Multiobjective Optimization, Scalarization, and Maximal Elements of Preorders

and Applied Analysis 3 (2) weakly Pareto optimal with respect to the function u = (u1, . . . , um) : X 󳨃→ R if for no x ∈ X it occurs that ui(x0) < ui(x) for all i ∈ {1, . . . , m}. Definition 3. The set of all (weakly) Pareto optimal elements with respect to the function u = (u1, . . . , um) : X 󳨃→ R will be denoted byXPar u (XwPar u , respectively). It is clear that XPar u ⊂ XwPar u for every positive integer m, every nonempty set X, and every function u = (u1, . . . , um) : X 󳨃→ R. Definition 4. Consider themultiobjective optimization problem (1). Then we introduce the preorders ≾u and ≾wu on X defined as follows for all x, y ∈ X: (1) x≾uy ⇔ [ui(x) ≤ ui(y) for all i ∈ {1, . . . , m}]. (2) x≾wuy ⇔ [(ui(x) = ui(y) for all i ∈ {1, . . . , m}) or (ui(x) < ui(y) for all i ∈ {1, . . . , m})]. Remark 5. Notice that the indifference relation ∼u and the strict part ≺u of the preorder ≾u, as well as the indifference relation ∼wu and the strict part ≺wu of the preorder ≾wu , are defined as follows, for all x, y ∈ X: x∼uy ⇐⇒ x∼wuy ⇐⇒ [ui (x) = ui (y) ∀i ∈ {1, . . . , m}] , x≺ux ⇐⇒ [ui (x) ≤ ui (y) ∀i ∈ {1, . . . , m}] and there exists i ∈ {1, . . . , m} such that ui (x) < ui (y) , x≺wuy ⇐⇒ [ui (x) < ui (y) ∀i ∈ {1, . . . , m}] . (7) Definition 6. Apreorder≾ on a topological space (X, τ) is said to be (1) upper semiclosed if i(x) = {z ∈ X | x ≾ z} is a closed subset ofX for every x ∈ X; (2) upper semicontinuous if l(x) = {z ∈ X | z ≺ x} is an open subset ofX for every x ∈ X. While it is guaranteed that a preorder ≾ on a compact topological space (X, τ) has a maximal element provided that ≾ is either upper semiclosed (seeWard Jr. [23,Theorem 1]) or upper semicontinuous (see the theorem in Bergstrom [24]), a characterization of the existence of a maximal element for a preorder on a compact topological space (X, τ)was presented by Rodŕıguez-Palmero and Garćıa-Lapresta [16]. Definition 7 (see Rodŕıguez-Palmero andGarćıa-Lapresta [16, Definition 4]). A preorder ≾ on a topological space (X, τ) is said to be transfer transitive lower continuous if for every element x ∈ X which is not a maximal element of ≾ there exist an element y ∈ X and a neighbourhoodN(x) of x such that y ≺ z implies thatN(x) ≺ z for all z ∈ X. Theorem 8 (see Rodŕıguez-Palmero and Garćıa-Lapresta [16, Theorem 3]). A preorder ≾ on a compact topological space (X, τ) has a maximal element if and only if it is transfer transitive lower continuous. We recall that a real-valued function u on a topological space (X, τ) is said to be upper semicontinuous if u(] − ∞, α[) = {x ∈ X: u(x) < α} is an open set for all α ∈ R. A popular theorem guarantees that an upper semicontinuous real-valued function attains its maximumon a compact topological space. As usual, for a real-valued function u on a nonempty set X, we denote by argmax u the set of all the points x ∈ X such that u attains its maximum at x (i.e., argmax u = {x ∈ X: u(z) ≤ u(x) for all z ∈ X}). 3. Existence of Maximal Elements and Pareto Optimality A finite familyU = {u1, . . . , um} of real-valued functions on a nonempty setX gives rise to a preorder ≾ onXwhich admits precisely the (Richter-Peleg) multiutility representation U. It is easy to relate the maximal elements of such a preorder ≾ to the solutions of the associated multiobjective optimization problem (1). Theorem 9. Let ≾ be a preorder on a setX. Then the following statements hold: (1) If ≾ admits a finite multiutility representationU = {u1, . . . , um} thenX u = X M. (2) If ≾ admits a finite Richter-Peleg multiutility representation U = {u1, . . . , um} thenX u = X M. Proof. Assume that the preorder ≾ on X admits a finite (Richter-Peleg) multiutility representation U = {u1, . . . , um}. In order to show that XPar u ⊂ X M (XwPar u ⊂ X M), consider, by contraposition, an element x0 ∈X M. Then there exists an element x ∈ X such that x0 ≺ x, or equivalently ui(x0) ≤ ui(x) for all i ∈ {1, . . . , m} with an index i ∈ {1, . . . , m} such that ui(x0) < ui(x) (respectively, ui(x0) < ui(x) for all i ∈ {1, . . . , m}). Then we have that x0 is not (weakly) Pareto optimal. In a perfectly analogous way it can be shown that X M ⊂ XPar u (X M ⊂ XwPar u ). Hence, the proof is complete. The following proposition is an immediate consequence of Definition 4. Proposition 10. Consider the multiobjective optimization problem (1). Then U = {u1, . . . , um} is a finite (Richter-Peleg) multiutility representation of the preorder≾u (≾wu , respectively). From Theorem 9 and Proposition 10, we immediately arrive at the following proposition. 4 Abstract and Applied Analysis Proposition 11. Consider the multiobjective optimization problem (1). The following conditions are equivalent on a point x0 ∈ X: (i) x0 is (weakly) Pareto optimal with respect to the function u = (u1, . . . , um). (ii) x0 is maximal with respect to the preorder ≾u (≾wu ) on X. 4. Multiobjective Optimization on


