A simple and efficient dichotomic search algorithm for multi-objective mixed integer linear programs

We present a simple and at the same time fficient algorithm to compute all nondominated extreme points in the outcome set of multi-objective mixed integer linear programmes in any dimension. The method generalizes the well-known dichotomic scheme to compute the set of nondominated extreme points in the outcome set of a bi-objective programme based on the iterative solution of weighted sum scalarizations. It uses as a main routine a convex hull algorithm. The algorithm is illustrated with, and numerically tested on, instances of multi-objective assignment and knapsack problems. Experimental results confirm the computational efficiency of the approach. Finally, an implementation in incremental polynomial time with respect to the number of computed nondominated extreme points is possible, under the assumption that the lexicographic version of the problem can be solved in polynomial time.

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1 Introduction

Even though efficient solution methods exist for many classes of single-objective integer linear programming problems (like, for example, knapsack and assignment problems), this is in general not the case if multiple objective functions have to be considered. Multi-objective integer linear programming problems are often intractable (and hence possess a possibly huge number of efficient solutions), and the majority of efficient solutions may be unsupported, (see, for example, Ehrgott and Gandibleux, 2002). To nevertheless obtain information on the structure of a given problem and on the related trade-offs, the iterative solution of weighted sum scalarizations is an indispensible tool in many algorithms. This includes, for example, two phase methods as well as branch and bound and dynamic programming based algorithms (Ehrgott et al., 2016)

. Weighted sum scalarizations principally allow the computation of all nondominated points on the boundary of the convex hull of the set of feasible outcome vectors, and in particular of its extreme points which will be referred to as

nondominated extreme points in the following. While the identification of the relevant weights is easy in bi-objective problems, this is in general not the case in higher dimensions. Existing methods are often computationally expensive and/or complicated to use as will be illustrated in the following sections.

The goal of this paper is to devise a simple and at the same time efficient algorithm to compute all nondominated extreme points of a multi-objective mixed integer linear programming problem in any dimension. To keep the exposition simple, we will focus on multi-objective integer linear programmes in the following. However, most of the results immediately transfer to the case of multi-objective mixed integer linear programmes. We will indicate this in the following whenever appropriate.

A multi-objective integer linear programme is written as

(MOIP)

where and . denotes the set of feasible solutions of the problem and is defined by

 X={x∈Zn:Ax=b, x≧0}, (1)

with and . Unless stated otherwise, we will assume that all data is integer. The outcome set is defined by .

We assume that no feasible solution minimizes all objective functions simultaneously and that the ideal point with components , , exists and is strictly positive, i.e. , . Note that this is not a restrictive assumption. We use the following notation for componentwise orders in . Let . We write if for , if and , and if . We define and analogously and .

Definition 1.

A feasible solution is efficient (weakly efficient) if there does not exist any other feasible solution such that (). If is efficient, then is a nondominated outcome vector (weakly nondominated outcome vector) or nondominated point (weakly nondominated point) for short. If are such that we say that dominates and dominates . Feasible solutions are equivalent if .

The set of all efficient solutions ( of all weakly efficient solutions) and the set of all nondominated outcome vectors ( of all weakly nondominated outcome vectors) are referred to as the efficient set (weakly efficient set) and the nondominated set (weakly nondominated set), respectively.

The nondominated set is bounded by two particular vectors: the ideal point with and the nadir point with , for .

Several classes of efficient solutions can be distinguished.

• Supported efficient solutions are optimal solutions of a weighted sum single objective problem

(MOIP)

for some weight . Their images in the objective space are supported nondominated points. We use the notations and , respectively. All supported nondominated points are located on the boundary of the convex hull of (), i.e., they are nondominated points of

• Nonsupported efficient solutions are efficient solutions that are not optimal solutions of (MOIP) for any . Nonsupported nondominated points are located in the interior of the convex hull of .

In addition we can distinguish two classes of supported efficient solutions, namely

• supported efficient solutions whose objective vectors are located on the vertex set of (we call these extremal supported efficient solutions, , and nondominated extreme points, , respectively) and

• those supported efficient solutions for which is located in the relative interior of a face of . For such a solution there exist extremal supported efficient solutions satisfying for all , , and with such that . The corresponding sets of non-extremal supported efficient solutions and their outcome vectors are denoted by and , respectively.

