On the Pareto compliance of the averaged hausdorff distance as a performance indicator

Univ. Sci. 23 (3): 333-355, 2018. doi: 10.11144/Javeriana.SC23-3.otpc Bogotá original article On the Pareto Compliance of the Averaged Hausdorff Distance as a Performance Indicator Andrés Vargas1, * Edited by Juan Carlos Salcedo-Reyes () 1. Departamento de Matemáticas, Pontificia Universidad Javeriana, Bogotá, Colombia. * Received: 20-08-2018 Accepted: 27-09-2018 Published on line: 28-09-2018 Citation: Vargas A. On the Pareto Compliance of the Averaged Hausdorff Distance as a Performance Indicator, Universitas Scientiarum, 23 (3): 333-354, 2018. doi: 10.11144/ Javeriana.SC23-3.otpc Funding: N.A. Electronic supplementary material: N.A. Abstract The averaged Hausdorff distance ∆ p is an inframetric, recently introduced in evolutionary multiobjective optimization (EMO) as a tool to measure the optimality of finite size approximations to the Pareto front associated to a multiobjective optimization problem (MOP). Tools of this kind are called performance indicators, and their quality depends on the useful criteria they provide to evaluate the suitability of different candidate solutions to a given MOP. We present here a purely theoretical study of the compliance of the ∆ p -indicator to the notion of Pareto optimality. Since ∆ p is defined in terms of a modified version of other well-known indicators, namely the generational distance GDp , and the inverted generational distance IGDp , specific criteria for the Pareto compliance of each one of them is discussed in detail. In doing so, we review some previously available knowledge on the behavior of these indicators, correcting inaccuracies found in the literature, and establish new and more general results, including detailed proofs and examples of illustrative situations. Keywords: averaged Hausdorff distance; generational distance; inverted generational distance; multiobjective optimization; Pareto optimality; performance indicator. Introduction A fundamental task in evolutionary multiobjective optimization (EMO) consists in the explicit computation of the set of solutions (known as the Pareto set) and their images (the Pareto front) corresponding to the problem of simultaneous optimization of multiple objective functions, or multiobjective optimization problem (MOP), for short. It is an important fact (see, e.g. [7]) that every non-trivial MOP admits more than one solution, i.e., there is no single point that simultaneously optimizes all the objective functions. A solution is called Pareto optimal if there is no objective function that can be improved without degrading the rest. Even though the Pareto set P of a MOP, Universitas Scientiarum, Journal of the Faculty of Sciences, Pontificia Universidad Javeriana, is licensed under the Creative Commons Attribution 4.0 International Public License 334 On the Pareto Compliance of Δp consisting of the set of all optimal solutions, turns out to be a compact subset of Rn in common situations, generally it cannot be calculated in a purely analytical way, and the use of numerical algorithms becomes essential. It is often desirable (and even necessary) to approximate P with a subset A ⊂ Rn , called archive, that resembles P and its properties as closely as possible. Archives are usually assumed to consist of a finite number of points that can be numerically found, and EMO algorithms are an important tool employed for achieving that aim. To measure their accuracy, the distance between an outcome archive A and the original Pareto set P should be defined in an appropriate sense, but several inequivalent notions of distance can be considered, and their values are not necessarily attained in a unique manner. This means that the set of candidate approximations is usually not unique. A readily available notion of distance between sets that can be used in this setting is the Hausdorff distance (see, e.g. [4]), but due to its definition, it allows for undesirable ambiguities, and heavily punishes single outlier solutions. Alternative performance indicators have been introduced in the literature (see, e.g. [10]) and among them, the averaged Hausdorff distance ∆ p was recently proposed in [9] by modifying the well-known generational (GD) and inverted generational (IGD) indicators, in such a way that their values correspond to useful averages. As a result, ∆ p does not punish individual but collective behavior, fixing some drawbacks of the standard Hausdorff distance. Other properties of ∆ p have been investigated in the literature, for example, from the theoretical side that we are interested here, explicit analytical calculations of optimal archives with ∆ p for particular Pareto fronts have been obtained in [8]. In this work, we establish conditions ensuring the Pareto compliance of the GDp and IGDp indicators by means of mathematical criteria involving the behavior of candidate solutions, and summarize at the end the consequences for the compliance of the averaged Hausdorff distance. The proofs require only simple properties of ∆ p (or the intermediate GDp and IGDp ) derived from their definitions, which will be recalled in the section on preliminaries. It is expected that these criteria for compliance with Pareto optimality can help to elucidate advantages and possible drawbacks of ∆ p as a performance indicator for the evaluation of MOEAs. This is a necessary step in view of the possibility to modify and generalize the averaged Hausdorff distance with the purpose of enhancing its usefulness for applications. In fact, promising generalizations are already appearing in the literature for both, the cases of finite and continuous approximations (see [12] and [1], respectively). Further Universitas Scientiarum Vol. 23 (3): 333-354 http://ciencias.javeriana.edu.co/investigacion/universitas-scientiarum 335 Vargas work in this direction requires a detailed understanding of the behavior of the original p-indicators, and the treatment presented here reveals also relevant aspects that should be considered. This paper is organized as follows: In Section 2, we briefly present the background and notation required for the understanding of the rest of the manuscript. The core of this work appears in Section 3, where different criteria for the Pareto compliance of all the indicators, GDp , IGDp and ∆ p are provided, including complete proofs, particular examples and important observations. Finally, conclusions and perspectives for future research are pointed out in Section 4. Preliminaries Throughout the document we will employ the abbreviations R∗ := RK{0} and R+ := [0, ∞) whenever necessary. Multiobjective Optimization Given a decision space X ⊂ Rn and a vector valued function F : X ⊂ Rn → Rk defined on it, a multiobjective optimization problem consists in the simultaneous minimization of its k components f1 , . . . , fk . A solution is called Pareto-optimal when the elements of the image, or objective space, Y = F (X ) are non-dominated in the sense of Pareto [7]. This notion is define (...truncated)