Lagrange Multipliers, Duality, and Sensitivity in Set-Valued Convex Programming via Pointed Processes

We present a new kind of Lagrangian duality theory for set-valued convex optimization problems whose objective and constraint maps are defined between preordered normed spaces. The theory is accomplished by introducing a new set-valued Lagrange multiplier theorem and a dual program with variables that are pointed closed convex processes. The pointed nature assumed for the processes is essential for the derivation of the main results presented in this research. We also develop a strong duality theorem that guarantees the existence of dual solutions, which are closely related to the sensitivity of the primal program. It allows extending the common methods used in the study of scalar programs to the set-valued vector case.


Introduction
Set-valued optimization is an expanding branch in applied mathematics that has attracted a great deal of attention in the last decades [1,13,14,23,24].This topic tackles optimization problems where the objective and/or the constraint maps are set-valued ones acting between abstract spaces.Set-valued optimization problems have been analysed according to different concepts of solution.Such optimal solutions have usually been defined by means of vector, set, or lattice approaches.
In the vector approach, the first to be developed, the concept of solution is based on the standard notion of Pareto minimal point and its many variants.This approach has been widely developed in the convex case and the corresponding literature is extensive.We refer the reader to [3,13,14,21] and the references therein.
However, solution concepts based on vector approaches are sometimes improper.In order to avoid this drawback, it is sometimes convenient to work with order relation for sets.The solution concept based on set approaches is based on a set order relation.This is obtained by extending the original preordered image vector space to its power set.This approach was introduced by Kuroiwa, see [15][16][17][18][19]. Important contributions were made later, see for example [8][9][10][11][12].
CONTACT Fernando García-Castaño Email: Fernando.gc@ua.esFinally, in lattice approaches, the concept of solution is based on a lattice structure on the power set of the image space.The corresponding infimum and supremum are sets related to the sets of weak optimal points.This view is useful for applications of set-valued approaches in the theory of vector optimization, especially in duality theory.We refer the reader to [20] for a comprehensive discussion and [7] for some subsequent extensions.
In this paper, we deal with the solution concept based on a vector approach.From this perspective, we establish a new Lagrange multiplier theorem.Then, we introduce a new kind of Lagrangian duality scheme for set-valued optimization programs whose objective and constraint maps are convex and defined between preordered normed spaces.We prove that the sensitivity of the primal program is closely related to the set of dual solutions.Although such solutions are usually continuous linear operators, the dual variables considered in this paper are pointed closed convex processes.The use of processes as dual variables is not entirely new in set-valued analysis.For instance, we find dual variables that are closed convex processes in [5][6][7]20].The processes in this work are also pointed.This property seems to improve the adaptability of the variables to the structure of convex set-valued vector problems.On the other hand, the arguments in this manuscript are direct and broadly geometric.Roughly speaking, we extend the methods used in the study of scalar programs in [22,Chapter 8] to the study of set-valued vector programs.In this way, our work includes the scalar case as a particular one.Finally, some results in our earlier paper [2] have been enhanced by those obtained in this work.
The paper is organized as follows.In Section 2 we state some preliminary terminology.In Section 3 we introduce the parametric constrained set-optimization problem (P (z)) to be analysed in the paper.Then we state a new Lagrange multiplier theorem for (P (0)).Such a result, Theorem 3.9, is stated in terms of nondominated points instead of minimal points.Hence, we do not force the primal program to reach its optimal points.In Section 4 we define a dual program.Theorem 4.2 guarantees the existence of a dual solution even if we do not assume the existence of a minimal solution in the primal program.Section 5 is devoted to the study of the sensitivity of the primal program.The formulas for sensitivity in Theorems 5.5 and 5.8 are expressed in terms of the Lagrange process introduced in [2].In Remark 6 we note that the sensitivity of the program is closely related to the solutions of the dual program.Finally, in Section 6, we present conclusions that summarize this work and we pose some open problems for further research.

