Discrete representation of the non-dominated set for multi-objective optimization problems using kernels

Highlights

• We give algorithms which produce discrete representations of the non-dominated set.

• These representations satisfy conditions of covering, spacing, and minimum size.

• We introduce the concept of kernel for this purpose and study its properties.

• In the bi-objective case we give exact and approximate algorithms to build kernels.

• For more than two objectives we give some positive and negative results.

Abstract

In this paper, we are interested in producing discrete and tractable representations of the set of non-dominated points for multi-objective optimization problems, both in the continuous and discrete cases. These representations must satisfy some conditions of coverage, i.e. providing a good approximation of the non-dominated set, spacing, i.e. without redundancies, and cardinality, i.e. with the smallest possible number of points. This leads us to introduce the new concept of (ε, ε′)-kernels, or ε-kernels when ${\varepsilon }^{\prime }=\varepsilon$ is possible, which correspond to ε-Pareto sets satisfying an additional condition of ε′-stability. Among these, the kernels of small, or possibly optimal, cardinality are claimed to be good representations of the non-dominated set.

We first establish some general properties on ε-kernels. Then, for the bi-objective case, we propose some generic algorithms computing in polynomial time either an ε-kernel of small size or, for a fixed size k, an ε-kernel with a nearly optimal approximation ratio $1+\varepsilon$. For more than two objectives, we show that ε-kernels do not necessarily exist but that (ε, ε′)-kernels with ${\varepsilon }^{\prime }\le \sqrt{1+\varepsilon }-1$ always exist. Nevertheless, we show that the size of a smallest (ε, ε′)-kernel can be very far from the size of a smallest ε-Pareto set.

Introduction

In multi-objective optimization, in opposition to single objective optimization, there is typically no optimal solution i.e. one that is best for all the objectives. The solutions of interest, called efficient solutions, are such that any solution which is better on one criterion is necessarily worse on at least one other criterion. In other words, a solution is efficient if its corresponding vector of criterion values is not dominated by any other vector of criterion values corresponding to a feasible solution. These vectors, associated to efficient solutions, are called non-dominated points. For many multi-objective optimization problems, one of the main difficulties is the large cardinality of the set of non-dominated points (or Pareto set). For problems with continuous variables, the set of non-dominated points is usually infinite. Even in the discrete case, it is well-known that most classic multi-objective combinatorial optimization problems, like shortest path, spanning tree, assignment, knapsack, …, are intractable, in the sense that they admit families of instances for which the number of non-dominated points is exponential in the size of the instance (Ehrgott, 2000). Therefore, in all these cases, it is necessary to determine a good representation of the Pareto set so as to provide decision makers with a tractable set of points describing as well as possible the different choices. The notion of representation is understood here in a broad sense, as in Vanderpooten, Weerasena, and Wiecek (2016), as any set of points being representative of the Pareto set. This is more general than the same notion defined, e.g., in Sayin (2000) where representations are supposed to be subsets of the set to be represented. Therefore we can accept that a representation of the Pareto set might include dominated points. Indeed, provided that a set of points satisfies conditions of coverage, spacing, and cardinality presented hereafter, it fully qualifies to be a good representation. In particular, adopting a broad definition allows us to consider representations whose elements are obtained through approximate optimization (in cases where exact optimization is not available or too costly). Clearly, when dominated points are present in a representation, they must be rather good (so as to satisfy the coverage property). Moreover, when possible, guaranteeing that a representation only contains non-dominated points is a desirable property. In this paper, we propose two algorithms for the bi-objective case. The first one, based on exact optimization, produces a representation consisting of non-dominated points only. The second one, based on approximate optimization, produces a representation that may contain dominated points. In both cases, however, a priori guarantees on the quality of the returned set are provided.

Measures of the quality of a discrete representation of the Pareto set have been discussed in Faulkenberg and Wiecek (2010), Sayin (2000). As outlined in these papers three dimensions are relevant:

• coverage which ensures that any non-dominated point is represented or covered by at least one point in the representation,

• spacing, also called uniformity, which ensures that any two points in the representation are sufficiently spaced, avoiding redundancies,

• cardinality which should be minimal so as to make the representation as tractable as possible.

Coverage is the most important dimension for the representation to be meaningful. However, it must be counterbalanced by the two other dimensions which favor a uniform and small cardinality representation, respectively. While coverage on the one hand and spacing and cardinality on the other hand are clearly conflicting, the relationship between spacing and cardinality is not obvious. At first sight it could seem that improving spacing will lead to a decrease of the number of points in the representation. It must be observed, however, that imposing spacing is an additional constraint that may impact negatively on the cardinality. An interesting result in our paper is that no negative impact is to be expected in the bi-objective case, but it is no longer true when dealing with at least three objectives. This shows the interest of considering all three dimensions.