For a given subset of the objective space , we denote by the set of all nondominated points relatively to , i.e. , and by the set of all weakly nondominated points relatively to , i.e. . Let be a face of a convex polytope of dimension . We say that is nondominated if , and that is weakly nondominated if . If is a facet, i.e. a -dimensional polytope, then is nondominated (weakly nondominated) if, and only if, every normal vector to pointing to the interior of is such that () in minimization problems, and () in maximization problems, for all . A maximal nondominated face is a nondominated face that is not contained in any other nondominated face.

The purpose of this paper is to develop a simple and at the same time efficient dichotomic search algorithm that determines one efficient solution for each nondominated extreme point, or equivalently, to generate all the nondominated extreme points in the outcome set of a multi-objective integer programme.

When this reduces to solving a sequence of single objective problems (MOIP) with , because the “natural” order of nondominated points (i.e. implies ) allows us to search by dichotomy. This dichotomic search provides the appropriate values of in a straightforward way, see e.g. Cohon (1978)

, who calls the procedure “noninferior set estimation method”, or

Aneja and Nair (1979). An extension of this procedure to dimensions was stated as a major challenge in Ehrgott and Gandibleux (2002), and has remained unsolved until recently with the methods proposed by Przybylski et al. (2010b), Özpeynirci and Köksalan (2010) and Bökler and Mutzel (2015), which can be applied to compute the nondominated extreme points of any MOIP. The proof of their correctness explicitly or implicitly relies on a decomposition of the weight set, i.e., the set of all relevant weights for the considered weighted sum scalarizations (MOIP). Using a dual interpretation, Bökler and Mutzel (2015) show in addition that, if the respective weighted sum scalarizations can be solved in polynomial time, their method runs in output polynomial time (with respect to the number of nondominated extreme points computed) for every fixed number of objectives.

The iterative solution of weighted sum scalarizations and the corresponding decomposition of the weight set that comprises all relevant weights play an important role also in the context of multi-objective linear programmes (MOLP), where is replaced by in (1). It is well known that every efficient solution of (MOLP) is supported, (see Isermann, 1974; Steuer, 1985). Algorithms for the solution of MOLPs are available since the seventies with the first kind of methods being multicriteria simplex algorithms (see, for example, Ehrgott and Wiecek, 2005, and references therein). More recently, another stream of research has been proposed: enumerating points in the objective space rather than solutions in the decision space . Benson (1998a, b) has argued that this is advantageous because in general is much smaller than . Weight set decomposition has been used to prove the correctness of both kinds of methods (see Yu and Zeleny, 1975, for the multi-objective simplex algorithm). Algorithms computing a weight set decomposition by determining simultaneously the set of nondominated extreme points of an MOLP have also been proposed. For example, Benson and Sun (2002) use a decomposition of the weight set described in Benson and Sun (2000). Weight set decomposition has also been the foundation for the extension of the primal-dual simplex algorithm to the multi-objective case, see Ehrgott et al. (2007). A dual variant of Benson’s outer approximation algorithm is suggested in Ehrgott et al. (2012). It is based on the formulation of dual problems that include information on weights describing the facets of the primal polyhedron (see Heyde and Löhne, 2008; Hamel et al., 2014), i.e., on the weight set decomposition. Since weight cells correspond to nondominated extreme points in the objective space, primal dual methods for MOLPs imply a double description of the nondominated set. This interrelation has beeen used in Csirmaz (2018) to further refine primal dual methods for MOLPs by combining a combinatorial enumeration strategy for the nondominated extreme points (yielding a provably best possible iteration count) with tailored single-objective LP-solver calls. Note that the approach of Heyde and Löhne (2008) and Hamel et al. (2014) can also be viewed as the basis of the work of Bökler and Mutzel (2015)

for multi-objective combinatorial optimization problems mentioned above.

In a different line of research, there exists a large variety of methods whose purpose is to approximate the set of nondominated points of convex or non-convex multi-objective problems (see Ruzika and Wiecek, 2005, for a review). In particular, methods for the approximation of multi-objective convex problems can be used for the approximation of the set of nondominated extreme points of MOIPs, see, for example, Schandl et al. (2002) and Rennen et al. (2011). For many of these methods, an exact representation containing all nondominated extreme points can be obtained by driving the quality indicator to zero. Even if it is not the original purpose of such approximation methods, this shows that many of them can be applied to compute the set of nondominated extreme points of MOIPs.

