Towards fast approximations for the hypervolume indicator for multi-objective optimization problems by Genetic Programming
Introduction
Real-world problems often require the simultaneous optimization of several competing objectives, leading to multi-objective optimization problems (MOPs). One important characteristic of MOPs is that their solution sets, the so-called Pareto sets, as well as their images, the Pareto fronts, typically form objects of dimension , where is the number of objectives considered in the given problem. For the numerical treatment of such problems, multi-objective evolutionary algorithms (MOEAs) have caught many researchers’ and practitioners’ interest during the last two decades. Reasons for this include that MOEAs are of global nature, very robust, require minimal assumptions on the model, and are capable of computing finite-size approximations of the entire Pareto set/front of the given MOP in a single run of the algorithm. Since the outcome of every MOEA is an entire set of candidate solutions (population) that ideally resembles the solution set (mainly the Pareto front), one question that naturally arises is how to measure the obtained approximation quality. This is needed to compare different solution sets and guide the MOEA towards the “best” Pareto front approximation.
One performance indicator that is widely used is the Hypervolume indicator (HV, [1], [2]). Although this indicator has several valuable properties [2], [3], [4], it has one critical weakness: the cost for evaluating the HV value of given candidate sets grows exponentially with the number of objectives. That is, while this cost is relatively low for bi-objective problems (compared to the overall cost of a MOEA), the computation of the HV values become the bottleneck for MOPs with more objectives, which represents a severe drawback for the applicability of the HV in modern applications. Since decision-making processes are getting more sophisticated, it is a natural consequence that also the related MOPs increase their number of optimization objectives. And this is not only valid for the quality assessment of a given solution set, but even for the correct functioning of those MOEAs that are based on computing thousands of HV values and HV contributions (i.e., the contribution of an individual of a given population to the HV value) within one run of the algorithm. A straightforward implementation of the HV value of a given set with a magnitude leads to a complexity of , while the best algorithm has a complexity of . Literature reports several methods that aim for a reduction of the computational cost for the HV. For instance, some methods proposed algorithms that reduce the complexity of the computation for specific cases [3], [5]; others employ techniques to approximate the values of the Hypervolume [6]; and finally, algorithms specialized on the Hypervolume contributions have also been proposed [7], [8], [9].
In this work, we propose using a machine learning regression technique that can produce relatively simple and efficient models that approximate the Hypervolume indicator’s behavior. The goal is to approximate the real indicator value, with minimal deviation, for any given problem. The modeling strategy considered is Genetic Programming (GP), which can produce models expressed as symbolic mathematical expressions. The GP system is set up to obtain efficient and straightforward models, avoiding unnecessarily large or complex structures. Thus, the resulting expressions’ main advantage is their computational complexity, which significantly speeds up the computational times (mostly runs in linear time) while keeping the quality in the obtained approximation.
Accordingly, we can summarize the main contributions of this study as follows.
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We pose the problem of deriving approximate models of the HV indicator as a supervised learning problem that we approach through GP regression.
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We show that the learned models are highly efficient, particularly when combined with an adequate updating process, achieving large speedups relative to the state-of-the-art.
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We present results that show that the evolved models effectively approximate the HV indicator, allowing them to be used in two common scenarios: (1) quality indicators and (2) guiding the search of an indicator-based MOEA, both tasks tested for 3-objectives, 4-objective, and 5-objective MOPs.
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The evolved models are quite general, since models trained on two benchmarks can be used to guide an indicator based MOEA on a variety of different MOPs.
The remainder of this paper is organized as follows. In Section 2, we briefly present some definitions and related work on multi-objective optimization and GP, respectively. In Section 3, we present the problem formulation, posing it as a supervised learning problem on which to apply GP. Afterwards, we outline our proposed approach in Section 4 and provide details of the main algorithms used. In Section 5 we present the main results of our study. Finally, Section 6 contains the conclusions and future work.
Section snippets
Background on multi-objective optimization and performance indicators
A continuous MOP can be mathematically defined as follows: Hereby, , is the objective function that is defined by the individual objectives . The domain of is defined by the subset of the that satisfies all inequality and equality constraints,
The optimality of a MOP is defined by the concept of dominance. Let , then we say that the vector is less than (), if for all
Computational problem statement
In this work, the goal of deriving a MOEA performance indicator is posed as a synthesis or learning problem instead of a traditional analytical or formal derivation. While this could be modeled in different ways, we propose defining a supervised machine learning problem to build a new model to compute a performance indicator. To do so, it is necessary to define a target functionality that a learning algorithm will attempt to match, contained in a set of training instances. From this, it is then
Methodology
This section outlines our proposed approach to derive models that can efficiently approximate the HV indicator. The proposed methodology includes the following main stages.
- 1.
Generate the learning dataset. From a set of widely used MO benchmarks, we obtain a sample of approximations to the Pareto front. We made this by running a MOEA on these problems, and used a heuristic sampling policy. To compute the HV for each data set (ground truth), we apply the WFG implementation.1
- 2.
Experiments and results
The experiments were carried out on a Dell R730 Power Edge Server with 2X Intel Xeon E5-2650 processors and 512 GB RAM running KVM virtual machines over Ubuntu Linux. The GPTIPS 2.0 software was downloaded from https://sites.google.com/site/gptips4matlab, running on MATLAB Version: 9.3.0.713579 (R2017b). To compute the HV indicator,we used the WFG implementation,3 the SMS-EMOA code was obtained from PlatEMO,4 with the
Conclusions and future work
In this work, we have presented a new methodology that allows approximating the Hypervolume (HV) values for MOPs. In particular, we have, for the first time in related literature, a supervised learning problem and used GP to evolve solutions for it. We have confirmed the reliability of our models through a comprehensive set of experimental evaluations. Numerical results show that our models approximate the real HV value of the selected benchmark problems, DTLZ and WFG, with great accuracy but
CRediT authorship contribution statement
Cristian Sandoval: Methodology, Software, Validation, Investigation, Data curation, Visualization. Oliver Cuate: Formal analysis, Investigation, Writing – review & editing, Methodology. Luis C. González: Conceptualization, Writing – original draft, Writing – review & editing, Supervision, Project administration, Funding acquisition. Leonardo Trujillo: Conceptualization, Methodology, Resources, Writing – original draft, Writing – review & editing, Supervision, Project administration, Funding
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The first author was supported by CONACYT (Mexico) doctoral scholarship with CVU number 789493. Oliver Cuate acknowledges Instituto Politécnico Nacional and funding from project SIP 20221947.
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