Abstract and Applied Analysis
Volume 2018 (2018), Article ID 3804742, 6 pages
https://doi.org/10.1155/2018/3804742

Research Article

Multiobjective Optimization, Scalarization, and Maximal Elements of Preorders

Paolo Bevilacqua,¹ Gianni Bosi,² and Magalì Zuanon³

¹DIA, Università di Trieste, 34127 Trieste, Italy
²DEAMS, Università di Trieste, 34127 Trieste, Italy
³DEM, Università 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

Academic Editor: Simeon Reich

Copyright © 2018 Paolo Bevilacqua et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We characterize the existence of (weak) Pareto optimal solutions to the classical multiobjective optimization problem by referring to the naturally associated preorders and their finite (Richter-Peleg) multiutility representation. The case of a compact design space is appropriately considered by using results concerning the existence of maximal elements of preorders. The possibility of reformulating the multiobjective optimization problem for determining the weak Pareto optimal solutions by means of a scalarization procedure is finally characterized.

1. 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]).

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 ; we use the approach with the maximum for the sake of convenience):

where is the choice set (or the design space), is the decision function (in this case a utility function) associated with the th individual (or criterion), and is the vector-valued function defined by for all .

An element is a (weak) Pareto optimal solution to problem (1), for every , if for every ; then for every index (respectively, for every , if for every ; then for at least one index ). In this case, the point is said to be (weakly) Pareto optimal or a (weakly) efficient point for (MOP). Usually, is a subset of and concavity restrictions are posed on the functions (see, e.g., Ehrgott and Nickel [9]). 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 [10]).

It should be noted that Pareto optimality can be also considered by starting from a family 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) 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 on defined, for all , by

and, for determining the weak Pareto optimal solutions, we refer to the preorder on defined, for all , by

The consideration that an element is a (weak) Pareto optimal solution to problem (1) if and only if is a maximal element for the preorder (, respectively) and the observation that the function is a (Richter-Peleg) multiutility representation of the preorder (, 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 , nor to the real-valued functions that are usually assumed to be concave in the literature.

2. 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 is said to be a maximal element of if for no it occurs that . In the sequel we shall denote by the set of all the maximal elements of a preordered set . Please observe that can be empty.

Denote by the incomparability relation associated with a preorder on a set (i.e., for all , if and only if ).

We recall that a function is said to be

(1)isotonic or increasing if for all ;(2)strictly isotonic or order-preserving if it is isotonic and, in addition, for all .

Strictly isotonic functions on are also called Richter-Peleg representations of in the economic literature (see, e.g., Richter [21] and Peleg [22]).

Definition 1. A family of (necessarily isotonic) functions is said to be

(1)a finite multiutility representation of the preorder on if, for all ,

(2)a finite Richter-Peleg multiutility representation of the preorder on if is a finite multiutility representation and in addition every function is a Richter-Peleg representation of .

Alcantud et al. [15, Remark ] noticed that a (finite) Richter-Peleg multiutility representation of a preorder on a set also characterizes the strict part of , in the sense that, for each ,

Definition 2. Consider the multiobjective optimization problem (1). Then a point is said to be

(1)Pareto optimal with respect to the function if for no it occurs that for all and at the same time for at least one index ;(2)weakly Pareto optimal with respect to the function if for no it occurs that for all .

Definition 3. The set of all (weakly) Pareto optimal elements with respect to the function will be denoted by (, respectively).

It is clear that for every positive integer , every nonempty set , and every function .

Definition 4. Consider the multiobjective optimization problem (1). Then we introduce the preorders and on defined as follows for all :

(1)(2)

Remark 5. Notice that the indifference relation and the strict part of the preorder , as well as the indifference relation and the strict part of the preorder , are defined as follows, for all :

Definition 6. A preorder on a topological space is said to be

(1)upper semiclosed if is a closed subset of for every ;(2)upper semicontinuous if is an open subset of for every .

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 ]) 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 of such that implies that 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 is an open set for all . 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 the set of all the points such that attains its maximum at (i.e., ).

3. Existence of Maximal Elements and Pareto Optimality

A finite family of real-valued functions on a nonempty set gives rise to a preorder on which admits precisely the (Richter-Peleg) multiutility representation . 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 . Then the following statements hold:

(1)If admits a finite multiutility representation then .(2)If admits a finite Richter-Peleg multiutility representation then .