In the following, we will combine ideas from approximation methods with the approaches of Przybylski et al. (2010b) and Özpeynirci and Köksalan (2010) to obtain a simple and efficient dichotomic scheme for the exact computation of all nondominated extreme points of MOIPs. Note that we could also make the reverse step, i.e., by combining our new algorithm with appropriate quality indicators, it can be easily converted into a method to approximate the convex hull of the nondominated set of an MOIP.

Since dichotomic search algorithms rely on the repeated solution of weighted sum scalarizations (MOIP), it is reasonable to suppose for a practical application that the single-objective problems (MOIP) can be solved efficiently. Therefore, the method we propose would most likely be applied to multi-objective combinatorial optimization problems. As in Przybylski et al. (2010b), we will use the assignment and knapsack problems with objectives, respectively (AP) and (KP), to illustrate and test our algorithms. The multi-objective assignment problem can be stated as

(AP)

where all objective function coefficients are non-negative integers and is the vector of decision variables. The multi-objective knapsack problem is given by

(KP)

where all objective function coefficients are non-negative integers, and are positive integers, and is the vector of decision variables.

The remainder of the paper is structured as follows. We review the bi-objective dichotomic scheme and the difficulties of its extension to the multi-objective case in Section 2. The methods proposed for the determination of nondominated extreme points of multi-objective (mixed-)integer linear programmes are reviewed in Section 3. Solution methods for computing a convex approximation of multi-objective optimization problems are reviewed in Section 4. New developments extending ideas proposed in the literature are proposed in Section 5, and these developments are immediately used to define two new solution methods. Finally, experimental results are provided on multi-objective assignment and knapsack problems in Section 6 and show the practical efficiency of the proposed methods.

2 Bi-objective dichotomic scheme and difficulties in its extension to the multi-objective case

The bi-objective dichotomic scheme has been designed using specific properties of the bi-objective case. As a consequence, its extension to the multi-objective case is not obvious.

2.1 Classical dichotomic scheme in the bi-objective case

The dichotomic scheme is based on the consideration of consecutive supported nondominated points and with respect to one objective, i.e. and . A weighted sum problem (MOIP) with and is solved to find new supported points located “between” and . The vector corresponds to a normal to the line joining and , as illustrated in Figure 2 with a negative multiple of to highlight the optimization sense. Following the solution of this weighted sum problem, a supported nondominated point is obtained and two cases are possible.

1. If , then is necessarily a new supported nondominated point and two new problems (MOIP) have to be solved, one with defined by and and one with defined by and (see Figure 2).

2. If , then the search stops, and is a part of an edge of .

This scheme is initialized with nondominated points minimizing respectively the first and the second objectives, and is usually implemented recursively. More detailed descriptions can be found in Aneja and Nair (1979) and Cohon (1978).

2.2 Difficulty in the extension to the multi-objective case

The dichotomic scheme is not immediate to generalize to more than two objectives, because as illustrated in Figures 2 and 2, it relies on the natural order of nondominated points in the objective space, i.e. that implies The absence of this natural order causes the following difficulties when (Przybylski et al., 2010b).

• To define a hyperplane in

, points are necessary. But there might be more than different nondominated extreme points (at most lexicographically optimal points) resulting from the initial single objective optimizations. It is unclear which points to choose to define a hyperplane to start the procedure.

• Even if the initialization yields exactly initial points, the normal to a hyperplane defined by nondominated points does not necessarily have positive components. Then the optimization of (MOIP) does not necessarily yield other supported nondominated points.

The following example has been used in Przybylski et al. (2010b) to illustrate how the dichotomic scheme can fail.

Example 1.

Consider an instance of the assignment problem with three objectives (Tenfelde-Podehl, 2003) where

 C1=⎛⎜ ⎜ ⎜⎝3645235435424536⎞⎟ ⎟ ⎟⎠, C2=⎛⎜ ⎜ ⎜⎝2354534352644525⎞⎟ ⎟ ⎟⎠, and C3=⎛⎜ ⎜ ⎜⎝4242424642632453⎞⎟ ⎟ ⎟⎠.

