# Multiobjective Optimization, Scalarization, and Maximal Elements of Preorders.

1. Introduction

It is very well known that multiobjective optimization (see, e.g., Miettinen  and Ehrgott ) 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  and Pietrzak ), portfolio selection (see, e.g., Xidonas et al. ), economics and risk-sharing (see, e.g., Chateauneuf et al.  and Barrieu an Scandolo ), and insurance theory (see, e.g., Asimit et al. ).

The multiobjective optimization problem (MOP) is usually formulated by means of the standard notation (needless to say, this formulation of the multiobjective optimization problem is equivalent, "mutatis mutandis," to [minx.sub.x[member of]X][[f.sub.1] (x), ..., [f.sub.m](x)] = [min.sub.x[member of]X]f(x), m [greater than or equal to] 2; we use the approach with the maximum for the sake of convenience):

[mathematical expression not reproducible], (1)

where X is the choice set (or the design space), [u.sub.i] is the decision function (in this case a utility function) associated with the ith individual (or criterion), and u : X [??] [R.sup.m] is the vector-valued function defined by u(x) = ([u.sub.1] (x),..., [u.sub.m](x)) for all x [member of] X.

An element [x.sub.0] [member of] X is a (weak) Pareto optimal solution to problem (1), for every x [member of] X, if [u.sub.i]([x.sub.0]) [less than or equal to] [u.sub.i](x) for every i [member of] {1,..., m}; then [u.sub.i]([x.sub.0]) = [u.sub.i](x) for every index i (respectively, for every x [member of] X, if [u.sub.i]([x.sub.0]) [less than or equal to] [u.sub.i](x) for every i [member of] {1,..., m}; then [u.sub.i]([x.sub.0]) = [u.sub.i](x) for at least one index i). In this case, the point [x.sub.0] [member of] X is said to be (weakly) Pareto optimal or a (weakly) efficient point for (MOP). Usually, X is a subset of [R.sup.n] and concavity restrictions are posed on the functions [u.sub.i] (see, e.g., Ehrgott and Nickel ). In this case, an appropriate scalarized problem can be considered to determine Pareto optimal solutions (see Miettinen [1, Theorems 3.4.5 and 3.5.4]). Further, robust multiobjective optimization has been also considered in the literature (see, e.g., Bokrantz and Fredriksson ).

It should be noted that Pareto optimality can be also considered by starting from a family [{[[??].sub.i]}.sub.i[member of]{1, ..., m}] of not necessarily total preorders on a set X (see, e.g., d'Aspremont and Gevers ).

In this paper we approach the multiobjective optimization problem (1) by referring to the preorders which are naturally associated with this problem. This means that, for determining the Pareto optimal solutions, we introduce the preorder [[??].sub.u] on X defined, for all x,y [member of] X, by

[mathematical expression not reproducible], (2)

and, for determining the weak Pareto optimal solutions, we refer to the preorder [[??].sup.w.sub.u] on X defined, for all x, y [member of] X, by

[mathematical expression not reproducible]. (3)

The consideration that an element [x.sub.0] [member of] X is a (weak) Pareto optimal solution to problem (1) if and only if [x.sub.0] [member of] X is a maximal element for the preorder [mathematical expression not reproducible], respectively) and the observation that the function u = ([u.sub.1], ..., [u.sub.m]): X [??] [R.sup.m] is a (Richter-Peleg) multiutility representation of the preorder [mathematical expression not reproducible], 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  and Evren and Ok , while Richter-Peleg multiutility representations were introduced by Minguzzi  and then studied by Alcantud et al. .

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 Rodriguez-Palmero and Garcia-Lapresta  and Bosi and Zuanon ). 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 , 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 X, which usually is assumed to coincide with [R.sup.n], nor to the real-valued functions [u.sub.i] that are usually assumed to be concave in the literature.

2. Notation and Preliminaries

Let X be a nonempty set (decision space) and denote by [??] a preorder (i.e., a reflexive and transitive binary relation) on X. If in addition [??] is antisymmetric, then it is said to be an order. As usual, [??] denotes the strict part of [??] (i.e., for all x,y [member of] X, x [??] y if and only if (x [??] y) and not (y [??] x)). Furthermore, ~ stands for the indifference relation (i.e., for all x,y [member of] X, x ~ y if and only if (x [??] y) and (y [??] x)). We have that ~ is an equivalence relation on X. We denote by [??] the quotient order on the quotient set [mathematical expression not reproducible] is the indifference class associated with x [member of] X).