Preliminaries and Notation
Let Y and Z be normed spaces with topological duals Y * and Z * .We denote by cl(A), int(A), bd(A), co(A), and cone(A) the closure, interior, boundary, convex hull, and cone hull of a set A ⊂ Y , respectively.Sometimes the parentheses will be omitted.We will consider the sum of two subsets in Y in the usual way adopting the convention Given a set-valued map F : Z ⇒ Y we may identify, in a natural way, F with its graph which is the set defined by Graph(F ) := {(z, y) ∈ Z × Y : y ∈ F (z)}.The domain of F is defined by Dom(F ) := {z ∈ Z : F (z) = ∅} and the image of F is defined by Im(F ) := ∪ z∈Dom(F ) F (z).The image by F of the set A ⊂ Dom(F ) is F (A) := ∪ a∈A F (a).A set-valued map F is said to be a process if Graph(F ) is a cone.A process F is said to be convex (resp.closed, pointed) if Graph(F ) is convex (resp.closed, pointed).We will denote by P (Z, Y ) the set of all closed convex processes F from Z into Y such that Dom(F ) = Z.Let Y + ⊂ Y be a convex cone and pick arbitrary y 1 , y 2 ∈ Y , we write y 1 ≤ y 2 if and only if y 2 −y 1 ∈ Y + .Then "≤" defines a reflexive and transitive relation (a preorder) on Y and the cone Y + is called the ordering cone on Y .A set-valued map F : C ⇒ Y defined on a non-empty convex subset C ⊂ Z, is said to be We say that a point y 0 ∈ Y is a minimal point of a set A ⊂ Y , written y 0 ∈ Min(A), if y 0 ∈ A and y 0 is nondominated by A.If Y + is pointed, then y 0 ∈ Min(A) if and only if A ∩ (y 0 − Y + ) = {y 0 }.Let us note that in the real line with the usual order minimal points become minima, and any nondominated point belonging to the closure of a set becomes its infimum.Analogously, we say that If the ordering cone Y + is solid, then we can introduce the following relation.For arbitrary y 1 , y 2 ∈ Y we write y 1 < y 2 if and only if y 2 − y 1 ∈ int(Y + ).We say that a point y 0 ∈ A is a weak minimal (resp.weak maximal) point of a set A ⊂ Y , written y 0 ∈ WMin(A) (resp.

Problem Formulation and Set-Valued Lagrange Multipliers
We begin this section by establishing the set-valued vector constrained problem (P (z)) which will be analysed throughout this paper.We next introduce the notion of nondominated point of (P (z)).Then we introduce the notion of Lagrange multiplier of (P (0)) associated with a nondominated point, and we show that such Lagrange multipliers always exist.We associate to each nondominated point y 0 of (P (0)), a non-empty set Γ y0 of pointed closed convex processes.The main result of the section, Theorem 3.9, assures that every element in Γ y0 is a Lagrange multiplier of (P (0)).
Here and subsequently we consider X, Y , and Z normed spaces.We assume that Y and Z are preordered normed spaces and that each corresponding ordering cone, Y + and Z + , is proper and solid.Let Ω ⊂ X be a convex set.Let us be given the set-valued maps F : Ω ⇒ Y and G : Ω ⇒ Z where F is Y + -convex and G is Z + -convex.These maps determine the following parametric set-optimization problem: We adopt the following notations: for every z ∈ Z we have S(z) With these notations, we obtain in addition four set-valued maps: In the terminology of scalar optimization, the map M is called the marginal function.In our setting, we will refer to M as the set-valued marginal map.An important question in connection with parametric optimization is the study of the derivative of such a map.Section 5 is devoted to such a topic (sensitivity).
Definition 3.1.Let z ∈ Z and consider the corresponding program (P (z)).(i) A pair (x z , y z ) ∈ S(z) × V (z) is called a minimizer of (P (z)) if Then y z is said to be a minimal point of (P (z)).(ii) A point y z ∈ Y is called a nondominated point of (P (z)), written y z ∈ N D(P (z)), if y z ∈ cl(V (z)) and y z is nondominated by the set V (z).
Remark 1.The notion of nondominated point is less restrictive than that of minimal point because a nondominated point is not required to be achieved at a feasible solution.This fact is significant because the optima of the problem can be chosen among the corresponding nondominated points.
The constraint in (P (z)) can be rewritten as, z ∈ G(x) + Z + , which leads us to the next notion.Definition 3.2.Let y 0 ∈ N D(P (0)).A closed convex process ∆ : Z ⇒ Y is said to be a Lagrange multiplier of (P (0)) at y 0 , if y 0 is a nondominated point of the program Min F (x) + ∆(G(x) + Z + ) such that x ∈ Ω. (
From now on and throughout the whole paper, we assume the usual Slater constraint qualification.It will appear in the statements of the main results in the paper guaranteeing that y * = 0 for every (z * , y * ) ∈ S ′ Y+ (y 0 ).

