ℝ is a real-valued function and K = [0, +∞[, then the above definition reduces to preinvexity, invexity, weak invexity, and invariant monotonicity, respectively, for real-valued functions, that has been investigated in Jabarootian and Zafarani (2006), Soleimani-damaneh (2010). Jabarootian and Zafarani (2009) A mapping F:Ω ⇉ Y from an invex set Ω with respect to to an ordered Banach space is said to enjoy Condition A if Condition C. Mohan and Neogy (1995) Let :X × X → X. Then, for any x, y ∈ X, ∈ [0, 1] Remark 2 By some computation, we can see that if Condition C holds, then for any x 1 , x 2 ∈ X and Let Ω be a convex subset of a vector space X. Then a mapping F: Lemma 2.10 (see e.g. Fakhar & Zafarani, 2005) Let Ω be a nonempty and convex subset of a Hausdorff topological vector space X. Suppose that Γ,Γ:Ω ⇉ Ω are two set-valued mappings such that the following conditions are satisfied:

(MVLI) and set-valued optimization problems
This section is devoted to get some relations between scalarized Minty variational-like inequalities and scalaraized set-valued optimization problems.
Suppose that F:X ⇉ Y is a set-valued map between Banach spaces. We consider the following set-valued optimization problem Definition 3.1 (i) Chen, Huang, and Yang (2005) A point x is said to be a weakly efficient solution of problem (1) iff there exists ȳ ∈ F(x) such that (ii) x is said to be a scalaraized solution of problem (1) (x is a solution of (SOP)) iff, for any y * ∈ K + �{0}, there exists ȳ ∈ F(x) such that Suppose that :X × X → X. Now, we consider the following scalarized Minty variational-like inequality (MVLI): Find a vector x ∈ Ω such that, for any x ∈ Ω and y * ∈ K + �{0}, there exist y ∈ M y * (x) and x * ∈ F(x, y)(y * ) such that F(x 1 ) ⊂ F(x 2 + (x 1 , x 2 )) + K, for all x 1 , x 2 ∈ Ω.
When F = f :X → Y is a vector-valued function, Santos et al. (2008) and Alshahrani et al. (2014) by using a similar way, considered nonsmooth Stampacchia variational-like inequality in which the limiting nonconvex subdifferntials was replaced by the convex Clarke's generalized directional derivative.

Lemma 3.3 Oveisiha and Zafarani (2014) Every solution of (SOP) is a weakly efficient solution of problem (1).
Theorem 3.4 Let X be an Asplund space, Ω ⊆ X be an invex set and F:Ω ⊆ X ⇉ Y. If F is weakly K-invex and x ∈ Ω is a solution of (SOP), then it is a solution of (MVLI).
Proof Assume that x is a solution of (SOP). Suppose to the contrary that x is not a solution of (MVLI). Hence, there exist x ∈ Ω and y * ∈ K + �{0} such that for all y ∈ M y * (x) and x * ∈ F(x, y)(y * ). Now, weak K-invexity of F implies that, there exists x * ∈ F(x, y)(y * ) such that Relations (3.1) and (3.2) contradicts that x is a solution of (SOP). ✷

Theorem 3.5 Let X be an Asplund space, Ω ⊆ X be an invex set and F:Ω ⊆ X ⇉ Y be epi-Lipschitz. Suppose that F satisfies Condition A and is K-invex, satisfies Condition C and x is a solution of (MVLI), then it is a solution of (SOP) and hence, a weakly efficient solution of problem (1).
Proof Assume that x ∈ Ω is a solution of (MVLI). Suppose to the contrary that x is not a solution of (SOP). Hence, there exists y * ∈ K + �{0} such that for all ȳ ∈ F(x) for a y ∈ F(x) that x ∈ Ω. Set x(t) =x + t (x,x). Choose t 0 ∈]0, 1[ arbitrary. Since F is epi-Lipschitz, by Lemma 2.6, f y * is Lipschitz on Ω. Now, by using mean value inequality for limiting subdifferential (Corollary 3.51 in Mordukhovich (2006)), there exist t 1 ∈]0, t 0 ] and 1 ∈ M f y * (x + t 1 (x,x)) such that Since F is K-invex and epi-Lipschitz, Theorem 2.7 implies that f y * is invex. Now, by a similar proof as that of Lemma 3.2 in Jabarootian and Zafarani (2006), invexity of f y * implies its preinvexity. Hence, we obtain (3.5) f y * (x + t 0 (x,x)) − f y * (x) ≤ t 0 (f y * (x) − f y * (x)).
Then, the normal subdifferential of F is which r is a positive real number. Then satisfies Condition C, F satisfies Condition A and is K-invex with respect to . Hence, by some computation we can see that all assumptions of Theorem 3.5 are fulfilled and x = 0 is a solution of (MVLI), therefore it is a solution of (SOP) and weakly efficient solution of problem (1).
Here, we obtain an existence theorem for the solution of (MVLI) and therefore a weak efficient solution of problem (1).

Then, one has
Proof Since the proof is direct, it is omitted. ✷ For normal subdifferential, we need the following condition to get an existence result for (MVLI).