For every x [member of] X, we set

[mathematical expression not reproducible]. (4)

Given a preordered set (X, [??]), a point [x.sub.0] [member of] X is said to be a maximal element of X if for no z [member of] X it occurs that [x.sub.0] [??] z. In the sequel we shall denote by [X.sup.[??].sub.M] the set of all the maximal elements of a preordered set (X, [??]). Please observe that [X.sub.M] can be empty.

Denote by m the incomparability relation associated with a preorder [mathematical expression not reproducible].

We recall that a function [mathematical expression not reproducible] is said to be

(1) isotonic or increasing if [mathematical expression not reproducible];

(2) strictly isotonic or order-preserving if it is isotonic and, in addition, [mathematical expression not reproducible].

Strictly isotonic functions on (X, [??]) are also called Richter-Peleg representations of [??] in the economic literature (see, e.g., Richter  and Peleg ).

Definition 1. A family U = {[u.sub.1],..., [u.sub.m]} of (necessarily isotonic) functions [mathematical expression not reproducible] is said to be

(1) a finite multiutility representation of the preorder [??] on X if, for all x, y [member of] X,

[mathematical expression not reproducible]; (5)

(2) a finite Richter-Peleg multiutility representation of the preorder [??] on X if U is a finite multiutility representation and in addition every function [u.sub.i] [member of] U is a Richter-Peleg representation of [??].

Alcantud et al. [15, Remark 2.3] noticed that a (finite) Richter-Peleg multiutility representation U of a preorder a on a set X also characterizes the strict part [mathematical expression not reproducible], in the sense that, for each x,y [member of] X,

[mathematical expression not reproducible]. (6)

Definition 2. Consider the multiobjective optimization problem (1). Then a point [x.sub.0] [member of] X is said to be

(1) Pareto optimal with respect to the function u = ([u.sub.1], ..., [u.sub.m]) : X [??] [R.sup.m] if for no x [member of] X it occurs that [u.sb.i] ([x.sub.0]) [less than or equal to] [u.sub.i](x) for all i [member of] {1,..., m} and at the same time [u.sub.i]([x.sub.0]) < [u.sub.i](x) for at least one index i;

(2) weakly Pareto optimal with respect to the function u = ([u.sub.1],..., [u.sub.m]) : X [??] [R.sup.m] if for no x [member of] X it occurs that [u.sub.i]([x.sub.0]) < [u.sub.i](x) for all i [member of] {1,..., m}.

Definition 3. The set of all (weakly) Pareto optimal elements with respect to the function u = ([u.sub.1],..., [u.sub.m]): X [??] [R.sup.m] will be denoted by [X.sup.Par.sub.u] ([X.sup.wPar.sub.u], respectively).

It is clear that [X.sup.Par.sub.u] [subset] [X.sup.wPar.sub.u] for every positive integer m, every nonempty set X, and every function u = ([u.sub.1],..., [u.sub.m]) : X [??] [R.sup.m].

Definition 4. Consider the multiobjective optimization problem (1). Then we introduce the preorders [mathematical expression not reproducible] on X defined as follows for all x, y [member of] X:

(1) [mathematical expression not reproducible].

(2) [mathematical expression not reproducible].

Remark 5. Notice that the indifference relation [~.sub.u] and the strict part [[??].sub.u] of the preorder [[??].sub.u], as well as the indifference relation [~.sup.w.sub.u] and the strict part [[??].sup.w.sub.u] of the preorder [[??].sup.w.sub.u], are defined as follows, for all x, y [member of] X:

[mathematical expression not reproducible] (7)

Definition 6. A preorder [??] on a topological space (X, [tau]) is said to be

(1) upper semiclosed if i(x) = {z [member of] X | x [??] z} is a closed subset of X for every x [member of] X;

(2) upper semicontinuous if l(x) = {z [member of] X | z < x} is an open subset of X for every x [member of] X.

While it is guaranteed that a preorder [??] on a compact topological space (X, [tau]) has a maximal element provided that [??] is either upper semiclosed (see Ward Jr. [23, Theorem 1]) or upper semicontinuous (see the theorem in Bergstrom ), a characterization of the existence of a maximal element for a preorder on a compact topological space (X, [tau]) was presented by Rodriguez-Palmero and Garcia-Lapresta .

