Multi-Objective Optimization by Means of Pairs of Functions

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Multi-Objective Optimization by Means of Pairs of Functions

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Gianni Bosi^*

,Daris Roberto,Magalì Zuanon

Gianni Bosi^*

,Daris Roberto,Magalì Zuanon

This version is not peer-reviewed

Submitted:

08 December 2023

Posted:

08 December 2023

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Abstract

We introduce and discuss a generalization of the classical multi-objective optimization to pairs of functions. This procedure is referred to as bi-multi-objective optimization. A justification of this general optimization procedure is presented, related both to multi-objective optimization under ambiguity concerning individual preferences and to Pareto optimality for a family of preferences with nontransitive indifference. Incidentally, the binary relation naturally associated to a bi-multi-objective optimization problem is represented by a finite bi-multi-utility, which generalizes to the nontransitive case the classical finite multi-utility representation. An important application is presented to Markowitz portfolio selection under ambiguity concerning both the vector of returns and the covariance matrix.

Keywords:

Subject: Computer Science and Mathematics - Mathematics

MSC: 90C29 (Primary); 91B10 (Secondary)

1. Introduction

It is very well known that multi-objective optimization (see e.g. Miettinen 1999 and Ehrgott 2005) is the most popular tool which allows to choose among different options in the presence of many agents (or criteria) represented by finitely many real-valued functions

u_{i}

(

i \in {1, . . ., m}

), which have to be maximized all at the same time by following the approach introduced by Pareto 1896, and therefore called Pareto optimality. Such notion appears in Debreu 1954 and Mas-Colell et al. 1995 in the framework of general equilibrium. Multi-objective optimization appears in different applications of mathematics, such as insurance theory (see e.g. Asimit et al. 2017), design engineering (see e.g. Das 1999), economics and risk-sharing (see e.g. Chateauneuf et al. 2015, Chanas and Kuchta 1996 and Barrieu and Scandolo 2008), and portfolio selection (see e.g. Fliege and Werner 2014 and Xidonas et al. 2017).

Recently, Bevilacqua et al. 2018 approached the classical multi-objective optimization problem by referring to the naturally associated preorders, in such a way that, since the Pareto optimal elements are precisely the maximal elements of such preorders, the classical results concerning for example the existence of maximal elements for not necessarily total preorders on compact topological spaces can be used (see e.g. the famous results of Bergstrom 1975, Ward 1954 and Rodríguez-Palmero and García-Lapresta 2002, and the recent results in Bosi and Zuanon 2017).

In this paper we introduce a generalization of multi-objective optimization, called bi-multi-objective optimization. Such a problem has the form

max_{x \in X} [(u_{1} (x), v_{1} (x)), . . ., (u_{m} (x), v_{m} (x))] = max_{x \in X} (u (x), v (x)), m \geq 2,

where the pair

(u_{i}, v_{i})

of real-valued functions on the set X associated with the i-th individual satisfies, for every

i \in {1, . . ., m}

, the condition

u_{i} \leq v_{i}

, and a point

x_{0} \in X

is said to be Pareto optimal if for no

x \in X

it occurs that

u_{i} (x_{0}) \leq v_{i} (x)

for all

i \in {1, . . ., m}

and

v_{\bar{i}} (x_{0}) < u_{\bar{i}} (x)

for at least one index

\bar{i}

Needless to say, bi-multi-objective optimization coincides with the classical multi-objective optimization when

u_{i} = v_{i}

for all

i \in {1, . . ., m}

We present an interpretation of bi-multi-objective optimization based on decision theory, since to every pair

(u_{i}, v_{i})

we can associate the interval order

≾_{i}

on X defined, for all

x \in X

, by

x ≾_{i} y \Leftrightarrow u_{i} (x) \leq v_{i} (y) .

Hence, the appearance and use of bi-multi-objective optimization can be related to the intransitivity of the indifference of the individual interval orders (see e.g. Bosi and Zuanon 2014,Bosi and Zuanon 2014).

