Low-Rank Multi-Objective Linear Programming

TL;DR

This paper presents a method to reduce the number of objectives in multi-objective linear programs by transforming them into equivalent vector linear programs with fewer objectives, leading to faster computation and better visualization.

Contribution

It introduces a novel transformation technique for MOLPs with non-full rank objective matrices, improving computational efficiency and interpretability.

Findings

Significant speedup in solving MOLPs.

Enhanced visualization and understanding of objectives.

Effective reduction of objectives without losing information.

Abstract

When solving multi-objective programs, the number of objectives essentially determines the computing time. This can even lead to practically unsolvable problems. Consequently, it is worthwhile to reduce the number of objectives without losing information. In this article, we discuss multi-objective linear programs (MOLP) with objective matrices that do not have a full row rank. We introduce a method to transform an MOLP into an equivalent vector linear program (VLP) with merely as many objectives as the rank of the original objective matrix. To this end, only a factorization of this matrix is needed to be calculated. One of the factors then forms a new ordering cone while the other factor remains as the objective matrix. Through a series of numerical experiments, we will show that this approach indeed leads to a significant speedup of computing time, and therefore provides a powerful technique for solving MOLPs in practice. As there are fewer objectives to consider, the approach additionally helps decision makers to get a better visualization as well as understanding of the actual problem. Moreover, we will point out that the equivalence of the MOLP and the corresponding VLP can be used to derive statements about the concept of nonessential objectives.

Equations

$${\le }_C:-\left\{(z^1,z^2)\in {\mathbb{R}}^q\times {\mathbb{R}}^q|z^2-z^1\in C\right\}\text{.}$$

$$\nexists z\in A:z{⪇}_C\bar{z}$$

$$\forall z\in A:z{\le }_C\bar{z}⟹z=\bar{z}\text{.}$$

$${\mathrm{Min}}_C(A)=\left\{\bar{z}\in A|A\cap (\bar{z}-C)=\left\{\bar{z}\right\}\right\}=\left\{\bar{z}\in A|A\cap (\bar{z}-C\setminus \left\{0\right\})=\varnothing \right\}\text{.}$$

$$\mathrm{Eff}(\text{hyperlink}P\text{P}):-P^{-1}[{\mathrm{Min}}_{C}P[S]]\cap S=\left\{\bar{x}\in S|\nexists x\in S:Px{⪇}_{C}P\bar{x}\right\}\text{.}$$

$$\forall z\in A:z{\le }_C\bar{z}⟹\bar{z}{\le }_{C}z\text{.}$$

$$\overline{\mathrm{Eff}}(\text{hyperlink}P\text{P}):-P^{-1}\left[{\overline{\mathrm{Min}}}_{C}P[S]\right]\cap S\text{.}$$

$${\overline{\mathrm{Min}}}_C(A)=\left\{\bar{z}\in A|A\cap (\bar{z}-C\setminus (C\cap (-C)))=\varnothing \right\}=\left\{\bar{z}\in A|A\cap (\bar{z}-C\setminus (-C))=\varnothing \right\}\text{.}$$

$$C:-\left\{z\in {\mathbb{R}}^k|Lz\ge 0\right\}$$

$$R{x}^1{\le }_{C}R{x}^2⟺P{x}^1\le P{x}^2\text{.}$$

$$\mathrm{Eff}(\text{hyperlink}molp\text{MOLP})=\mathrm{Eff}(\text{hyperlink}vlp\text{VLP})\text{mbox}and\mathrm{Min}P[S]=L[{\mathrm{Min}}_{C}R[S]]\text{.}$$

$$\mathrm{Eff}(\text{hyperlink}vlpmethodstart{\text{VLP}}^{\prime })=\mathrm{Eff}(\text{hyperlink}vlpmethodend{\text{VLP}}^{\prime \prime })\text{mbox}and{\mathrm{Min}}_{C^{\prime }}P[S]=L[{\mathrm{Min}}_{C^{\prime \prime }}R[S]]\text{.}$$

$$\tilde{C}:-\left\{z\in {\mathbb{R}}^k|Lz⪈0\right\}\cup \left\{0\right\}=(C\setminus (-C))\cup \left\{0\right\}\text{.}$$

$$\mathrm{Eff}(\text{hyperlink}molp\text{MOLP})=\overline{\mathrm{Eff}}(\text{hyperlink}vlp\text{VLP})\text{mbox}and\mathrm{Min}P[S]=L[{\overline{\mathrm{Min}}}_{C}R[S]]\text{.}$$

$$p\in \mathrm{cone}(\mathcal{P}\setminus \left\{p\right\})\text{,}$$

$$\forall p\in {\mathcal{P}}^{\ast }:p\in \mathrm{cone}(\mathcal{P}\setminus {\mathcal{P}}^{\ast })\text{.}$$

Full text

Keywords: Multi-objective programming; Linear vector optimization; Objective space reduction; Criteria reduction; Nonessential objectives

Mathematics Subject Classification: 90C29; 90C05

1 Introduction

For multi-objective linear programs (MOLP), algorithms like the Benson algorithm [12] primarily operate in the objective space. Thus, computing time increases rapidly with the number of objectives and quickly exceeds what can be calculated. However, optimization problems with at least ten objectives are common in practice (whether linear or nonlinear). An example of a linear program is given in [6] that employs multi-objective optimization to map an entropy region of jointly distributed random variables. One example of a nonlinear program is deciding on the most environmentally friendly design for photovoltaic systems [19]. Due to the large number of objectives, it is worthwhile to reduce the number of objectives without losing information.