Definition 7 (see Rodriguez-Palmero and Garcia-Lapresta [16, Definition 4]). A preorder [??] on a topological space (X, [tau]) is said to be transfer transitive lower continuous if for every element x [member of] X which is not a maximal element of a there exist an element y [member of] X and a neighbourhood N(x) of x such that y [??] z implies that N (x) [??] z for all z [member of] X.

Theorem 8 (see Rodriguez-Palmero and Garcia-Lapresta [16, Theorem 3]). A preorder [??] on a compact topological space (X, [tau]) 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, [tau]) is said to be upper semicontinuous if [u.sup.-1](] - [infinity], [alpha][) = {x [member of] X: u(x) < [alpha]} is an open set for all [alpha] [member of] 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 u on a nonempty set X, we denote by arg max u the set of all the points x [member of] X such that u attains its maximum at x (i.e., arg max u = {x [member of] X: u(z) [less than or equal to] u(x) for all z [member of] X}).

3. Existence of Maximal Elements and Pareto Optimality

A finite family U = {[u.sub.1],..., [u.sub.m]} of real-valued functions on a nonempty set X gives rise to a preorder [??] on X which 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 set X. Then the following statements hold:

(1) If [??] admits a finite multiutility representation [mathematical expression not reproducible].

(2) If a admits a finite Richter-Peleg multiutility representation [mathematical expression not reproducible].

Proof. Assume that the preorder a on X admits a finite (Richter-Peleg) multiutility representation [mathematical expression not reproducible], consider, by contraposition, an element [x.sub.0] [not member of] [X.sup.[??].sub.M]. Then there exists an element [mathematical expression not reproducible]. Then we have that [x.sub.0] is not (weakly) Pareto optimal. In a perfectly analogous way it can be shown that [mathematical expression not reproducible]. 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 = {[u.sub.1],..., [u.sub.m]} is a finite (Richter-Peleg) multiutility representation of the preorder [mathematical expression not reproducible], respectively).

From Theorem 9 and Proposition 10, we immediately arrive at the following proposition.

Proposition 11. Consider the multiobjective optimization problem (1). The following conditions are equivalent on a point [x.sub.0] [member of] X:

(i) [x.sub.0] is (weakly) Pareto optimal with respect to the function u = ([u.sub.1], ..., [u.sub.m]).

(ii) [x.sub.0] is maximal with respect to the preorder [mathematical expression not reproducible].

4. Multiobjective Optimization on Compact Spaces

The following theorem provides a characterization of the existence of Pareto optimal solutions to the multiobjective optimization problem (1) in terms of compactness of the choice set and appropriate semicontinuity conditions of the strict parts of the naturally associated preorders.

Theorem 12. Consider the multiobjective optimization problem (1). The following conditions are equivalent:

(i) [X.sup.Par.sub.u] ([X.sup.wPar.sub.u]) is nonempty.

(ii) There exists a compact topology [tau] on X and an upper semiclosed preorder [mathematical expression not reproducible].

(iii) There exists a compact topology [tau] on X such that [mathematical expression not reproducible] is upper semicontinuous.

Proof. (i) [??] (ii). Since [X.sup.Par.sub.u] ([X.sup.wPar.sub.u]) is nonempty, we have that [mathematical expression not reproducible] is nonempty by Proposition 11. Therefore, condition (ii) is verified by Bosi and Zuanon [17, Corollary 3.2, (i) [??] (ii)].

(ii) [??] (i). Since [??] is an upper semiclosed preorder on compact topological space (X, [tau]), [??] has a maximal element from Ward Jr. [23, Theorem 1]. Therefore, also [mathematical expression not reproducible] has a maximal element due to the fact that [mathematical expression not reproducible].

(i) [??] (iii). See Alcantud [25, Theorem 4, (a) [??] (b)]. Hence, the proof is complete.

Corollary 13. Consider the multiobjective optimization problem (1) where X is endowed with a compact topology [tau]. Then [X.sup.Par.sub.u] is nonempty provided that there exist a positive integer n and a function u' = ([u'.sub.1],...,[u'.sub.n]) : X [??] [R.sup.n] with all the real-valued functions [u'.sub.i] (i [member of] {1, ..., n}) upper semicontinuous, such that the following condition is verified:

(i) For all x,y [member of] X, x [[??].sub.u] y implies that [u.sub.i](x) [less than or equal to] [u.sub.i](y) for all i [member of] {1,..., n} and there exists i [member of] {1,..., n} such that [u.sub.i](x) < [u.sub.i](y).