Incidentally, we observe that we can associate to every bi-multi-objective optimization problem

{max}_{x \in X} (u (x), v (x))

the reflexive binary relation ≾ on X defined by

x ≾ y \Leftrightarrow [u_{i} (x) \leq v_{i} (y) \forall i \in {1, . . ., m}] .

This is a finite bi-multi-utility representation of a not necessarily transitive binary relation ≾ on a set X, which is performed by means of a finite family

{(u_{i}, v_{i})}_{i = 1, . . . m}

of pairs

(u_{i}, v_{i})

of real valued functions on X such that

u_{i} \leq v_{i}

for all

i \in {1, . . ., m}

. Needless to say, this bi-multi-utility representation generalizes to the nontransitive case the classical finite multi-utility representation

{(u_{i}}}_{i = 1, . . . m}

of a preorder ≾ on X (see e.g. Bevilacqua et al. 2018 and Kaminski 2007), according to which

x ≾ y \Leftrightarrow [u_{i} (x) \leq u_{i} (y) \forall i \in {1, . . ., m}] .

Another interpretation of the bi-multi-utility optimization consists of ambiguity concerning the individual preferences, in the spirit of Haskell et al. 2016. Indeed,

(u, v)

can be interpreted as a range of utility functions

w

such that

u \leq w \leq v

. The set of all the weak Pareto optimal solutions to the bi-multi-objective optimization problem corresponding to the pair

(u, v)

include all the weak Pareto optimal solutions to the multi-objective optimization problems corresponding to all the functions

w

between

u

and

v

(see Theorem 1 below).

Compared to other interval optimization methods proposed in the literature (see e.g. Ishibuchi and Tanaka 1990, Chateauneuf et al. 2015 and Wu 2009), our approach appears simpler, since it does not require any particular choice among the possible orderings of intervals. On the other hand, bi-multi-objective optimization, which can be also applied to portfolio choice, is close to multi-objective optimization with interval-valued objective functions (see e.g. Wu et al. 2013).

The paper is structured as follows. Section 2 contains the notation, the basic definitions and the preliminary results. Section 3 presents the characterization of the solutions to the bi-multi-objective optimization problem in terms of the individual interval orders and of the maximal elements of the naturally associated reflexive binary relation. Section 4 contains an application to Markowitz portfolio selection under ambiguity. The section of the conclusions finishes the paper.

2. Notation and Preliminaries

The classical definitions relative to multi-objective optimization that we are going to present are the same as those found in Miettinen 1999 and Ehrgott 2005.

Definition 1.

The multi-objective optimization problem (MOP) is formulated by means of the standard notation1

max_{x \in X} [u_{1} (x), . . ., u_{m} (x)] = max_{x \in X} u (x), m \geq 2,

(1)

where X is the choice set (or the decision space ),

u_{i}

is the decision function (in this case a utility function ) associated with the i-th individual (or criterion), and

u : X \mapsto R^{m}

is the vector-valued function defined by

u (x) = (u_{1} (x), . . ., u_{m} (x))

for all

x \in X

Definition 2.

Consider the multi-objective optimization problem (1). Then a point

x_{0} \in X

is said to be

1.: Pareto optimal with respect to the function $u = (u_{1}, . . ., u_{m}) : X \mapsto R^{m}$ if for no $x \in X$ it occurs that $u_{i} (x_{0}) \leq u_{i} (x)$ for all $i \in {1, . . ., m}$ and at the same time $u_{\bar{i}} (x_{0}) < u_{\bar{i}} (x)$ for at least one index $\bar{i}$ ;
2.: weakly Pareto optimal with respect to the function $u = (u_{1}, \dots, u_{m}) : X \mapsto R^{m}$ if for no $x \in X$ it occurs that $u_{i} (x_{0}) < u_{i} (x)$ for all $i \in {1, . . ., m}$ .

Following Bevilacqua et al. 2018, Definition 2.3, the set of all (weakly) Pareto optimal elements with respect to the function

u = (u_{1}, . . ., u_{m}) : X \mapsto R^{m}

will be denoted by

X_{u}^{P a r}

(

X_{u}^{w P a r}

, respectively). It is clear that

X_{u}^{P a r} \subset X_{u}^{w P a r}

for every positive integer m, every nonempty set X and every function

u = (u_{1}, . . ., u_{m}) : X \mapsto R^{m}

We now introduce the bi-multi-objective optimization problem, and then the associated concepts of Pareto optimal and weakly Pareto optimal point.