Introduction
It is very well known that multiobjective optimization (see, e.g., Miettinen [1] and Ehrgott [2]) allows choosing among various available options in the presence of more than one agent (or criterion), and therefore it represents a popular and important tool which appears in many different disciplines.This is the case, for example, of design engineering (see, e.g., Das [3] and Pietrzak [4]), portfolio selection (see, e.g., Xidonas et al. [5]), economics and risk-sharing (see, e.g., Chateauneuf et al. [6] and Barrieu an Scandolo [7]), and insurance theory (see, e.g., Asimit et al. [8]).
It should be noted that Pareto optimality can be also considered by starting from a family {≾  } ∈{1,...,} of not necessarily total preorders on a set  (see, e.g., d' Aspremont and Gevers [11]).
In this paper we approach the multiobjective optimization problem (1) The consideration that an element  0 ∈  is a (weak) Pareto optimal solution to problem (1) if and only if  0 ∈  is a maximal element for the preorder ≾ u (≾  u , respectively) and the observation that the function u = ( 1 , . . .,   ) :   → R  is a (Richter-Peleg) multiutility representation of the preorder ≾ u (≾  u , respectively) allow us to present various results concerning the existence of solutions to the multiobjective optimization problem, also in the classical case when the design space is a compact topological space.We recall that the concept of a (finite) multiutility representation of a preorder was introduced and studied by Ok [12] and Evren and Ok [13], while Richter-Peleg multiutility representations were introduced by Minguzzi [14] and then studied by Alcantud et al. [15].
The consideration of a compact design space allows us to use classical results concerning the existence of maximal elements for preorders on compact spaces (see Rodríguez-Palmero and García-Lapresta [16] and Bosi and Zuanon [17]).We also address the scalarization problem by using classical results in Decision Theory related to potential optimality of maximal elements (see Podinovski [18,19]).In particular, we refer to a classical theorem of White [20], according to which every maximal element for a preorder is determined by maximizing an order-preserving function (provided that an order-preserving function exists).In particular, we show that when considering the multiobjective optimization problem (1) in order to determine the weak Pareto optimal solution, this problem can be reformulated as an equivalent one in a such a way that every weak Pareto optimal solution is determined by maximizing an objective function.
It should be noted that the results presented are fairly general, and we do not impose any restrictions neither to the choice set , which usually is assumed to coincide with R  , nor to the real-valued functions   that are usually assumed to be concave in the literature.

Notation and Preliminaries
Let  be a nonempty set (decision space) and denote by ≾ a preorder (i.e., a reflexive and transitive binary relation) on .If in addition ≾ is antisymmetric, then it is said to be an order.As usual, ≺ denotes the strict part of ≾ (i.e., for all ,  ∈ ,  ≺  if and only if ( ≾ )   ( ≾ )).Furthermore, ∼ stands for the indifference relation (i.e., for all ,  ∈ ,  ∼  if and only if ( ≾ ) and ( ≾ )).We have that ∼ is an equivalence relation on .We denote by ≾ |∼ the quotient order on the quotient set  |∼ (i.e., for all ,  ∈ , []≾ |∼ [] if and only if  ≾ , where [] = { ∈ :  ∼ } is the indifference class associated with  ∈ ).
For every  ∈ , we set Given a preordered set (, ≾), a point  0 ∈  is said to be a maximal element of  if for no  ∈  it occurs that  0 ≺ .
In the sequel we shall denote by  ≾  the set of all the maximal elements of a preordered set (, ≾).Please observe that   can be empty.
Definition 6.A preorder ≾ on a topological space (, ) is said to be While it is guaranteed that a preorder ≾ on a compact topological space (, ) has a maximal element provided that ≾ is either upper semiclosed (see Ward Jr. [23, Theorem 1]) or upper semicontinuous (see the theorem in Bergstrom [24]), a characterization of the existence of a maximal element for a preorder on a compact topological space (, ) was presented by Rodríguez-Palmero and García-Lapresta [16].
Definition 7 (see Rodríguez-Palmero and García-Lapresta [16,Definition 4]).A preorder ≾ on a topological space (, ) is said to be transfer transitive lower continuous if for every element  ∈  which is not a maximal element of ≾ there exist an element  ∈  and a neighbourhood N() of  such that  ≺  implies that N() ≺  for all  ∈ .
Theorem 8 (see Rodríguez-Palmero and García-Lapresta [16,Theorem 3]).A preorder ≾ on a compact topological space (, ) has a maximal element if and only if it is transfer transitive lower continuous.
We recall that a real-valued function  on a topological space (, ) is said to be upper semicontinuous if  −1 (] − ∞, [) = { ∈ : () < } is an open set for all  ∈ R. A popular theorem guarantees that an upper semicontinuous real-valued function attains its maximum on a compact topological space.
As usual, for a real-valued function  on a nonempty set , we denote by arg max  the set of all the points  ∈  such that  attains its maximum at  (i.e., arg max  = { ∈ : () ≤ () for all  ∈ }).
From Theorem 9 and Proposition 10, we immediately arrive at the following proposition.