Assumption 3.4 (Slater constraint qualification
).There exists In what follows, we will make use of the set-valued maps introduced by Hamel in [5].For every (z * , y Definition 3.5.Let y 0 ∈ N D(P (0)) and (z * , y * ) ∈ S ′ Y+ (y 0 ).We define (i) The set of processes associated to (z * , y * ) by (ii) The set of processes associated to y 0 by Lemma 3.8 below states that assumption 3.4 implies Γ y0 = ∅.Then we are guaranteed that such a set is non-empty for every nondominated point of (P (0)).Lemma 3.6.Let y 0 ∈ N D(P (0)) and ∆ ∈ Γ y0 .Then ∆ is a closed, convex, and pointed process.
Let us finish showing (2).Let u ∈ ∆(G(x 0 )+Z + )∩(−Y + ).Since y 0 is a minimal point of (P [∆]) achieved at x 0 , we have that y 0 ∈ F (x 0 ) and (F ( Remark 3. In the proof of inclusion (2), we only assume that ∆ is a Lagrange multiplier.Therefore, such an inclusion still holds if we consider Lagrange multipliers of (P (0)) at y 0 which do not belong to Γ y0 .
Next, we adapt Example 4.4 in [2] to our current set-valued context to check that the proper inclusion Graph(∆) \ {(0, 0)} ⊆ int(Graph(−S (z * ,y * ) )) from Definition 3.5 becomes decisive to detect the optimal point in (P [∆]).The inclusion is directly related to the property that ∆ is pointed.Let us note that additional requirement is assumed neither on the optimal point (such as some type of proper efficiency) nor on the ordering cone.

Duality
This section is devoted to the study of a dual problem for the program (P (0)).We develop a geometric duality approach analogous to the scalar case approach.In our setting, we make the pointed closed convex processes play the same role as the linear continuous operators play in the scalar case.Optimal points of the dual problem are weak maximal points.This is an interesting feature from a practical point of view because weak maximal points can be obtained via linear scalarizations.
We define the auxiliary set-valued map Ψ : P (Z, Y ) ⇒ Y and the dual set-valued map Φ : Then the image of a process ∆ by the dual map is the set of points in the border of Ψ(∆) which are nondominated by Ψ(∆).Theorem 3.9 above states that if N D(P (0)) = ∅, then Φ(∆) = ∅ for every ∆ ∈ Γ y0 .The dual problem of (P (0)) is defined by WMax Φ(∆) such that ∆ ∈ P (Z, Y ).
The first sentence in the following result is on weak duality.However, the second one is on strong duality (based on Theorem 3.9).Theorem 4.2.Let y 0 ∈ N D(P (0)).The following statements hold.