Definition 3.

The bi-multi-objective optimization problem (BIMOP) is formulated as follows:

max_{x \in X} [(u_{1} (x), v_{1} (x)), . . ., (u_{m} (x), v_{m} (x))] = max_{x \in X} (u (x), v (x)), m \geq 2,

(2)

where the pair

(u_{i}, v_{i})

of decision functions associated with the i-th individual satisfies, for every

i \in {1, \dots, m}

, the condition

u_{i} \leq v_{i}

Definition 4.

Consider the bi-multi-objective optimization problem (2). Then a point

x_{0} \in X

is said to be

1.: Pareto optimal with respect to the function $(u, v) : X \mapsto R^{m} \times R^{m}$ if for no $x \in X$ it occurs that $u_{i} (x_{0}) \leq v_{i} (x)$ for all $i \in {1, \dots, m}$ and at the same time $v_{\bar{i}} (x_{0}) < u_{\bar{i}} (x)$ for at least one index $\bar{i}$ ;
2.: weakly Pareto optimal with respect to the function $(u, v) : X \mapsto R^{m} \times R^{m}$ if for no $x \in X$ it occurs that $v_{i} (x_{0}) < u_{i} (x)$ for all $i \in {1, \dots, m}$ .

Definition 5.

The set of all (weakly) Pareto optimal elements with respect to the function

(u, v) : X \mapsto R^{m} \times R^{m}

will be denoted by

X_{(u, v)}^{P a r}

(

X_{(u, v)}^{w P a r}

, respectively).

Remark 1.

It is clear that

X_{(u, v)}^{w P a r} \supset X_{(u, v)}^{P a r}

. Indeed, consider any element

x_{0} \notin X_{(u, v)}^{w P a r}

. Then it happens that, for some element

x \in X

u_{i} (x_{0}) \leq v_{i} (x_{0}) < u_{i} (x) \leq v_{i} (x)

for all

i \in {1, \dots, m}

, which clearly implies that

x_{0} \notin X_{(u, v)}^{P a r}

, since

u_{i} (x_{0}) < v_{i} (x)

for all

i \in {1, \dots, m}

and at the same time

v_{\bar{i}} (x_{0}) < u_{\bar{i}} (x)

for some

\bar{i}

Remark 2.

Everyone immediately checks that

max_{x \in X} u (x) = max_{x \in X} (u (x), u (x)),

i.e., problem (2) coincides with problem (1) in case that

u_{i} = v_{i}

for every

i \in {1, . . ., m}

Example 1.

Consider the case when

X = [0, 2]

(i.e., X is the closed real interval with extremes 0 and 2),

m = 2

u_{1} (x) = x

v_{1} (x) = x + 1

u_{2} (x) = x^{2} - 2

v_{2} (x) = x

. Observe that

u_{i} \leq v_{i}

for

i = 1, 2

, so that we can consider the bi-multi-objective optimization problem

max_{x \in [0, 2]} [(u_{1} (x), v_{1} (x)), (u_{2} (x), v_{2} (x))] = max_{x \in [0, 2]} [(x, x + 1), (x^{2} - 2, x)] .

We have that 2 is the unique Pareto optimal point, since for every

x_{0} \in [0, 2 [

it happens that

x_{0} \leq 2 + 1 = 3

, and on the other hand

x_{0} < 2^{2} - 2 = 2

. The set of all weakly Pareto optimal points is

[1, 2]

, since

x_{0} + 1 < 2

if and only if

x_{0} < 1

, while

x_{0} < 2^{2} - 2

for every

x_{0} < 2

3. Bi-Multi-Objective Optimization

An important result can be established, which relates the weakly Pareto optimal solutions to the bi-multi-objective optimization problem with respect to the function

(u (x), v (x))

and the weakly Pareto optimal solutions to the multi-objective optimization problem with respect to any function

w = (w_{1}, . . ., w_{m})

such that

u_{i} \leq w_{i} \leq v_{i}

for every

i \in {1, . . ., m}

Theorem 1.