Many approaches have been investigated to reduce the dimension of the objective space for nonlinear programs. For instance, [1, 2, 3, 23, 20] examined which objectives are essential for describing the set of minimal points and which can be eliminated. [17, 4, 11, 19] simplified the problem by constructing a smaller set of objectives from the actual problem.

Moreover, [8, 7, 18, 24] studied the concept of nonessential objectives in the context of multi-objective linear programming. An important result is that objectives can be deleted if they can be represented as a conic combination of the other objectives. Thus, objectives can be ignored when they are linearly dependent in a particular manner. However, not all linear combinations are conic combinations, and therefore we ask whether it is possible to eliminate linear dependence completely.

Given a vector linear program (VLP) whose ordering cone is polyhedral and therefore determined by a matrix, we can multiply the objective matrix by this matrix and replace the ordering cone with the standard ordering cone. We then obtain an equivalent MOLP. This well-known method is presented in Lemma 2.3.4 of [22]. We can also do this in the converse direction: Starting with a factorization of the objective matrix of an MOLP yields an equivalent VLP by defining the ordering cone through the one factor and using the other one as the objective matrix. If the objective matrix does not have a full row rank, a suitable factorization results in a problem with merely as many objectives as the rank of the original objective matrix. This approach eliminates linear dependence but introduces a new partial ordering. Moreover, with this approach, we can simplify and generalize the results about nonessential objectives in [8, 7, 18, 24].

This article is structured as follows. In Section˜2, we introduce the notions of an MOLP and a VLP in $\mathbb{R}^{q}$ as well as the concepts of minimality and generalized minimality. Section˜3 presents the aforementioned transformation approach. We prove that both the MOLP and the VLP are indeed equivalent problems and explain how the VLP can be interpreted. Additionally, we examine the case when the factorization is not rank-minimal and state a more advanced theorem on the equivalence of MOLP and VLP in a generalized sense. Section˜4 focuses on the concept of nonessential objectives. We prove the statement on conic combinations of objectives using the reformulation of the MOLP into an equivalent VLP with the power of the concept of generalized minimality from Section˜3. We relate the elimination of nonessential objectives to facet or vertex enumeration of the ordering cone. In Section˜5, we demonstrate the effectiveness of the transformation approach through a series of numerical experiments and therefore show that it provides a powerful technique for solving MOLPs in practice.

2 Problem Setting and Preliminaries

In this section, we introduce multi-objective linear programs (MOLP) and—as a generalization—vector linear programs (VLP) in $\mathbb{R}^{q}$. To this end, we first need to clarify the notion of minimality.

Consider the preordering $\leq_{C}$ on $\mathbb{R}^{q}$ generated by a convex cone $C\subseteq\mathbb{R}^{q}$ [9, 10, 12], i.e.,

[TABLE]

We will always write $z^{1}\leq_{C}z^{2}$ instead of $(z^{1},z^{2})\in\ \leq_{C}$. We omit the index if $C$ equals the cone $\mathbb{R}_{+}^{q}\coloneq\{z\in\mathbb{R}^{q}\;|\;\forall\,j=1,\ldots,q:z_{j}\geq 0\}$; we then call $\leq$ the natural ordering on $\mathbb{R}^{q}$. When we say $z^{1}\lneq_{C}z^{2}$, we mean $z^{1}\leq_{C}z^{2}$ and $z^{1}\neq z^{2}$. Note that $\leq_{C}$ is a partial ordering if and only if $C$ is pointed, i.e., $C\cap(-C)=\left\{0\right\}$.

Given a nonempty subset $A\subseteq\mathbb{R}^{q}$, a point $\bar{z}\in A$ is called minimal w.r.t. $\leq_{C}$ (or $C$-minimal) in $A$ if

[TABLE]

or, equivalently,

[TABLE]

The set of all minimal points of $A$ is denoted by $\operatorname{Min}_{C}(A)$. Note that a point $\bar{z}\in A$ is a minimal point of $A$ if and only if there is no distinct point $z\in A$ being an element of $\bar{z}-C$ (with $\bar{z}-C$ denoting the Minkowski sum of $\left\{\bar{z}\right\}$ and $-C$). Therefore, the set of minimal points can be written as

[TABLE]

In Figure˜2.1, some examples are given.