Let us try to apply the usual dichotomic scheme to this instance. The three single objective assignment problems with objective coefficients , , yield three points:

:

with and objective vector is the unique optimal solution minimizing the first objective.

:

with and objective vector is the unique optimal solution minimizing the second objective.

:

with and objective vector is the unique optimal solution minimizing the third objective.

The normal to the plane defined by , and is either or . In both cases not all components are positive. Solving (MOIP) we get the dominated point for with as optimal solution. Solving (MOIP) we obtain , or . Therefore, the dichotomic scheme stops without finding any further supported nondominated points. However, for with is a supported nondominated point that can be found solving (MOIP) with .

Consequently, a straight-forward transposition of the bi-objective case is not possible.

3 Methods for the multi-objective case

Three methods have been proposed for the computation of the set of nondominated extreme points of MOIP.

3.1 The method by Przybylski et al. (2010b)

The method by Przybylski et al. (2010b) is based on the computation of a weight set decomposition. The weight set is defined by

 W0:={λ∈Rp:λk>0 for k∈{1,…,p},λp=1−p−1∑k=1λk}, (2)

and can be seen as a normalized weight space. is a polytope of dimension and in particular, it is bi-dimensional in the three-objective case (very helpful for illustration purposes).

Given a supported nondominated point , the set is defined by

 W0(y):={λ∈W0:λTy=min{λTy′:y′∈convY}} (3)

and corresponds to the subset of weights for which is the image of an optimal solution of (MOIP). The method by Przybylski et al. (2010b) is based on the following results.

Proposition 1 (Przybylski et al., 2010b).

Let be a supported nondominated point.

1. is a convex polytope.

2. Nondominated point is a nondominated extreme point of if and only if has dimension .

3. Let be a set of supported nondominated points. Then

 YSN1⊆S⟺W0=⋃y∈SW0(y).
Definition 2 (Przybylski et al., 2010b).

Two nondominated extreme points and are called adjacent if and only if their common facet in the weight set is a polytope of dimension .

Given the set of nondominated extreme points , the decomposition of in sets is given by Proposition 1(4). Figure 3 illustrates this decomposition for the case of Example 1.

Proposition 1(1) provides a way to compute the sets knowing the set . However, is not known at the beginning of an iterative algorithm since it is the set that should be computed simultaneously with the weight set decomposition. Przybylski et al. (2010b) have thus proposed to consider the directly computable sets

 W0S(y):={λ∈W0:λTy≤λTy′ % for all y′∈S∖{y}}

for subsets of supported nondominated points . Similar to the dichotomic scheme for the bi-objective case, a subset of supported nondominated points (containing possibly non-extreme points) is known at each step of the algorithm. Proposition 1(5) is used as an optimality condition in the algorithm, in the sense that it allows to show that . At any step of the algorithm, for all and . Each new explored supported nondominated point can be used to update the sets . At termination of the algorithm, for all and we have thus by application of Proposition 1(5).

The principle of the method is to consider each known supported point , and to show either that or to identify new supported nondominated points allowing next an update of . To show that (or not) is done by showing that all facets of both polytopes are (not) common.

More precisely, given two nondominated extreme points that are adjacent w.r.t. , the question is whether the common facet of and is also the common facet of and or not. The facet is investigated in order to identify new supported points or to confirm that . In particular if , is an edge and its investigation is realized by the computation of the nondominated extreme points of a bi-objective problem. If , nondominated extreme points of -objective problems must be computed, which is implemented by a recursive application of the method.

3.2 The method by Bökler and Mutzel (2015)

Motivated by the recent work on a dual Benson’s algorithm (see Ehrgott et al., 2012; Hamel et al., 2014), Bökler and Mutzel (2015) take a dual perspective on the weight set decomposition of Przybylski et al. (2010b). The following interpretation is based on these references and adapted to the notation from Section 3.1. Let

be the closure of (see (2)), that includes weights with components equal to . For , consider the linear programming relaxation of (MOIP) given by

 min{¯λTCx:Ax=b,x≧0}

and the corresponding dual linear program

 max{bTu:ATu≦CT¯λ}.