Consider the bi-multi-objective optimization problem (2) with respect to the function

(u (x), v (x))

and let

w : X \mapsto R^{m}

be any function such that

u_{i} \leq w_{i} \leq v_{i}

for every

i \in {1, . . ., m}

. Then

X_{w}^{w P a r} \subset X_{(u, v)}^{w P a r}

Proof.

Consider the function

(u (x), v (x)) : X \mapsto R^{m} \times R^{m}

with

u_{i} \leq v_{i}

for every

i \in {1, . . ., m}

, and let

w : X \mapsto R^{m}

be any function such that

u_{i} \leq w_{i} \leq v_{i}

for every

i \in {1, . . ., m}

. By contraposition, if

x_{0} \in X

is such that

x_{0} \notin X_{(u, v)}^{w P a r}

, then, according to Definition 4, it happens that there exists

x \in X

such that, for every

i \in {1, . . ., m}

w_{i} (x_{0}) \leq v_{i} (x_{0}) < u_{i} (x) \leq w_{i} (x)

, implying that

w_{i} (x_{0}) < w_{i} (x)

for every

i \in {1, . . ., m}

. This means that

x_{0} \notin X_{w}^{w P a r}

(see Definition 2). This consideration completes the proof.□

Remark 3.

The above Theorem 1 can be interpreted in terms of ambiguity concerning the criteria to be adopted in a classical multi-objective optimization problem. Indeed, the weakly Pareto optimal solutions to a bi-multi-objective optimization problem (2) include all the weakly Pareto optimal solutions to every multi-objective optimization problem (1) corresponding to a set of utilities

w

in the assigned range

(u, v)

Remark 4.

Since in problem (2) we require that

u_{i} \leq v_{i}

for every

i \in {1, . . ., m}

, we have that m (possibly degenerate) closed real intervals

[u_{i} (x), v_{i} (x)]

are naturally associated to every

x \in X

In order to present a characterization and interpretation of the bi-multi-objective optimization problem (2) based on decision theory, let us introduce some classical definitions relative to binary relations and in particular to interval orders. While these definitions are classical (see e.g. Fishburn 1985), the reader can refer for example to Bosi and Zuanon 2014,Bosi and Zuanon 2014 for a deeper discussion concerning the existence of representations by means of pairs of upper semicontinuous real valued functions.

In the sequel, the symbol ≾ will stand for a reflexive binary relation on a set X (i.e.,

x ≾ x

for every

x \in X

). For all

x, y \in X

x ≾ y

has to be read as “the alternative y is at least as preferable as the alternative x”. The strict part (or asymmetric part) of a binary relation ≾ will be denoted by ≺ (i.e., for all

x, y \in X

x ≺ y

if and only if

(x ≾ y) a n d n o t (y ≾ x)

). The indifference relation∼ associated to ≾ is defined, for all

x, y \in X

, as

x \sim y

if and only if

(x ≾ y) a n d (y ≾ x)

). Notice that ∼ is an equivalence on X when ≾ is transitive (i.e., for all

x, y, z \in X

(x ≾ y) a n d (y ≾ z) \Rightarrow x ≾ z

Definition 6.

A preorder ≾ on an arbitrary nonempty set X is a binary relation on X which is reflexive and transitive.

Definition 7.

An interval order ≾ on an arbitrary nonempty set X is a binary relation on X which is reflexive and in addition verifies the following condition for all

x, y, z, w \in X

(x ≾ z) a n d (y ≾ w) \Rightarrow (x ≾ w) o r (y ≾ z) .

This latter property is often referred to as the Ferrers property (see e.g. Bosi and Zuanon 2014).

Since it is easily seen that an interval order is total (i.e., for all

x, y \in X

either

x ≾ y

y ≾ x

), we have that actually

x ≺ y

if and only if

n o t (y ≾ x)

(

x, y \in X

) when ≾ is an interval order. It is well known that an interval order ≾ is not transitive in general, while its strict part ≺ is always transitive.