Proof. Assume that the preorder on admits a finite (Richter-Peleg) multiutility representation . In order to show that , consider, by contraposition, an element . Then there exists an element such that , or equivalently for all with an index such that (respectively, for all ). Then we have that is not (weakly) Pareto optimal. In a perfectly analogous way it can be shown that (). Hence, the proof is complete.

The following proposition is an immediate consequence of Definition 4.

Proposition 10. Consider the multiobjective optimization problem (1). Then is a finite (Richter-Peleg) multiutility representation of the preorder (, 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 :

(i) is (weakly) Pareto optimal with respect to the function .(ii) is maximal with respect to the preorder () on .

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) is nonempty.(ii)There exists a compact topology on and an upper semiclosed preorder on such that .(iii)There exists a compact topology on such that is upper semicontinuous.

Proof. (i) (ii). Since () is nonempty, we have that is nonempty by Proposition 11. Therefore, condition (ii) is verified by Bosi and Zuanon [17, Corollary , (i) (ii)].
(ii) (i). Since is an upper semiclosed preorder on compact topological space , has a maximal element from Ward Jr. [23, Theorem ]. Therefore, also () has a maximal element due to the fact that ().
(i) (iii). See Alcantud [25, Theorem , (a) (b)].
Hence, the proof is complete.

Corollary 13. Consider the multiobjective optimization problem (1) where is endowed with a compact topology . Then is nonempty provided that there exist a positive integer and a function with all the real-valued functions upper semicontinuous, such that the following condition is verified:

(i)For all , implies that for all and there exists such that .

Proof. By Theorem 9 and Proposition 11, we have that is nonempty provided that there exists an upper semiclosed preorder on such that . Let be a function with the indicated properties. Define a preorder on by

The preorder is upper semiclosed on since is upper semicontinuous for all and is a (finite) multiutility representation of . Condition (i) precisely means that . 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 is endowed with a compact topology and the real-valued functions are all upper semicontinuous. Then is nonempty.

Proof. This is a particular case of the above Corollary 13, when and .

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 , is an element of , and is a subset of , the scripture    stands for “ for all ” (respectively “ for all ”).

Theorem 15. Consider the multiobjective optimization problem (1) where is endowed with a compact topology . Then is nonempty if and only if for every element which is not Pareto optimal there exist an element and a neighbourhood of such that, for all , if , then for all .

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 [20]. Given any maximal element relative to a preorder on a set , it guarantees the existence of some order-preserving function attaining its maximum at .

Theorem 16 (see White [20, Theorem  1]). Let be a preordered set and assume that there exists an order-preserving function . If is nonempty, then for every there exists a bounded order-preserving function such that .

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 :

(i).(ii)There exists a bounded real-valued function on which is order-preserving for the preorder on such that .

Proof. Without loss of generality, we can assume that the functions appearing in the multiobjective optimization problem (1) are all bounded. Since is a finite (Richter-Peleg) multiutility representation of the preorder (, respectively) by Proposition 10, it is easily seen that the function is order-preserving for the preorder (, 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 and , if , then .

As usual, if is any nonempty subset of , we denote by the cardinality of .

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

(i)There exist a positive integer and a function satisfying the following conditions:

(a);(b) for all .

(ii).

Proof. The implication “(i) (ii)” is clear. Let us show that also the implication “(ii) (i)” holds true. Let and . Following the proof of White [20, Theorem ], we can define, for every and ,

where are positive real numbers. Further, define for . In this way, the real-valued functions are all order-preserving for such that for .
It is clear that for all and that . It remains to show that . To this aim, by Lemma 18 it suffices to show that or equivalently that the following property holds for all elements :

Three cases have to be considered.

(1) and . We have that, for every , and . Hence, the above property is obviously verified.(2) and . In this case there exists such that , , and therefore we have that . On the other hand, from the fact that is contradictory, we have that there exists such that . Hence, property is verified also in this case.(3) and . Clearly, we must have that either or . In the first case, it is clear that property holds with all equalities on both sides of the equivalence. In the second case, since , there exist such that and . On the other hand, the definition of the function implies the existence of such that , and, therefore, for that , we have that . Analogously, there exists such that , and, therefore, for that , we have that . 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.

Conflicts of Interest

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

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