Now, let $C$ be a polyhedral convex cone, i.e., $C=\{z\in\mathbb{R}^{q}\;|\;Bz\geq 0\}$ for some matrix $B\in\mathbb{R}^{r\times q}$. Furthermore, let $S\subseteq\mathbb{R}^{n}$ be a polyhedron and $P\in\mathbb{R}^{q\times n}$ a matrix. Then, we call

[TABLE]

a vector linear program (VLP). $S$ is referred to as the feasible set, and its image $P[S]$ under $P$ is named the image of (P). The task is to find feasible points $\bar{x}\in S$ whose images under $P$ are $C$-minimal in the image of (P), i.e., $P\bar{x}\in\operatorname{Min}_{C}P[S]$. These points are called efficient and form the set

[TABLE]

The transposed rows of $P$ are referred to as the objectives of the program. A VLP is called multi-objective linear program (MOLP) if one minimizes w.r.t. the natural ordering. For an MOLP, efficient points can be considered as the best compromises over the objectives. Note that maximization problems can be simply obtained by either considering $-P$ as the objective matrix or $-C$ as the ordering cone.

As can be seen in Figure˜1(c), it is possible that even nonempty compact sets do not have minimal points. This may occur if and only if the ordering cone is not pointed, i.e., if the ordering is not antisymmetric. To fix this, one can introduce the so-called generalized minimality. Note that there is no unique name for this concept. For instance, [10] directly introduces minimality in the generalized form. We decided to distinguish both concepts, as [9] do.

For a nonempty subset $A\subseteq\mathbb{R}^{q}$, apoint $\bar{z}\in A$ is called generalized minimal w.r.t. $\leq_{C}$ in $A$ if

[TABLE]

The set of all minimal points of $A$ is denoted by $\operatorname{\overline{Min}}_{C}(A)$. Analogous to Equation˜2.6, one can extend VLPs to determine generalized efficient points, i.e.,

[TABLE]

Obviously, minimal points are always generalized minimal points. Additionally, one can easily see that generalized minimality coincides with minimality if $C$ is pointed as $\leq_{C}$ then is antisymmetric. The set of generalized minimal points can alternatively be expressed as

[TABLE]

By this generalization, the elements of the dashed set in Figure˜1(c) can be considered minimal.

3 Reduction of the Number of Objectives

When determining efficient points of an MOLP

[TABLE]

algorithms—like the Benson algorithm [12]—primarily operate in the image space of the problem. In practice, one can observe that the computation time grows rapidly when increasing the number of objectives. Consequently, it is of huge interest to reduce this number.

If the objective matrix does not have a full row rank, i.e., $k\coloneq\operatorname{rk}(P)<q$, the objectives are linear dependent. This means that the set of objectives contains less information than initially assumed. Hence, one can ask whether the number of objectives can be reduced to $k$ without losing information. In this section, we present an approach to achieve this aim for MOLP with a low-rank objective matrix.

As a first step, we decompose the objective matrix $P$ into a product of two matrices $L\in\mathbb{R}^{q\times k}$ and $R\in\mathbb{R}^{k\times n}$ where $\operatorname{rk}(L)=\operatorname{rk}(R)=k$, i.e., $P=LR$. This is always possible, e.g., by using Gauß’ algorithm or a singular value decomposition. Now we define the set

[TABLE]

which is a convex cone as $z\in C$ and $\lambda\geq 0$ imply $L(\lambda z)=\lambda(Lz)\geq 0$ and therefore $\lambda z\in C$. Since $C\cap(-C)=\{z\in\mathbb{R}^{k}\;|\;Lz=0\}=\left\{0\right\}$ holds by the linear independence of the columns of $L$, $C$ is pointed.

As mentioned in Section˜2, $C$ defines a partial ordering $\leq_{C}$. For any $x^{1},x^{2}\in\mathbb{R}^{n}$ we have

[TABLE]

Thus, we can compare two feasible points under $R$ in $\mathbb{R}^{k}$ instead of $P$ in $\mathbb{R}^{q}$. Using $R$ as the objective matrix and $C$ as the ordering cone of a new VLP

[TABLE]

the number of objectives decreases from $q$ to $k$.

The following theorem states that (MOLP) and (VLP) are equivalent in the sense that their sets of efficient points coincide. Furthermore, the minimal points are related just by applying $L$.

Theorem 3.1**:**

Consider the problem (MOLP) with a decomposition of the objective matrix $P=LR$ into the matrices $L\in\mathbb{R}^{q\times k}$ and $R\in\mathbb{R}^{k\times n}$ where $k=\operatorname{rk}(P)=\operatorname{rk}(L)=\operatorname{rk}(R)$. Then, for the problem (VLP) the following statements hold:

[TABLE]

• Proof.

The second equation follows directly from the first one by Equation˜2.6 and $P=LR$. To prove the first equation, take some $\bar{x}\in\operatorname{Eff}(\hyperlink{vlp}{\text{VLP}})$. According to Equation˜2.6, there is no point $x\in S$ with both $Rx\neq R\bar{x}$ and $Rx\leq_{C}R\bar{x}$. Due to Equation˜3.3, one can rewrite the inequality as $LRx\leq LR\bar{x}$. As, by assumption, $L$ is injective, $Rx\neq R\bar{x}$ is equivalent to $LRx\neq LR\bar{x}$. The equivalence of $\bar{x}\in\operatorname{Eff}(\hyperlink{vlp}{\text{VLP}})$ and $\bar{x}\in\operatorname{Eff}(\hyperlink{molp}{\text{MOLP}})$ then follows from $P=LR$ and, again, Equation˜2.6. ∎

Next to the aforementioned speedup in computing time, which we will discuss in more detail in Section˜5, another crucial practical benefit arises from Theorem˜3.1. It becomes much easier for decision makers to get a feeling for the actual problem since the number of objectives decreases, and plotting the problem (e.g., via net diagrams) gets simpler. In particular, the objectives’ linear independence means that the objectives contain less information than one primarily has supposed. Due to linear dependencies, the image of the feasible set just lies in a $k$-dimensional subspace of $\mathbb{R}^{q}$, and the transformation into a VLP is nothing else than a projection onto $\mathbb{R}^{k}$. Nevertheless, particular linear combinations of objectives might be of special practical interest—although they actually are redundant for the solution, one can output them additionally.