Coverage and spacing may be implemented in several ways. A distance-oriented perspective is used in Faulkenberg and Wiecek (2010), Sayin (2000). The quality of coverage is then measured by the distance between the points in the Pareto set and the points in the representation (to be minimized) while the quality of spacing is measured by the distance between the points in the representation (to be maximized). Various definitions of distances are possible leading to different types of representations, but the Euclidean norm is often used. Although natural, this geometric vision is not directly related to the decision maker’s preferences. Consider a representation containing a point y but not y′, based on the fact that y covers y′. In a distance-oriented perspective, this is justified by the fact that y and y′ are close enough. In preference-oriented perspective, the justification is that y is preferred to, or at least as good as, y′. We note that in the second case the comparison is oriented, which cannot be represented by a distance. As to spacing, points y and y′′ belong to the representation since they are far enough in a distance-oriented perspective, whereas the justification is that they are incomparable in a preference-oriented perspective. In a preference-oriented perspective, the definition of a preference relation is required. When aiming at representing the whole non-dominated set, this relation should generalize the standard Pareto dominance relation, without favoring any type of solution. A natural candidate is the $(1+\varepsilon )$-dominance relation which is an extension of the Pareto dominance relation including a tolerance threshold. Given ε > 0, which represents a tolerance on each objective, this relation is defined as follows between any two points y and y′: y$(1+\varepsilon )$-dominates y′ if y is at least as good as y′ within a factor $1+\varepsilon$ for all the objectives. This leads us to consider that y covers y′ if y$(1+\varepsilon )$-dominates y′, that is if y is at least as good as y′ considering the tolerance ε. Moreover, given a tolerance ε′, y and y′′ are sufficiently spaced if neither y$(1+{\varepsilon }^{\prime })$-dominates y′′ nor y′′ $(1+{\varepsilon }^{\prime })$-dominates y, that is there is no reason to discard any of the points y and y′′ since none of them can be considered at least as good as the other one.

This idea of coverage leads to the concept of an ε-Pareto set, introduced in Papadimitriou and Yannakakis (2000), which is a set P_ε of points such that for any non-dominated point y′, there exists a point y ∈ P_ε which $(1+\varepsilon )$-dominates y′. Note that there may exist many ε-Pareto sets, some of which can include redundancies and some of which can have a more or less small size. An interesting problem introduced in Vassilvitskii and Yannakakis (2005) and further studied in Diakonikolas and Yannakakis (2009) is the efficient construction of ε-Pareto sets of size as small as possible.

In this paper, we focus on the same issue but including also the spacing dimension. Therefore, the ε-Pareto sets studied in this work, called (ε, ε′)-kernels, are required to satisfy an additional property of stability which imposes that the points in an (ε, ε′)-kernel have to be pairwise independent relatively to the $(1+{\varepsilon }^{\prime })$-dominance relation, thus controlling spacing.

A variety of methods have been proposed taking coverage, spacing, and/or cardinality into account (see Faulkenberg, Wiecek, 2010, Ruzika, Wiecek, 2005 for surveys). Two broad classes of methods can be distinguished: (i) algorithms which generate a set of points satisfying some properties with respect to some of the quality measures, (ii) filtering techniques which start from an initial set of given points - possibly the whole Pareto set - and retain a subset of these so as to ensure properties with respect to some of the quality measures. Among recent references that are not cited in the two previous surveys, we mention Bazgan, Hugot, and Vanderpooten (2009), Diakonikolas and Yannakakis (2009), Ehrgott, Shao, and Schóbel (2011), Eusébio, Figueira, and Ehrgott (2014), Filippi and Stevanato (2013), Shao and Ehrgott (2016) and Vaz, Paquete, Fonseca, Klamroth, and Stiglmayr (2015) as examples of methods of type (i) and type (ii), respectively.

Methods of type (i) are often based on exact or approximate iterative optimizations which generate the points forming the representation. They are either generic like Diakonikolas and Yannakakis (2009), Filippi and Stevanato (2013) or specific to a class of problems like Shao and Ehrgott (2016), Ehrgott et al. (2011), Bazgan et al. (2009), and Eusébio et al. (2014) which deal respectively with multi-objective linear programming, multi-objective nonlinear convex problems, multi-objective knapsack problems, and bi-objective cost flow problems. Generic algorithms can also be used as methods of type (ii), where optimizations are simply performed by scanning an explicit list of given points. It should be observed that most methods are specific to some problems and/or restricted to the bi-objective case.

Among the previously mentioned references, Ehrgott et al. (2011), Eusébio et al. (2014), Shao and Ehrgott (2016), Vaz et al. (2015) are distance-oriented methods. They use a Euclidean norm to define their distance. References (Bazgan, Hugot, Vanderpooten, 2009, Diakonikolas, Yannakakis, 2009, Filippi, Stevanato, 2013) are preference-oriented methods. All of them use the $(1+\varepsilon )$-dominance relation. However, they only ensure coverage, and sometimes cardinality, but do not consider spacing.

The algorithms we are proposing are generic preference-oriented methods of type (i). These algorithms can be applied to discrete or continuous, linear or nonlinear, bi-objective optimization problems, depending on the availability of some problem-dependent routines. Besides providing a priori guarantees on the three quality measures, we also guarantee that our generic algorithms are polynomial when the routines are polynomial.

Our paper is organized as follows. In the next section, we define the basic concepts, formalize the notion of (ε, ε′)-kernels, and recall some results of previous related works. In Section 3, we study the bi-objective case. We show some general results and present generic polynomial time algorithms to construct small (ε, ε′)-kernels under some conditions. In Section 4, we study the case of three or more objectives, pointing out specific difficulties. Section 5 presents some experimental results which demonstrate the practical applicability of our approach. We conclude with some possible extensions to this work.