Then we can associate with every weight (that is uniquely determined by its first components) a set of dual feasible solutions with corresponding dual objective values . This information is comprised in the dual polyhedron

(4)

introduced in Heyde and Löhne (2008). We are particularly interested in maximizing for different scalarizations , that is, we are interested in the upper envelope of w.r.t. the last component. In other words, we are looking for a maximal subset of w.r.t. the cone . Using linear programming duality, this -maximal subset of can be written as

(5)

where in this context, .

Heyde and Löhne (2008) showed that the -maximal facets of correspond to the (weakly) nondominated extreme points of (note that all extreme points of are in fact nondominated), i.e., to the nondominated extreme points in , and vice versa. Note that this observation is also reflected in Proposition 1(3), i.e., there is a one-to-one correspondence between the -maximal facets of and the sets with . Note also that remains optimal for all weights and thus changes linearly with on . Consequently, there is also a one-to-one correspondence between the extreme points of and the extreme points of the closure of the weight sets , .

From an algorithmic point of view, this dual interpretation gives rise to an alternative way to (implicitly) compute the weight set decomposition. Following the description in Bökler and Mutzel (2015), the dual method also works with a partial list of nondominated extreme points of that define a subset of the facet describing inequalities of . In order to test whether an extreme point of the current intermediate dual polyhedron (given by the facet describing inequalities) is also an extreme point of , a weighted sum problem with weight is solved. Bökler and Mutzel (2015) suggest to apply a lexicographic weighted sum scalarization in order to avoid non-extreme supported efficient solutions and also dominated solutions in the case of weights with components equal to . If the optimal objective value equals , the above question is answered positively, and otherwise a new facet describing inequality is found and added to the intermediate description of .

Bökler and Mutzel (2015) noticed that this algorithm can also be applied to multi-objective combinatorial optimization problems. Indeed, for all combinatorial optimization problems (and more generally for all mixed-integer linear programmes), there exists a linear programming formulation, which could be used in the definition of the dual Polyhedron (4). We emphasize that to know the linear formulation of the problem is in fact not necessary, since only optimal objective values of weighted sum problems are required in the method, and these optimal values do not depend on the formulation of the problem, as can be seen for example in formulation (5).

In comparison to the algorithm of Przybylski et al. (2010b), the dual method avoids the time-consuming analysis of common facets of weight sets between pairs of nondominated extreme points, c.f. Section 3.1.

Under the assumptions that the number of objective functions is fixed, the weighted sum scalarizations (MOIP) can be solved in polynomial time, and the polytope is computed with a dual algorithm to a statical convex hull algorithm (e.g. Chazelle, 1993), Bökler and Mutzel (2015) show that the dual Benson method runs in output polynomial time, i.e. its running time is bounded by a polynomial in the input and the output size. Moreover, if the lexicographic weighted sum problems can still be solved in polynomial time, the dual Benson method runs in incremental polynomial time, i.e., the -th delay (the running time between the output of the -th and the -st solution) is bounded by a polynomial in the input and .

3.3 The method by Özpeynirci and Köksalan (2010)

The method by Özpeynirci and Köksalan (2010) is performed directly in the objective space. In order to compute all nondominated extreme points, Özpeynirci and Köksalan (2010) consider the convex hull of the outcome set , and in particular the nondominated frontier. As is a polytope, Özpeynirci and Köksalan (2010) define the nondominated frontier as the union of all of its nondominated faces. However, if in the bi-objective case all maximal nondominated faces are nondominated edges, with objectives there can be maximal nondominated faces of dimension 1, 2,, . Özpeynirci and Köksalan (2010) illustrate this fact in the three-objective case using the numerical instance of Example 1. There is indeed one maximal nondominated face of dimension 2 (facet) given by the weight defined by the intersection of the sets , and , and one maximal nondominated face of dimension 1 given by any weight in the interior of the edge (see Figure 3).

Özpeynirci and Köksalan (2010) introduce dummy points in order to modify the structure of the nondominated frontier. These points are defined by

 mq:=Meq for q=1,…,p

where is the -th unit vector, and is a large positive constant. The set of dummy points is defined by . As Özpeynirci and Köksalan (2010) make the assumption that for all , dummy points are thus nondominated points of . must be chosen large enough so that nondominated extreme points of remain nondominated extreme points of . Lower bound values for that guarantee this, under the assumption that objective coefficients and variables are integer, are given in (Özpeynirci, 2008). The following properties are then verified.