Proof. By Theorem 9 and Proposition 11, we have that [X.sup.Par.sub.u] = [X.sup.[??]a.sub.M] is nonempty provided that there exists an upper semiclosed preorder [mathematical expression not reproducible]. Let u' = ([u.sub.1], ..., [u'.sub.n]) : X [??] [R.sup.n] be a function with the indicated properties. Define a preorder [??] on X by

[mathematical expression not reproducible]. (8)

The preorder [??] is upper semiclosed on (X, [tau]) since [u'.sub.i] is upper semicontinuous for all i [member of] {1,..., n} and U = {[u.sub.1], ..., [u.sub.n]} is a (finite) multiutility representation of [??]. Condition (i) precisely means that [mathematical expression not reproducible]. Hence, Theorem 12, (ii) [??] (i), applies, and the corollary is proved.

Corollary 14 (see Ehrgott [2, Theorem 2.19]). Consider the multiobjective optimization problem (1) where X is endowed with a compact topology t and the real-valued functions [u.sub.i] (i [member of] {1, ..., m}) are all upper semicontinuous. Then [X.sup.Par.sub.u] is nonempty.

Proof. This is a particular case of the above Corollary 13, when m = n and u' = u.

As an application of Theorem 8, let us finally present a characterization of the existence of Pareto optimal solution to the multiobjective optimization problem (1) on a compact space. In case that [??] is a preorder on a set X, x is an element of X, and A is a subset of X, the scripture [mathematical expression not reproducible].

Theorem 15. Consider the multiobjective optimization problem (1) where X is endowed with a compact topology [tau]. Then [X.sup.Par.sub.u] is nonempty if and only if for every element x [member of] X which is not Pareto optimal there exist an element y [member of] X and a neighbourhood N(x) of x such that, for all [mathematical expression not reproducible].

5. Scalarization and the Representation of All Pareto Optimal Elements

In this paragraph we address the scalarization of the multiobjective optimization problem under fairly general conditions.

The following theorem was proved by White . Given any maximal element [x.sub.0] relative to a preorder [??] on a set X, it guarantees the existence of some order-preserving function u attaining its maximum at [x.sub.0].

Theorem 16 (see White [20, Theorem 1]). Let (X, [??]) be a preordered set and assume that there exists an order-preserving function [mathematical expression not reproducible] is nonempty, then for every [x.sub.0] [member of] [X.sup.[??].sub.M] there exists a bounded order-preserving function [mathematical expression not reproducible].

The following corollary is an easy consequence of Theorem 16.

Corollary 17. Consider the multiobjective optimization problem (1). The following conditions are equivalent on a point [x.sub.0] [member of] X:

(i) [x.sub.0] [member of] [X.sup.Par.sub.u] ([x.sub.0] [member of] [X.sup.wPar.sub.u]).

(ii) There exists a bounded real-valued function [mathematical expression not reproducible] on X which is order-preservingfor the preorder [mathematical expression not reproducible].

Proof. Without loss of generality, we can assume that the functions [u.sub.i] appearing in the multiobjective optimization problem (1) are all bounded. Since U = {[u.sub.1], ..., [u.sub.m]} is a finite (Richter-Peleg) multiutility representation of the preorder [mathematical expression not reproducible], respectively) by Proposition 10, it is easily seen that the function [u.sup.*] := [[summation].sup.m.sub.h=1] [u.sub.h] is order-preserving for the preorder [mathematical expression not reproducible], respectively). Then we are ready for applying Theorem 16.

The simple proof of the following lemma is left to the reader.

Lemma 18. For any two functions [mathematical expression not reproducible].

As usual, if A is any nonempty subset of X, we denote by [absolute value of A] the cardinality of A.

Theorem 19. Consider the multiobjective optimization problem (1). Then the following conditions are equivalent:

(i) There exist a positive integer p [less than or equal to] m and a function u' = ([u'.sub.1],..., [u'.sub.p], [u'.sub.p+1],..., [u'.sub.m]) : X [??] [R.sup.m] satisfying the following conditions:

(a) [mathematical expression not reproducible];

(b) [mathematical expression not reproducible].

