Modelling the Pareto-optimal set using B-spline basis functions for continuous multi-objective optimization problems

Bhardwaj, Piyush; DasGupta, Bhaskar; Deb, Kalyanmoy

doi:10.1080/0305215x.2013.812727

Cited by 11 publications

(5 citation statements)

References 39 publications

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“…Optimizing only the control points of the Bézier curve, that define its curvature, enforces the decision variables of solutions in the approximation set to vary in a smooth, continuous fashion, thereby likely improving intuitive navigability of the approximation set. Previous work on parameterizations of the approximation set has been applied mainly in a post-processing step after optimization, or was performed in the objective space [3,15,19], but this does not aid in the navigability of the approximation set in decision space. Moreover, fitting a smooth curve through an already optimized set of solutions might result in a bad fit, resulting in a lower-quality approximation set.…”

Section: Introductionmentioning

confidence: 99%

Ensuring Smoothly Navigable Approximation Sets by Bézier Curve Parameterizations in Evolutionary Bi-objective Optimization

Maree

Alderliesten

Bosman

2020

Lecture Notes in Computer Science

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The aim of bi-objective optimization is to obtain an approximation set of (near) Pareto optimal solutions. A decision maker then navigates this set to select a final desired solution, often using a visualization of the approximation front. The front provides a navigational ordering of solutions to traverse, but this ordering does not necessarily map to a smooth trajectory through decision space. This forces the decision maker to inspect the decision variables of each solution individually, potentially making navigation of the approximation set unintuitive. In this work, we aim to improve approximation set navigability by enforcing a form of smoothness or continuity between solutions in terms of their decision variables. Imposing smoothness as a restriction upon common domination-based multi-objective evolutionary algorithms is not straightforward. Therefore, we use the recently introduced uncrowded hypervolume (UHV) to reformulate the multi-objective optimization problem as a single-objective problem in which parameterized approximation sets are directly optimized. We study here the case of parameterizing approximation sets as smooth Bézier curves in decision space. We approach the resulting single-objective problem with the genepool optimal mixing evolutionary algorithm (GOMEA), and we call the resulting algorithm BezEA. We analyze the behavior of BezEA and compare it to optimization of the UHV with GOMEA as well as the domination-based multi-objective GOMEA. We show that high-quality approximation sets can be obtained with BezEA, sometimes even outperforming the domination-and UHV-based algorithms, while smoothness of the navigation trajectory through decision space is guaranteed.

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Section: Introductionmentioning

confidence: 99%

Ensuring Smoothly Navigable Approximation Sets by Bézier Curve Parameterizations in Evolutionary Bi-objective Optimization

Maree

Alderliesten

Bosman

2020

Lecture Notes in Computer Science

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show abstract

“…Optimizing only the control points of the Bézier curve, that define its curvature, enforces the decision variables of solutions in the approximation set to vary in a smooth, continuous fashion, thereby likely improving intuitive navigability of the approximation set. Previous work on parameterizations of the approximation set has been applied mainly in a post-processing step after optimization, or was performed in the objective space [17,3,24], but this does not aid in the navigability of the approximation set in decision space. Moreover, fitting a smooth curve through an already optimized set of solutions might result in a bad fit, resulting in a lower-quality approximation set.…”

Section: Introductionmentioning

confidence: 99%

Ensuring smoothly navigable approximation sets by Bezier curve parameterizations in evolutionary bi-objective optimization -- applied to brachytherapy treatment planning for prostate cancer

Maree,

Alderliesten,

Bosman

2020

Preprint

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The aim of bi-objective optimization is to obtain an approximation set of (near) Pareto optimal solutions. A decision maker then navigates this set to select a final desired solution, often using a visualization of the approximation front. The front provides a navigational ordering of solutions to traverse, but this ordering does not necessarily map to a smooth trajectory through decision space. This forces the decision maker to inspect the decision variables of each solution individually, potentially making navigation of the approximation set unintuitive. In this work, we aim to improve approximation set navigability by enforcing a form of smoothness or continuity between solutions in terms of their decision variables. Imposing smoothness as a restriction upon common domination-based multi-objective evolutionary algorithms is not straightforward. Therefore, we use the recently introduced uncrowded hypervolume (UHV) to reformulate the multi-objective optimization problem as a single-objective problem in which parameterized approximation sets are directly optimized. We study here the case of parameterizing approximation sets as smooth Bézier curves in decision space. We approach the resulting single-objective problem with the gene-pool optimal mixing evolutionary algorithm (GOMEA), and we call the resulting algorithm BezEA. We analyze the behavior of BezEA and compare it to optimization of the UHV with GOMEA as well as the domination-based multi-objective GOMEA. We show that high-quality approximation sets can be obtained with BezEA, sometimes even outperforming the domination-and UHV-based algorithms, while smoothness of the navigation trajectory through decision space is guaranteed.

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“…On the other hand, GP-based MOO has recently generated a substantial activity in the statistical and optimization communities, with focuses either on sampling strategies (Ponweiser, Wagner, Biermann, and Vincze 2008;Wagner, Emmerich, Deutz, and Ponweiser 2010;Svenson 2011;Emmerich, Deutz, and Klinkenberg 2011;Picheny 2015;Zuluaga, Sergent, Krause, and Püschel 2013) or on uncertainty quantification (Bhardwaj, Dasgupta, and Deb 2014;Calandra, Peters, and Deisenroth 2014;Binois, Ginsbourger, and Roustant 2015a). GPareto aims at filling this gap by making most of the recent approaches available in a unified implementation to both MOO experts and end-users.…”

Section: Introductionmentioning

confidence: 99%

GPareto: An R Package for Gaussian-Process-Based Multi-Objective Optimization and Analysis

Binois¹,

Picheny²

2019

J. Stat. Soft.

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The GPareto package for R provides multi-objective optimization algorithms for expensive black-box functions and an ensemble of dedicated uncertainty quantification methods. Popular methods such as efficient global optimization in the mono-objective case rely on Gaussian processes or kriging to build surrogate models. Driven by the prediction uncertainty given by these models, several infill criteria have also been proposed in a multi-objective setup to select new points sequentially and efficiently cope with severely limited evaluation budgets. They are implemented in the package, in addition with Pareto front estimation and uncertainty quantification visualization in the design and objective spaces. Finally, it attempts to fill the gap between expert use of the corresponding methods and user-friendliness, where many efforts have been put on providing graphical postprocessing, standard tuning and interactivity.

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Modelling the Pareto-optimal set using B-spline basis functions for continuous multi-objective optimization problems

Cited by 11 publications

References 39 publications

Ensuring Smoothly Navigable Approximation Sets by Bézier Curve Parameterizations in Evolutionary Bi-objective Optimization

Ensuring Smoothly Navigable Approximation Sets by Bézier Curve Parameterizations in Evolutionary Bi-objective Optimization

Ensuring smoothly navigable approximation sets by Bezier curve parameterizations in evolutionary bi-objective optimization -- applied to brachytherapy treatment planning for prostate cancer

GPareto: An R Package for Gaussian-Process-Based Multi-Objective Optimization and Analysis

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