We recall that a total preorder ≾ on a set X is represented by a real-valued function u on X if, for all

x \in X

x ≾ y \Leftrightarrow u (x) \leq u (y) .

In this case u is said to be a utility function for ≾.

Definition 8.

A pair

(u, v)

of real-valued functions on X is said to represent an interval order ≾ on X if, for all

x, y \in X

x ≾ y \Leftrightarrow u (x) \leq v (y) .

It is clear that, if

(u, v)

is a representation of an interval order ≾ on X, then

u (x) \leq v (x)

for every

x \in X

due to the fact that ≾ is reflexive. Hence, we have that

u \leq v

The classical definition of a maximal element is needed.

Definition 9.

Let ≾ be a reflexive binary relation on a set X. A point

x_{0} \in X

is said to be a maximal element of

(X, ≾)

if for no

x \in X

it happens that

x_{0} ≺ x

It should be noted that actually a point

x_{0} \in X

is a maximal element for an interval order ≾ on a set X if and only if

x ≾ x_{0}

for every

x \in X

Let us introduce the concept of Pareto optimality with respect to a (finite) family of preferences (see e.g. d’Aspremont and Gevers 2002).

Definition 10.

A point

x_{0} \in X

is said to be Pareto optimal with respect to the family

{≾_{i}}_{i \in {1, . . ., m}}

of interval orders if for no point

x \in X

it occurs that

x_{0} ≾_{i} x

for all

i \in {1, . . ., m}

, with at least one index

\bar{i}

such that

x_{0} ≺_{\bar{i}} x

We are now ready to present a characterization of the solutions to the bi-multi-objective optimization problem.

Theorem 2.

Consider the bi-multi-objective optimization problem (2) with respect to the function

(u, v) : X \mapsto R^{m} \times R^{m}

with

u_{i} \leq v_{i}

for every

i \in {1, . . ., m}

. Then the following conditions are equivalent on a point

x_{0} \in X

1.: $x_{0} \in X_{(u, v)}^{P a r}$ ;
2.: $x_{0}$ is Pareto optimal with respect to the family ${≾_{i}}_{i \in {1, . . ., m}}$ of interval orders on X such that, for every $i \in {1, . . ., m}$ , $≾_{i}$ is represented by the pair $(u_{i}, v_{i})$ ;
3.: $x_{0}$ is a maximal element of the binary relation $≾ = ⋂_{i = 1}^{m} ≾_{i}$ where, for every $i \in {1, . . ., m}$ , the interval order $≾_{i}$ is represented by the pair $(u_{i}, v_{i})$ ;
4.: $x_{0}$ is a maximal element for the reflexive binary relation ≾ on X defined as follows for all $x, y \in X$ :

$x ≾ y \Leftrightarrow [u_{i} (x) \leq v_{i} (y) \forall i \in {1, . . ., m}] .$

(3)

Proof.

In order to show that 1 ⇒ 2, just consider that if

x_{0}

is not Pareto optimal with respect to the given family

{≾_{i}}_{i \in {1, . . ., m}}

of interval orders on X, then there exists

x \in X

such that

u_{i} (x_{0}) \leq v_{i} (x)

for all

i \in {1, . . ., m}

, with at least one index

\bar{i}

such that

v_{\bar{i}} (x_{0}) < u_{\bar{i}} (x)

, which precisely means that

x_{0} \notin X_{(u, v)}^{P a r}

The proofs that 2 ⇒ 3 and 3 ⇒ 4 are rather simple and therefore they are left to the reader.

Finally, to show that 4 ⇒ 1, consider that if

x_{0} \notin X_{(u, v)}^{P a r}

, then the existence of

x \in X

such that

u_{i} (x_{0}) \leq v_{i} (x) \Leftrightarrow x_{0} ≾_{i} x

for all

i \in {1, . . ., m}

and at the same time

v_{\bar{i}} (x_{0}) < u_{\bar{i}} (x) \Leftrightarrow x_{0} ≺_{\bar{i}} x

for at least one index

\bar{i}

precisely means that

x_{0}

is not a maximal element of the binary relation ≾ defined in (3). This consideration completes the proof. □

Remark 5.