Proposition 2 (Özpeynirci and Köksalan, 2010).

The introduction of dummy points in the set of nondominated extreme points has the following consequences.

1. For , we denote by the set of weights for which is the image of an optimal solution of , modified by the introduction of dummy points. The weight set decomposition becomes , and for all .

2. Every point is adjacent to at least points in .

3. Any pair of dummy points , are adjacent for .

4. If and , then every point is adjacent to at least points in .

5. All maximal nondominated faces of are facets. Thus, is fully described by a union of facets.

The method by Özpeynirci and Köksalan (2010) consists in the computation of , the extreme points of which are . According to Proposition 2(5), all maximal nondominated faces are facets. It is therefore possible to design an algorithm that identifies these facets.

The algorithm starts with the computation of . A set is iteratively computed until all facets of are identified.

At each iteration of the algorithm, a subset of points (initially ) called stage, is considered with the normal of the hyperplane it defines. If , then the solution of (MOIP) allows to obtain a supported nondominated point . If then is either a facet or a part of a facet of . Otherwise, is added to and new stages are generated , ,…, and considered next if not yet visited.

However, despite Proposition 2(5), an arbitrary stage does not necessarily have a normal (note that in this case, is not part of ). In this case, the weighted sum problem (MOIP) is not solved. Nevertheless, to observe such a stage cannot be a stopping condition for the algorithm, as the enumeration would be incomplete (see Example 1). In order to continue the enumeration, the authors use implicitly the fact that is a union of facets. Consequently, other (unvisited) stages can be defined using points in . Özpeynirci and Köksalan (2010) have proposed to choose a point together with such that for all . Such a point necessarily exists since for all . The stages , , , are next generated to continue the execution of the algorithm.

4 Approximation Algorithms

There is an extensive literature on the approximation (with a guarantee of quality) of the set of nondominated points of convex or non-convex multi-objective problems (see Ruzika and Wiecek, 2005, for a review). We are interested here only in methods that generate convex approximations of multi-objective problems.

The quality of the approximation is usually measured according to an indicator that can be very different for different methods. An a priori quality can be defined by fixing a target value for the quality indicator. In particular by fixing the quality indicator to 0, the obtained approximation becomes exact, i.e. all nondominated extreme points are found. Even if it is not the purpose of these methods, this shows that methods able to compute the set of nondominated extreme points of any MOIP have been proposed before the methods proposed in (Przybylski et al., 2010b) and (Özpeynirci and Köksalan, 2010). Of course, we can expect some weaknesses in the use of approximation methods used with an exact purpose, due to their initial design for another purpose. Our aim here is not to give a complete overview of these methods. We will just review properties that will be useful for the (exact) method we propose next, or that can be seen as related to our work.

4.1 The method by Schandl et al. (2002)

Schandl et al. (2002) have proposed methods for the approximation of the nondominated set of a multi-objective programme. For convex problems, their method can be interpreted as an extension of the dichotomic search to higher dimensions: starting from an initial approximation given by the lexicographic minima and using the nadir point (or an approximation of the nadir point) as reference point , the convex hull of all these points is computed. In each iteration of the procedure, a cone spanned by and a facet of this convex hull is selected for further refinement based on a problem specific error measure. In this cone, the normal of the defining facet is used to define weights for a subproblem (MOIP). The solution of this subproblem (MOIP) leads to a new point that is included in the convex hull for the next iteration. The procedure stops as soon as the approximation error falls below a prespecified threshold. In this way, a polyhedral approximation of the set is generated.

Note that if this method is applied in order to generate all nondominated extreme points of (MOIP), the reference point has to be selected to satisfy to ensure that .

The difficulties indicated in Section 2.2 are also noted in Schandl et al. (2002). Nevertheless, weights are used in this procedure to generate points that are not necessarily nondominated, but useful for the update of the convex hull. This method is thus not stopped because of the presence of weights with negative components. Applying this idea in the case of Example 1, we could keep the point obtained by the solution of (MOIP) and compute in order to obtain new facets with normals that can be used to define new weights for later iterations. Finally, the whole set or a subset of it, can be computed in this way and can be deduced by a filtering step.