Considering the problem

[TABLE]

which is equivalent to (VLP) due to Theorem˜3.1, decision makers become able to interpret the problem (VLP) as follows. First, note that a feasible point $\bar{x}$ is efficient for (VLP), i.e., $\bar{z}=R\bar{x}$ is $C$-minimal, if and only if $\bar{z}=R\bar{x}$ is efficient for ($\text{MOLP}^{\circ}$). Hence, at an efficient point, it is not possible to decrease any objective $\ell^{i}$ over $R[S]$ without increasing another objective $\ell^{j}$, where $\ell^{1},\ldots,\ell^{q}$ denote the transposed rows of $L$. This relationship can be used to visualize (VLP); an example is illustrated in Figure˜3.1. Since the rows of $L$ and $P$ correspond one-to-one due to the injectivity of $R^{\mathsf{T}}$, this behavior translates directly to the objectives of (MOLP), that is, minimizing an objective $\ell^{i}$ over $R[S]$ minimizes the corresponding objective $p^{i}$ over $S$, and vice versa.

One can implement the method into a solution algorithm for MOLP using the following distinction of cases, provided the rank of $P$ has already been calculated. If $k=n<q$, a trivial decomposition (i.e., $L=P$ and $R$ being the identity matrix in $\mathbb{R}^{n\times n}$) leads to a reduction in the number of objectives. If $P$ has full rank and $k=q\leq n$, the procedure has no advantage and can be omitted. In any other case in which $P$ has not a full rank, one should always compute a decomposition of $P$ and solve the corresponding VLP.

Note that the cone $C$ might have an empty interior (or might even be the trivial cone $\{0\}$), depending on $L$. This occurs when the conic hull of some objectives forms a linear subspace of $\mathbb{R}^{k}$. The solution algorithm must be able to handle these cases.

Remark 3.2**:**

The method is not limited to MOLP but can also be applied to VLP in general. Consider the following VLP

[TABLE]

where $C^{\prime}=\{z^{\prime}\in\mathbb{R}^{q}\;|\;Qz^{\prime}\geq 0\}$ for some $Q\in\mathbb{R}^{r\times q}$, and $P\in\mathbb{R}^{q\times n}$ with $\operatorname{rk}(P)=k$. Take a decomposition of the objective matrix into the matrices $L\in\mathbb{R}^{q\times k}$ and $R\in\mathbb{R}^{k\times n}$ where $k=\operatorname{rk}(P)=\operatorname{rk}(L)=\operatorname{rk}(R)$. Then, for the problem

[TABLE]

with ordering cone $C^{\prime\prime}=\{z^{\prime\prime}\in\mathbb{R}^{q}\;|\;QLz^{\prime\prime}\geq 0\}$, the following statements hold:

[TABLE]

This follows similarly to the proof of Theorem˜3.1.

In the remaining part of this section, we want to discuss the requirement of injectivity of $L$ and provide a generalization of Theorem˜3.1. Decomposing $P$ into $L\in\mathbb{R}^{q\times k}$ and $R\in\mathbb{R}^{k\times n}$ such that $\operatorname{rk}(P)\leq\operatorname{rk}(L)<k$ yields a cone $C$ that is not pointed since $C\cap(-C)=\ker(L)$ is a proper superset of $\{0\}$. The proof of Theorem˜3.1 shows that $\operatorname{Eff}(\hyperlink{vlp}{\text{VLP}})\subseteq\operatorname{Eff}(\hyperlink{molp}{\text{MOLP}})$ always holds as the injectivity of $L$ is not necessary for the proof of this inclusion. One can easily construct examples where the equality is not true. For instance, we can take a bounded feasible set $S$ and a matrix $L$ such that $C\cap(-C)$ is parallel to a non-vertex face of $R[S]$. Then, if $C$ has the right orientation, no point of this face is $C$-minimal due to the missing antisymmetry of the ordering, cf. Figure˜1(c). Moreover, there are no other minimal points, so $\operatorname{Eff}(\hyperlink{vlp}{\text{VLP}})=\emptyset$. As $S$ is compact, $P[S]=L[R[S]]$ has minimal points w.r.t. the natural ordering. Thus, $\operatorname{Eff}(\hyperlink{molp}{\text{MOLP}})\neq\emptyset$. In conclusion, the injectivity of $L$ is necessary for Theorem˜3.1.