Sensitivity Analysis
In this section, we analyse the sensitivity of (P (0)) by means of the contingent derivative.In our analysis, the concept of Lagrange process plays a crucial role.But we do not use its original definition given in [2,Definition 3.7].Instead of that, we introduce it by employing the concept of adjoint process.The adjoint of a process was introduced in [25], and it is also known as the transpose of a process [1,Definition 2.5.1].
From now on, we assume that the ordering cone Y + on the preordered normed space Y is pointed.Then, the preorder on Y induced by Y + becomes an order.
It is clear that the processes P ⋆ and Q ⋆ above are convex and closed.
Remark 4. The adjoint of a process is closely related to the set-valued maps S (z * ,y * ) introduced in Section 3. Indeed, given a convex process Q : Let us fix a point y 0 ∈ ND(P (0)).We define the process S Y+ (y 0 ) : Y * ⇒ Z * by the following abuse of notation.For every y * ∈ Y * , the image S Y+ (y 0 )(y * ) is the set of z * ∈ Z * such that (z * , y * ) belongs to the set S Y+ (y 0 ) in Definition 3.3.In this way, we use the same symbol S Y+ (y 0 ) for this process and the set in Definition 3.3.The adjoint that process appears in the following definition.Definition 5.2.Let y 0 ∈ N D(P (0)).We define the Lagrange process of (P (0)) at y 0 as the closed convex process Remark 5.In general, a Lagrange process is not necessarily a Lagrange multiplier for the same program (see [2,Example 4.4]).On the other hand, any condition (a)-(d) in the statement of Theorem 5.8 below assure that the Lagrange process considered there becomes a Lagrange multiplier.
Next, we state some terminology on tangent cones and set-valued derivatives (see [1] for further details).Next, we introduce the contingent derivative.Its usual symbol is D, and it has not to be confused with that used to denote the dual problem introduced in Section 4.
Next, our first result on sensitivity.The polar cone of a set will be used in the corresponding proof.The negative polar cone of a set Theorem 5.5.Let y 0 be a minimal point of (P (0)) and L y0 the Lagrange process of (P (0)) at y 0 .Assume that there exists Next, we introduce a derivative proposed by Shi in [26].We will use it to prove the following lemma.Definition 5.6.Let F : Z ⇒ Y be a set-valued map and (z 0 , y 0 ) ∈ Graph(F ).The S-derivative of F at (z 0 , y 0 ) is the set-valued map D S F (z 0 , y 0 ) : Z ⇒ Y defined in the following way: for any direction z ∈ Z, the point y belongs to D S F (z 0 , y 0 )(z) if and only if there exist two sequences n=1 converges to (z, y), the sequence {h n z n } ∞ n=1 converges to 0, and In what follows, we denote by S Y the unit sphere in Y .Let us recall that the marginal set-valued map M : Z ⇒ Y is defined by M (z) := Min(V (z)), ∀z ∈ Z. Lemma 5.7.Let y 0 be a minimal point of (P (0)) and L y0 the Lagrange process of (P (0)) at y 0 .Assume that the set Y + ∩ S Y is compact and that L y0 is a Lagrange multiplier of (P (0)) at y 0 .Then Min DV (0, y 0 )(z) = Min D(V +Y + )(0, y 0 )(z), ∀z ∈ Z.
In the statement of the following result, we assume domination property which is usually required in sensitivity analysis.We say that the set-valued map Theorem 5.8.Let y 0 ∈ V (0) be a minimal point of (P (0)) and L y0 be the Lagrange process of (P (0)) at y 0 .Assume the following: there exists Next, we see how our approach contains the conventional scalar case.

Conclusions
In this work, we provide a new set-valued extension of the classical Lagrange multiplier theorem for a constrained convex set-valued optimization problem.In previous approaches, the Lagrange multipliers were usually linear continuous operators, but in this manuscript, the Lagrange multipliers are pointed closed convex processes.We set a dual program whose dual variables are also pointed closed convex processes.The property of being pointed is essential for the main results in the paper.We prove that the Lagrange multipliers are solutions of the dual program.We check that the sensitivity of the problem is closely related to the set of solutions of the dual program.The arguments followed in this work are based on geometric principles and similar to those used in the scalar case.
We present some issues for further research.Since each minimal point of the primal program has associated many Lagrangian multipliers, it is of interest to determine which may be the most appropriate.So, we can pose a first question: how does the graph of a Lagrangian multiplier ∆ influence the program (P [∆]) and its corresponding solutions?On the other hand, pointed closed convex processes in set-valued analysis can be interpreted as the natural analogues to sublinear functions in the scalar case.

Definition 5 . 3 .
Let Y be a normed space, A ⊂ Y a non-empty set, y ∈ cl(A), and d the metric given by the norm on Y .The contingent cone to A relative at y, T A (y), is the cone defined by T A (y) := v ∈ Y : lim inf