It is clear that the finite representation (3) of a (reflexive) binary relation ≾ generalizes the classical finite multi-utility representation of a preorder (see e.g. Kaminski 2007 and Bevilacqua et al. 2018) according to which, for a given function

u : X \mapsto R^{m}

and for all

x, y \in X

x ≾ y \Leftrightarrow [u_{i} (x) \leq u_{i} (y) \forall i \in {1, . . ., m}] .

(4)

Compared to the this latter kind of representation, which is possible only when ≾ is a preorder, the representation (3) has the advantage that it is compatible with intransitive indifference, i.e. situations when there exist

x, y, z \in X

such that

x \sim y

and

y \sim z

, but

n o t (x \sim z

). Needless to say, representation (3) coincides with the representation of an interval order by means of two real-valued functions in the case when

m = 1

(see Definition 8).

As an immediate corollary of Theorem 2 (see Remark 2), we get a characterization of the solutions to the multi-objective optimization problem.

Corollary 1.

Consider the multi-objective optimization problem (1) with respect to the function

u : X \mapsto R^{m}

. Then the following conditions are equivalent on a point

x_{0} \in X

1.: $x_{0} \in X_{u}^{P a r}$ ;
2.: $x_{0}$ is Pareto optimal with respect to the family ${≾_{i}}_{i \in {1, . . ., m}}$ of total preorders on X such that, for every $i \in {1, . . ., m}$ , $≾_{i}$ is represented by the function $u_{i}$ ;
3.: $x_{0}$ is a maximal element of the binary relation $≾ = ⋂_{i = 1}^{m} ≾_{i}$ where, for every $i \in {1, . . ., m}$ , the total preorder $≾_{i}$ is represented by function $u_{i}$ ;
4.: $x_{0}$ is a maximal element for the preorder ≾ on X defined as follows for all $x, y \in X$ :

$x ≾ y \Leftrightarrow [u_{i} (x) \leq u_{i} (y) \forall i \in {1, . . ., m}] .$

(5)

We finish this section by presenting a topological condition guaranteeing the existence of solutions to the bi-multi-objective optimization problem.

We first recall that a real-valued function u on a topological space

(X, τ)

is said to be upper semicontinuous if

u^{- 1} (] - \infty, α [) = {x \in X : u (x) < α}

is an open set for all

α \in R

. The well known Weierstrass extreme value theorem guarantees that an upper semicontinuous real-valued function attains its maximum on a compact topological space.

We now present a sufficient condition for the existence of a Pareto optimal solution for the bi-multi-objective optimization problem (2). As a corollary, we obtain a well known result concerning the existence of a Pareto optimal solution to the multi-objective optimization problem.

Theorem 3.

X_{(u, v)}^{P a r} \neq \emptyset

provided that X is endowed with a compact topology τ and one of the following conditions is verified:

1.: the functions $u_{i}$ ( $i \in {1, . . ., m}$ ) are upper semicontinuous;
2.: the functions $v_{i}$ ( $i \in {1, . . ., m}$ ) are upper semicontinuous.

Proof.

1. For every

i \in {1, . . ., m}

, consider a point

x_{i} \in arg max u_{i}

. These points exist since

τ

is a compact topology on X, and all the functions

u_{i}

are upper semicontinuous on the topological space

(X, τ)

. Further, let the index

i^{*} \in {1, . . ., m}

be such that

u_{i^{*}} (x_{i^{*}}) = max {u_{1} (x_{1}), u_{2} (x_{2}), . . ., u_{m} (x_{m})}

. We have that

x_{i^{*}} \in X_{(u, v)}^{P a r}

, since for no

x \in X

and

i \in {1, . . ., m}

it may happen that

u_{i^{*}} (x_{i^{*}}) \leq v_{i^{*}} (x_{i^{*}}) < u_{i} (x)

2. We proceed in a perfectly analogous way, by defining the index

i^{*} \in {1, \dots, m}

in such a way that

v_{i^{*}} (x_{i^{*}}) = max {v_{1} (x_{1}), v_{2} (x_{2}), . . ., v_{m} (x_{m})}

, with

x_{i} \in arg max v_{i}

for every

i \in {1, \dots, m}

. We have that

x_{i^{*}} \in X_{(u, v)}^{P a r}

.□

Corollary 2.(Ehrgott 2005, Theorem 2.19)

X_{u}^{P a r} \neq \emptyset

provided that X is endowed with a compact topology τ and the real-valued functions

u_{i}

(

i \in {1, \dots, m}

) are all upper semicontinuous.