In this sketch of an example, considering generalized minimal points will fix the problem. Indeed, this approach resolves it in general. We will prove this in Theorem˜3.3 below. In Proposition 1 of [24], the problem is resolved by a slight modification of the ordering cone:

[TABLE]

The so-induced notion of minimality coincides with the generalized minimality w.r.t. $\leq_{C}$ introduced in Equation˜2.7. From a theoretical point of view, it is possible to treat a matrix $L$ that does not have a full rank by either using $\tilde{C}$ as the ordering cone or the concept of generalized minimality, respectively. Theorem˜3.3 therefore may be proven as [24] does. We will present an alternative proof similar to the proof of Theorem˜3.1 that directly uses the definition of generalized minimality.

Theorem 3.3**:**

Consider the problem (MOLP) with a decomposition of the objective matrix $P=LR$ into the matrices $L\in\mathbb{R}^{q\times k}$ and $R\in\mathbb{R}^{k\times n}$ where $k\geq\operatorname{rk}(P)$. Then, for the problem (VLP) the following statements hold:

[TABLE]

• Proof.

Take some $\bar{x}\in\operatorname{Eff}(\hyperlink{molp}{\text{MOLP}})$. As the ordering cone $\mathbb{R}_{+}^{q}$ of (MOLP) is pointed, this is equivalent to $\bar{x}\in\operatorname{\overline{Eff}}(\hyperlink{molp}{\text{MOLP}})$. According to Equation˜2.7, for any $x\in S$ satisfying $Px\leq P\bar{x}$ follows $P\bar{x}\leq Px$. Note that Equation˜3.3 also holds if $L$ is not injective since only the definition of $\leq_{C}$ was used there. Thus, $Rx\leq_{C}R\bar{x}$ implies $R\bar{x}\leq_{C}Rx$ for all $x\in S$. This is equivalent to $\bar{x}\in\operatorname{\overline{Eff}}(\hyperlink{vlp}{\text{VLP}})$. ∎

Theorem˜3.1 can be viewed as a corollary of Theorem˜3.3 since efficient and generalized efficient points coincide for pointed ordering cones. Remark˜3.2 also applies to a statement similar to Theorem˜3.3 regarding the equivalence of suitable VLP.

From a practical point of view, Theorem˜3.3 is not relevant since the number of objectives of (VLP) is not minimal in this case. However, the generalization is a valuable proving technique demonstrated in the next section.

4 Nonessential Objectives

It is possible that some objectives of (MOLP) are not relevant for determining efficient points—they are obsolete. Figure˜1(a) gives an example where one can remove, for instance, the objective $p^{2}$ without affecting the set of efficient points. This concept of so-called nonessential (or redundant) objectives has been developed by Thomas Gal and Heiner Leberling [8, 7]. In this section we aim to present this concept based on [8, 7, 24] and relate it to the previous section. First, the following introduces the terms of nonessential objective as well as nonessential system of objectives.

Consider the problem (MOLP) and $\mathcal{P}=\{p^{1},\ldots,p^{q}\}$ as the set of its objectives. The objective $p^{\,j}\in\mathcal{P}$ is called nonessential if for the problem

[TABLE]

$\operatorname{Eff}(\hyperlink{molpnonessential}{\text{MOLP}^{*}})=\operatorname{Eff}(\hyperlink{molp}{\text{MOLP}})$ holds. Otherwise, $p^{\,j}$ is essential. A subset $\mathcal{P}^{*}\subseteq\mathcal{P}$ is called a nonessential system of objectives if all objectives in $\mathcal{P}^{*}$ can be removed from (MOLP) simultaneously in the above sense, i.e., without altering $\operatorname{Eff}(\hyperlink{molp}{\text{MOLP}})$.

It is important to note that a system of nonessential objectives does not have to be nonessential. For the example in Figure˜1(a), $\{p^{2},p^{3},p^{4}\}$ and $\{p^{3},p^{4}\}$ are not nonessential, even though the individual objectives are nonessential themselves. However, the systems $\{p^{2},p^{3}\}$ and $\{p^{2},p^{4}\}$ are nonessential.

In general, it depends on the feasible set which objectives or systems of objectives, respectively, are nonessential. For instance, consider the problem in Figure˜1(b) that has the same objectives but a slightly modified feasible set. Here, $p^{4}$ is essential. This dependency also holds for the concepts of absolutely and relatively redundant systems of objectives, cf. [8].

However, a statement not depending on the feasible set ist more valuable. This is because in usual cases we do not know the feasible set exactly. It is a hard task to figure it out, and therefore this does not lead to a notable speedup in computing time. The following theorem shows that a statement independent of the feasible set can be derived for objectives that can be represented by conic combinations of the others. This result has already been investigated in previously mentioned studies in [8, 24]. Here, we present a proof that uses the reformulation of the MOLP into an equivalent VLP, employing the concept of generalized minimality from Section˜3. This proof is similar to the one in [24] that uses the cone from Equation˜3.10, but it uses a polyhedral cone instead. In [8], part˜i of the theorem has been proven via the concept of scalarization, cf. Theorem 4.2 in [12]. Part˜ii then follows from iterating part˜i over the elements of $\mathcal{P}^{*}$. Note that we do not use the concept of scalarization in our proof.