(ii) [mathematical expression not reproducible].

Proof. The implication "(i) [??] (ii)" is clear. Let us show that also the implication "(ii) [??] (i)" holds true. Let [mathematical expression not reproducible]. Following the proof of White [20, Theorem 1], we can define, for every h [member of] {1, ..., p} and x [member of] X,

[mathematical expression not reproducible], (9)

where [[delta].sub.1],..., [[delta].sub.p] are positive real numbers. Further, define [u'.sub.k] = [u.sub.k] for k = p + 1,..., m. In this way, the p real-valued functions [u'.,sub.1],..., [u'.sub.p] are all order-preserving for [[??].sup.w.sub.u] such that argmax [u'.sub.1] = [[x.sub.h]] for h = 1,..., p.

It is clear that [mathematical expression not reproducible].

It remains to show that [X.sup.wPar.sub.u] = [X.sup.wPar.sub.u']. To this aim, by Lemma 18 it suffices to show that [mathematical expression not reproducible] or equivalently that the following property holds for all elements x,y [member of] X:

(*) [mathematical expression not reproducible].

Three cases have to be considered.

(1) [mathematical expression not reproducible]. We have that, for every h [member of] {1,...,m}, [u'.sub.h](x) = [u.sub.h](x) and [u'.sub.h](y) = [u.sub.h](y). Hence, the above property (*) is obviously verified.

(2) [mathematical expression not reproducible]. In this case there exists h [member of] {1,..., p} such that x [member of] [[x.sub.h]], y [not member of] [[x.sub.h]], and therefore we have that [u.sub.h](y) = [u'.sub.h](y) < [u'.sub.h](x) = sup [u.sub.h](X) + [[delta].sub.h]. On the other hand, from the fact that x [[??].sup.w.sub.u] y is contradictory, we have that there exists k [member of] {1,..., p} such that [u.sub.k](y) < [U.sub.k](x). Hence, property (*) is verified also in this case.

(3) [mathematical expression not reproducible]. Clearly, we must have that either [mathematical expression not reproducible]. In the first case, it is clear that property (*) holds with all equalities on both sides of the equivalence. In the second case, since [mathematical expression not reproducible]. On the other hand, the definition of the function u' implies the existence of h [member of] {1,...,p} such that x [member of] arg max [u.sub.h], and, therefore, for that h, we have that [u'.sub.h] (y) < [u'.sub.h] (x). Analogously, there exists k e {1,..., p} such that y [member of] arg max [u.sub.k], and, therefore, for that k, we have that [u'.sub.k](x) < [u'.sub.k](y). This consideration completes the proof.

6. Conclusions

We approach the multiobjective optimization problem by using the preorders which are naturally associated with the concepts of Pareto optimal and, respectively, weakly Pareto optimal solutions, in the sense that the Pareto optimal and the weakly Pareto optimal solutions are precisely the maximal elements of these preorders. This interpretation gives us the possibility of using all the theorems concerning the maximal elements of the preorders (in particular on compact spaces) in order to guarantee the existence of solutions to the multiobjective optimization problem. This reinterpretation allows us to state a scalarization result under fairly general conditions. Our analysis does not require any particular requirement concerning the functions appearing in the multiobjective optimization problem or the choice set.

https://doi.org/10.1155/2018/3804742

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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Paolo Bevilacqua, (1) Gianni Bosi (iD), (2) and Magali Zuanon (3)

(1) DIA, Universita di Trieste, 34127 Trieste, Italy

(2) DEAMS, Universita di Trieste, 34127 Trieste, Italy

(3) DEM, Universita di Brescia, 25122 Brescia, Italy

Correspondence should be addressed to Gianni Bosi; gianni.bosi@deams.units.it

Received 31 July 2017; Accepted 17 December 2017; Published 28 January 2018

Title Annotation: Printer friendly Cite/link Email Feedback Research Article Bevilacqua, Paolo; Bosi, Gianni; Zuanon, Magali Abstract and Applied Analysis Report Jan 1, 2018 4513 Exact Null Controllability, Stabilizability, and Detectability of Linear Nonautonomous Control Systems: A Quasisemigroup Approach. The Implementation of Milstein Scheme in Two-Dimensional SDEs Using the Fourier Method. Existence theorems Mathematical optimization Mathematical research Optimization theory Ordered sets