4.2 The method by Rennen et al. (2011)

Rennen et al. (2011) develop another method to approximate the nondominated set of multi-objective programmes with convex objective functions and feasible sets. Their method is also based on the computation of the convex hull of a growing set of points . Interestingly enough, they add dummy points to the set of points that constitutes the current approximation of the nondominated set. The aim of this is to partially cope with the difficulties related to dominated facets. Namely, for each extreme point of , dummy points are defined in the following way:

 dik(y)={pyubk+θif k=iykotherwise

for all , where is an upper bound on the th component value of any nondominated point and is a positive constant.

It is then shown (Rennen et al., 2011, Lemma 2) that all relevant facets, i.e. facets that have at least one non-dummy point as extreme point, are weakly nondominated in the sense that any normal vector to any such facet pointing to the interior of is in .

Compared to Özpeynirci and Köksalan (2010), this approach defines a large number of dummy points (even if some are redundant and therefore filtered) that increase the numbers of extreme points and facets of .

5 A new Exact Method

In this section, we extend some properties proposed in (Przybylski et al., 2010b), (Özpeynirci and Köksalan, 2010) and (Schandl et al., 2002), and we propose next a new method.

5.1 Further analysis

We suppose that we know a subset of supported nondominated points, and we assume that the non-extreme points are filtered from as soon as possible. If then is a full-dimensional polytope. The facets of this polytope can be computed and updated (when points are added to ) using a convex hull algorithm. We consider the sets and for all as defined in (Przybylski et al., 2010b). In the following, it will be important to notice that is an open polytope, and for any supported point , and may be closed polytopes, or polytopes that are neither closed nor open. These polytopes may have open vertices and open faces. These open faces and open vertices are located on the boundary of .

Knowing a weight set decomposition, Proposition 3 below gives a characterization of the nondominated facets of the convex hull of a subset of supported points. This result has been stated with a different formulation and a different proof in (Przybylski et al., 2010b) and (Przybylski, 2006).

Proposition 3.

Let be a set of supported nondominated points, then there is a one-to-one correspondence between weights given by extreme points of that are located in , for , and weights associated to facets of .

Proof.

For all , can be described by a minimal set of (non-redundant) constraints for all where (plus possibly some of the constraints , and ). Given an extreme point of located in , satisfies at least of these constraints with equality as is a polytope of dimension . Therefore, there is a subset with such that for all . As , for all we have , and thus for all , . In other words, and is a part of a facet of . Since is a normal to pointing to the interior of and , is a part of a nondominated facet of .

Conversely, given a nondominated facet of , it is immediate that its associated weight belongs to and that there is thus a positive multiple in . Moreover, this weight is necessarily located at the intersection of the sets where the ’s are the extreme points of . ∎

The weights associated to other (i.e. dominated) facets of cannot be defined using a decomposition of as such weights do not belong to . However, weights with negative components are considered in (Schandl et al., 2002), in order to avoid a premature termination of the dichotomic scheme, i.e., to find appropriate weights in later iterations. In the following, we analyze under what conditions negative components in a weight associated to points in occur.

Example 2.

We consider the instance of the Assignment Problem given in Example 1. Suppose we start again by computing the set of solutions that minimize each objective. Computing (not full-dimensional as ), we get a single facet with a corresponding weight that is not suitable to continue the classical dichotomic algorithm. We analyze the cause for this using the weight space.

We consider the computation of the sets and we obtain the decomposition of given by Figure 5. We can note that the extreme points of the polytopes are not located in and that consequently there is no facet in . In other words, all facets of are dominated. To find the weight associated to such a facet of , we must extend the facets (here edges) of the sets outside of . We obtain the weight as an intersection of the extensions of , and (see Figure 5).

In order to analyse the observation of Example 2, we relax the constraints of strict positivity of the components of weights in the definition of and we obtain the extended weight set

 W0ext:={λ∈Rp with ∑pk=1λk=1}.

It should be noted that if includes weights with negative components, it does not include weights such that or . We obviously have . We accordingly extend the definition of the sets to for all by defining

 W0ext(y) :={λ∈W0ext:λTy≤λTy′ for all y′∈YSN1∖{y}} W0Sext(y) :={λ∈W0ext:λTy≤λTy′ for all y′∈S∖{y}}.

Proposition 4 provides the main properties of these new sets.

Proposition 4.

Let be a supported nondominated point.