Theorem 4.1**:**

Consider the problem (MOLP), and let $\mathcal{P}$ be the set of its objectives.

1. (i)

The objective $p\in\mathcal{P}$ is nonessential if

[TABLE]

where $\operatorname{cone}$ denotes the convex conic hull. 2. (ii)

The subset $\mathcal{P}^{*}\subseteq\mathcal{P}$ is a nonessential system of objectives if

[TABLE]

• Proof.

As the objective in a nonessential system $\mathcal{P}^{*}=\{p\}$ with only one element is, by definition, always nonessential, part˜i follows directly from part˜ii. Therefore, it suffices to show part˜ii.

Take a trivial decomposition of the objective matrix $P$, i.e., $L=P$ and $R$ being the $n$-dimensional identity matrix. From $L$, one obtains the cone $C$ via Equation˜3.2 and thus the associated problem (VLP) according to Equation˜3.4. Theorem˜3.3 then yields $\operatorname{\overline{Eff}}(\hyperlink{vlp}{\text{VLP}})=\operatorname{Eff}(\hyperlink{molp}{\text{MOLP}})$.

Now remove all rows from $P$ which are contained in $\mathcal{P}^{*}$ and obtain the submatrix $P^{\prime}\in\mathbb{R}^{q^{\prime}\times k}$. As we will prove below, $P^{\prime}$ also generates $C$. Thus, the VLP will not be altered when replacing $P$ by $P^{\prime}$. However, one can apply Theorem˜3.3 again to the simplified VLP. It remains an MOLP

[TABLE]

with just $q-\#\mathcal{P}^{*}$ of the $q$ original objectives, namely the elements from $\mathcal{P}\setminus\mathcal{P}^{*}$. Due to $\operatorname{Eff}(\hyperlink{molpreduced}{\text{MOLP}^{\prime}})=\operatorname{\overline{Eff}}(\hyperlink{vlp}{\text{VLP}})=\operatorname{Eff}(\hyperlink{molp}{\text{MOLP}})$, $\mathcal{P}^{*}$ forms a nonessential system of objectives.

It remains to show that the cone $C^{\prime}=\{z\in\mathbb{R}^{k}\,|\,P^{\prime}z\geq 0\}$ equals $C$. Note that $P^{\prime}$ only contains rows from $P$. Therefore, $C\subseteq C^{\prime}$. To prove the converse inclusion, take some $z\in C^{\prime}$. For $p\in\mathcal{P}\setminus\mathcal{P}^{*}$ (i.e., the rows of $P^{\prime}$) one directly obtains $p^{\mathsf{T}}z\geq 0$. According to the assumption $\mathcal{P}^{*}\subseteq\operatorname{cone}(\mathcal{P}\setminus\mathcal{P}^{*})$, for $p\in\mathcal{P}^{*}$ (i.e., all remaining rows of $P$) there exists $\lambda\in\mathbb{R}_{+}^{q^{\prime}}$ such that $p=(P^{\prime})^{\mathsf{T}}\lambda$ holds. Thus, it follows $p^{\mathsf{T}}z=(\lambda^{\mathsf{T}}P^{\prime})z=\lambda^{\mathsf{T}}(P^{\prime}z)\geq 0$. Summarized, this yields $Pz\geq 0$, and therefore $z\in C$. ∎

Remark 4.2**:**

Alternatively, one can prove the statement $C=C^{\prime}$ via the polar cones $C^{*}=\operatorname{cone}(\mathcal{P})$ and $(C^{\prime})^{*}=\operatorname{cone}(\mathcal{P}\setminus\mathcal{P}^{*})$, respectively. Note that the bi-polar cone of a nonempty closed convex cone is the cone itself [21]. Therefore, it suffices to show $C^{*}=(C^{\prime})^{*}$. $\mathcal{P}\setminus\mathcal{P}^{*}\subseteq\mathcal{P}$ directly yields $C^{*}\supseteq(C^{\prime})^{*}$. Conversely, the assumption $\mathcal{P}^{*}\subseteq\operatorname{cone}(\mathcal{P}\setminus\mathcal{P}^{*})$ together with the trivial inclusion $\mathcal{P}\setminus\mathcal{P}^{*}\subseteq\operatorname{cone}(\mathcal{P}\setminus\mathcal{P}^{*})$ implies $\mathcal{P}\subseteq\operatorname{cone}(\mathcal{P}\setminus\mathcal{P}^{*})$, and thus $C^{*}\subseteq(C^{\prime})^{*}$.

According to the theorem, for the examples in Figure˜4.1 the system $\{p^{2},p^{3}\}$ is non-essential independent of the feasible set since $p^{2}$ and $p^{3}$ can be expressed as conic combinations of the other objectives $p^{1}$ and $p^{4}$. Note that Theorem˜4.1 only gives a sufficient but not necessary condition [8]; in the example in Figure˜1(a), the objective $p^{4}$ is non-essential even though it is not a conic combination of the others. For a both necessary and sufficient condition, we refer to Theorem 2.4 in [8], but this result is rather theoretical since it depends on the feasible set as one can see from the example in Figure˜4.1.