Condition D.
Let F:X ⇉ Y and 0 ≠ y * ∈ K * . Then, for any x ∈ domF and ȳ 1 ,ȳ 2 ∈ M y * (x), we have Theorem 3.8 Let F:X ⇉ Y be K-invex and satisfy Condition D. Assume that the following conditions are satisfied: (1) is affine and continuous in the first argument and skew.
(2) There are a nonempty compact set M ⊂ X and a nonempty compact convex set B ⊂ X such that for each x � ∈ X�M, there exists x ∈ B and y * ∈ K + �{0} such that for any y ∈ M y * (x) and Then, (MVLI) has a solution. Also, the set of (MVLI) solutions is compact.
Proof Define two set-valued mappings Γ,Γ:X ⇉ X by for each x ∈ X. Γ(x) and Γ (x) are nonempty because they contain x. The proof is divided in the following steps.
(i) Γ is a KKM mapping on X. Suppose that Γ is not a KKM mapping. Then, there exist {x 1 , for each i = 1, … , m. Therefore, for any y 0 ∈ M y * (x 0 ) and x * ∈ F(x 0 , y 0 )(y * ), one has which yields a contradiction. Hence, Γ is a KKM mapping.
(iv) From condition 2, there exists a nonempty compact convex set B, such that cl �⋂ x∈B Γ(x) � is compact.
(v) Thus, all of the conditions of Lemma 2.10 are fulfilled by mapping Γ. Therefore, Hence, there exists x such that for any x ∈ X and y * ∈ K + �{0} there exist y ∈ M y * (x) and x * ∈ F(x, y)(y * ) such that Thus, (MVLI) has a solution. From (iii), Γ is closed valued and therefore, the set of solutions of (MVLI), i.e. ⋂ x Γ(x) is closed. Now, from condition 2, the set of solutions must be contained in the compact set M, hence it is compact. ✷

Approximation of weakly efficient solutions
In this section, we present the concept of approximate efficiency for set-valued maps that is a generalization of the same notion due to Kutateladze (1979).
Definition 4.1 Let v ∈ Y�{0} and ≥ 0. It is said that (x 0 , y 0 ) ∈ grF is an v-efficient solution of problem (1), denoted by (x 0 , y 0 ) ∈ WE(F, Ω, v, K), if we have When = 0, we get the definition of weakly efficient solution and denote WE(F, Ω, K) to be all weakly efficient points associated to the problem (1). We are using the lim sup →̄W E(F, Ω, v, K) to be the upper limit of the set-valued map ↦ WE (F, Ω, v, K) in the Painlevé-Kuratowski sense (see Mordukhovich, 2006).
In the next theorem, we present some properties of approximate efficient solutions.
Theorem 4.2 Assume that q ∈ intK. The following properties hold.
Also, it is clear that and therefore WE(F, Ω,̄q, K) is closed. Now, we show that if 0 >̄, then Suppose that (x i , y i ) → (x, y), with (x i , y i ) ∈ WE(F, Ω, i q, K) and i →̄. Then, there exists i 0 such that (x i , y i ) ∈ WE(F, Ω, 0 q, K) for any i ≥ i 0 . Since WE(F, Ω, 0 q, K) is closed, it follows that (x, y) ∈ WE(F, Ω, 0 q, K).
Finally, by using part (2), we obtain that and the proof is complete. ✷

Corollary 4.3 Assume that grF is closed and consider q ∈ intK. Then
Now, consider the following scalarization mapping :Y → ℝ and ≥ 0. Set Notice that (4.1) is the set of optimal solutions with error of the set-valued optimization problem •F.
The first part of the next proposition gives a sufficient condition for approximate efficient solutions of problem (1) by scalarization and the second part a necessary condition. For each y 0 ∈ F(Ω), let Proposition 4.4 Consider v ∈ Y�(− intK) and ≥ 0.
Proof Because the proof follows directly from definitions, it is omitted. ✷
(2) If we want to use the both parts of Proposition 4.4, we need a mapping :Y → ℝ such that where v ∉ − intK.
Now, by using the Tammer-Weidner nonlinear separation functional (Gerth & Weidner, 1990) satisfy this property and we denote it by e :Y → ℝ defined by e ∈ intK and The main properties of the scalar function e are given in the next lemma.
Theorem 4.6 Consider v ∈ Y�(− intK), ≥ 0 and the mapping :Y → ℝ, (y) = e (y + v), for all y ∈ Y. Then Proof By using Lemma (4.5), we have and by Proposition 4.4, we deduce that Also, notice that (0) = e ( v) = e (v), and the proof is complete. ✷ Let Z be a real locally convex Hausdorff topological linear space, D ⊂ Z be a proper convex cone with nonempty topological interior and consider a mapping G:X ⇉ Z and the set Ω D = {x ∈ X:G(x) ∩ −D ≠ �}. Now, we will characterize the q-efficient solutions of the set-valued optimization problem where q ∈ intK, F is K-preinvex and G is D-preinvex and the Slater constraint qualification is satisfied, i.e. there exists x ∈ Ω D such that G(x) ∩ − intD ≠ �. (1) We have that z 0 ∈ G(x 0 ) ∩ −D.
(2) Consider q ∈ intK and suppose that F, G are K-preinvex and D-preinvex, respectively, with respect to the same and the Slater constraint qualification is satisfied. Then that z 0 ∈ G(x 0 ) ∩ −D.