Proof.

This is a particular application of the above Theorem 3, when

u = v

. □

4. An Application to Markowitz Portfolio Selection

The Markowitz 1952 portfolio selection problem in the absence of short sales appears in the following form when expressed in terms of a multi-objective optimization problem:

max_{x \in X} [μ^{T} x, - x^{T} Σ x],

(6)

where

m = 2

is the number of criteria,

X = {x \in R_{+}^{n} : \sum_{i = 1}^{n} x_{i} = 1}

is the set of all portfolios,

n > 0

is the number of risky assets considered,

μ \in R^{n}

is a vector of expected returns, and Σ is a covariance matrix of the returns (see e.g. Fliege and Werner 2014). It can be therefore generalized by the following bi-multi-objective optimization problem:

max_{x \in X} [(μ_{1}^{T} x, μ_{2}^{T} x), (- x^{T} Σ_{2} x, - x^{T} Σ_{1} x)],

(7)

where

μ_{1}^{T} \leq μ_{2}^{T}

are two vectors of returns, and

Σ_{1} \leq Σ_{2}

are two covariance matrices (see also Wu et al. 2013).

From Theorem 1, we can now state the following proposition.

Proposition 1.

Given two vectors

μ_{1}^{T} \leq μ_{2}^{T}

of expected returns, and given two covariance matrices

Σ_{1} \leq Σ_{2}

, consider the Markowitz bi-multi-objective optimization problem

max_{x \in X} [(μ_{1}^{T} x, μ_{2}^{T} x), (- x^{T} Σ_{2} x, - x^{T} Σ_{1} x)], X = {x \in R_{+}^{n} : \sum_{i = 1}^{n} x_{i} = 1}

(8)

We have that, for every vector of expected returns

μ_{1}^{T} \leq μ^{T} \leq μ_{2}^{T}

and for every covariance matrix

Σ_{1} \leq Σ \leq Σ_{2}

, the weakly Pareto optimal solutions to the corresponding multi-objective optimization problem

max_{x \in X} [μ^{T} x, - x^{T} Σ x], X = {x \in R_{+}^{n} : \sum_{i = 1}^{n} x_{i} = 1},

(9)

are also weakly Pareto optimal for the above bi-multi-objective problem.

Therefore, it is easy to incorporate ambiguity concerning both the vector of returns and the covariance matrix in the classical Markowitz portfolio selection problem, since our bi-multi-objective optimization model takes into account a range of expected returns and covariance matrices.

5. Conclusions

The bi-multi-objective optimization problem is an extension of the classical multi-objective optimization problem, which on one hand is simple to implement, and on the other hand can be considered interesting since it can be thought of as a useful tool for incorporating ambiguity in the individual preferences.

In this paper, we have presented the basic theory concerning this optimization problem, whose associated Pareto optimal elements can be interpreted as the maximal elements of the intersection of the interval orders representing the individual preferences.

Hopefully, we shall develop the full potential of our proposal in a future paper.

Conflicts of Interest

The authors guarantee that they have no conflict of interest.

1	Needless to say, this formulation of the multi-objective optimization problem is equivalent, “mutatis mutandis”, to $min_{x \in X} [f_{1} (x), . . ., f_{m} (x)] = min_{x \in X} f (x), m \geq 2$ .

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Multi-Objective Optimization by Means of Pairs of Functions

Abstract

1. Introduction

2. Notation and Preliminaries

3. Bi-Multi-Objective Optimization

4. An Application to Markowitz Portfolio Selection

5. Conclusions

Conflicts of Interest

References

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