There are plenty of approaches to determine nonessential objectives and minimal generating systems, respectively, cf. [8, 7, 18, 24]. Alternatively, one can use the transformation of the MOLP into a VLP according to Theorem˜3.1 and the above Theorem˜4.1. To see how this can be achieved, we first introduce the following corollary [8, 7]. Here, a subset $D^{\prime}\subseteq D$ of a finite set $D\subseteq\mathbb{R}^{q}$ is called a minimal spanning system of $\operatorname{cone}(D)$ if $\operatorname{cone}(D^{\prime})=\operatorname{cone}(D)$ and for any proper subset $D^{\prime\prime}\subsetneq D^{\prime}$ one has $\operatorname{cone}(D^{\prime\prime})\subsetneq\operatorname{cone}(D^{\prime})$.

Corollary 4.3**:**

Consider the problem (MOLP), and let $\mathcal{P}$ be the set of its objectives. Moreover, let $\mathcal{P}^{\prime}\subseteq\mathcal{P}$ be a minimal spanning system of $\operatorname{cone}(\mathcal{P})$. Then, $\mathcal{P}\setminus\mathcal{P}^{\prime}$ is a nonessential system of objectives.

• Proof.

Regarding Theorem˜4.1, it suffices to show $\mathcal{P}\setminus\mathcal{P}^{\prime}\subseteq\operatorname{cone}(\mathcal{P}\setminus(\mathcal{P}\setminus\mathcal{P}^{\prime}))$. This follows immediately from $\mathcal{P}\setminus\mathcal{P}^{\prime}\subseteq\operatorname{cone}(\mathcal{P})$, the assumption $\operatorname{cone}(\mathcal{P})=\operatorname{cone}(\mathcal{P}^{\prime})$, and $\mathcal{P}^{\prime}=\mathcal{P}\setminus(\mathcal{P}\setminus\mathcal{P}^{\prime})$. ∎

If $\operatorname{cone}(\mathcal{P})$ is pointed (i.e., it has trivial lineality space $\{0\}$), the extreme directions $\operatorname{ext\,dir}(\operatorname{cone}(\mathcal{P}))$ of $\operatorname{cone}(\mathcal{P})$ form a minimal spanning system. From the corollary then follows that it suffices to determine those extreme directions and reduce the objective matrix to the rows corresponding to them. Note that if $\operatorname{cone}(\mathcal{P})$ has nontrivial lineality space $T\neq\{0\}$, one can decompose $\operatorname{cone}(\mathcal{P})$ via $\operatorname{cone}(\mathcal{P})=T+(\operatorname{cone}(\mathcal{P})\cap T^{\perp})$ [21], determine the extreme directions of $\operatorname{cone}(\mathcal{P})\cap T^{\perp}$, and then add the generating directions of $T$ in order to obtain a minimal spanning system.

When using the transformation of the MOLP into a VLP according to Theorem˜3.1 in practice, one can calculate a minimal H-representation $C^{\prime}=\{z\in\mathbb{R}^{k}\;|\;L^{\prime}z\geq 0\}$ of the ordering cone $C$ where $L^{\prime}$ is a submatrix of $L$ up to scaling of rows. Similar to the proof of Theorem˜4.1, a re-transformation of the VLP yields an equivalent MOLP with objective matrix $P^{\prime}=L^{\prime}R$. In fact, the determination of a minimal H-representation is the same as the removal of nonessential objectives due to Corollaray˜4.3 and the above paragraph.

To show this, we denote the transposed rows of $L$ by $\mathcal{L}=\{\ell^{1},\ldots,\ell^{q}\}$. Without loss of generality, we assume that the rows of $L$ and $L^{\prime}$ are both ordered and scaled such that the transposed rows of $L^{\prime}$ form the set $\mathcal{L}^{\prime}=\{\ell^{1},\ldots,\ell^{q^{\prime}}\}$. Furthermore, we set $\mathcal{L}^{*}=\mathcal{L}\setminus\mathcal{L}^{\prime}$. Due to the injectivity of $R^{\mathsf{T}}$, the elements of $\mathcal{L}$ and $\mathcal{L}^{*}$ uniquely correspond to the elements of $\mathcal{P}=R^{\mathsf{T}}[\mathcal{L}]$ and $\mathcal{P}^{*}=R^{\mathsf{T}}[\mathcal{L}^{*}]$, respectively. By construction, the polar cones of $C$ and $C^{\prime}$ are equal, i.e., $\operatorname{cone}(\mathcal{L})=\operatorname{cone}(\mathcal{L}\setminus\mathcal{L}^{*})$. Thus, $\mathcal{L}^{*}$ is a subset of $\operatorname{cone}(\mathcal{L}\setminus\mathcal{L}^{*})$. Applying $R^{\mathsf{T}}$ to this inclusion, one obtains $\mathcal{P}^{*}\subseteq\operatorname{cone}(\mathcal{P}\setminus\mathcal{P}^{*})$, which means that $\mathcal{P}^{*}$ is nonessential. Conversely, if $\mathcal{P}^{*}=\mathcal{P}\setminus\operatorname{ext\,dir}(\operatorname{cone}(\mathcal{P}))$, the injectivity of $R^{\mathsf{T}}$ yields $\mathcal{L}^{*}=\mathcal{L}\setminus\operatorname{ext\,dir}(\operatorname{cone}(\mathcal{L}))$. As $\operatorname{ext\,dir}(\operatorname{cone}(\mathcal{L}))$ is a minimal spanning system of $\operatorname{cone}(\mathcal{L})$ if $\operatorname{cone}(\mathcal{P})$ is pointed, the polar cone of $\operatorname{cone}(\operatorname{ext\,dir}(\operatorname{cone}(\mathcal{L})))$ must be given by a minimal H-representation of $C$. If $\operatorname{cone}(\mathcal{P})$ is not pointed, a similar statement follows using the decomposition of $\operatorname{cone}(\mathcal{P})$ mentioned above.

Therefore, if the used VLP solver inherently calculates a minimal H-representation (or similarly, a V-representation), the determination of nonessential objectives is obsolete. Otherwise, it is worth a thought to execute such a facet or vertex enumeration manually. This is because one then searches for nonessential objectives in $\mathbb{R}^{k}$ instead of $\mathbb{R}^{n}$, and the ordering cone becomes simpler.

5 Examples

In order to verify that the above-described approach indeed provides the desired advantages, we analyze in an example how the computing time depends on the number $q$ of objectives in the objective matrix, while holding all the other parameters constant. For the construction of the example, we use the bensolve tools package [14, 5] in version 1.3 for GNU Octave version 9.4.0, and for solving, we apply the BENSOLVE VLP solver [16, 15] in version 2.1.0. The CPU used is an Apple M2 MacBook Air chip.

The problem set is constructed from Example 13 of [13] in $\mathbb{R}^{12}$. Let $P_{0}$ be a regular simplex in $\mathbb{R}^{12}$ with edge length $1$, symmetrically placed around the origin (generated via the simplex function from bensolve tools). Recursively, define $P_{\ell}$ as the Minkowski sum of $P_{\ell-1}$ and its polar $\{y\in\mathbb{R}^{12}\;|\;\forall\,x\in P_{\ell-1}:y^{\mathsf{T}}x\leq 1\}$. From this sequence, we choose $P_{5}$. We calculate a P-representation of $P_{5}$ with bensolve tools, i.e., a matrix $M\in\mathbb{R}^{12\times n}$ and a polyhedron $S=\{x\in\mathbb{R}^{n}\;|\;a\leq Ax\leq b,l\leq x\leq u\}$ with $A\in\mathbb{R}^{m\times n}$ such that $P=\{Mx\;|\;x\in S\}$. The coefficient matrix $A$ is of size $m=1421$ times $n=2800$.

Since $P_{5}$ is full-dimensional, we have $\operatorname{rk}(M)=12$. We now construct a matrix $T$ supposed to reduce the rank of $M$ to $2$ as follows. In the first two columns of $T$, all entries are set to $-\sqrt{3}/6$, except the entries in rows $36912$ of the first column and in rows $14710$ of the second column which are set to zero. Column $j=3,\ldots,12$ is equal to the normalized vector of $(j-8)$ times the first plus $(j-7)$ times the second column. Finally, we choose the first $q$ rows of $TM$ as the objective matrix $P$, where $3\leq q\leq 12$. By construction of $T$, the rank of $P$ is $2$ for each such $q$.

We then apply the BENSOLVE VLP solver to solve the resulting MOLPs for every $q$ via the molpsolve function from bensolve tools. Moreover, for every test problem, we calculate an LU decomposition of $P$ by the lu function from Octave, form the cone $C$ as a polyh object and apply the VLP solver function vlpsolve from bensolve tools to the resulting VLP. The computing times of solving each MOLP and VLP (including the decomposition step) are measured. Each test problem is run seven times in succession. However, only the last five runs are used to calculate the average computing time.

Figure˜5.1 plots the average computing times against the number of objectives. The computing time for the MOLP increases exponentially with $q$. In contrast, the time for solving the VLP, including decomposition, remains nearly constant. This proves the effectiveness of the stated approach.

6 Conclusion

In this article, we examined an approach that reduces the number of objectives in a multi-objective linear program (MOLP) to the rank of the objective matrix. This is achieved by transforming the MOLP into an equivalent vector linear program (VLP). The objective matrix is decomposed into two factors: one forms a new ordering cone, and the other factor remains as the objective matrix. Examples demonstrate the value of this approach, as it can significantly reduce computing time. Moreover, it becomes much easier for decision makers to understand and interpret the problem and its solutions. This method is not limited to MOLP but can also be applied to VLP in general.

This approach can only handle rank-minimal factorizations. Using the concept of generalized minimality, we expanded the approach to the case when the factorization is not rank-minimal and stated a more advanced theorem on the equivalence of MOLP and VLP. As a corollary, we obtained Theorem 2.11 in [8] on nonessential objectives. We illustrated that, after eliminating nonessential objectives, some redundant information (i.e., linear dependence) may remain in the problem. This occurs because linear combinations might not be conic combinations. With our approach, linear dependence is eliminated.