On continuation methods for non-linear bi-objective optimization: towards a certified interval-based approach

Martin, BC; Goldsztejn, Alexandre; Granvilliers, Laurent; Jermann, Christophe

doi:10.1007/s10898-014-0201-3

Cited by 23 publications

(11 citation statements)

References 57 publications

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“…Another sub-class of the MP techniques is given by continuation-like methods that take advantage of the fact that the Pareto set forms-at least locally-a manifold. Methods of this kind start from a given initial solution and perform a search along the solution manifold [6,[27][28][29][30][31][32][33].…”

Section: Background and Related Workmentioning

confidence: 99%

A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems

et al. 2020

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Multi-objective optimization problems (MOPs) naturally arise in many applications. Since for such problems one can expect an entire set of optimal solutions, a common task in set based multi-objective optimization is to compute N solutions along the Pareto set/front of a given MOP. In this work, we propose and discuss the set based Newton methods for the performance indicators Generational Distance (GD), Inverted Generational Distance (IGD), and the averaged Hausdorff distance Δp for reference set problems for unconstrained MOPs. The methods hence directly utilize the set based scalarization problems that are induced by these indicators and manipulate all N candidate solutions in each iteration. We demonstrate the applicability of the methods on several benchmark problems, and also show how the reference set approach can be used in a bootstrap manner to compute Pareto front approximations in certain cases.

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Section: Background and Related Workmentioning

confidence: 99%

A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems

et al. 2020

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“…Via use of a clever sequence of these resulting scalar objective optimization problems, a suitable finite size approximation of the entire Pareto front can be computed in certain cases [43][44][45][46]. Multi-objective continuation methods take advantage of the fact that the Pareto set at least locally forms a manifold [47][48][49][50][51][52]. Starting with an initial (local) solution, further candidates are computed along the Pareto set of the given MOP.…”

Section: Multi-objective Optimizationmentioning

confidence: 99%

The Averaged Hausdorff Distances in Multi-Objective Optimization: A Review

2019

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A brief but comprehensive review of the averaged Hausdorff distances that have recentlybeen introduced as quality indicators in multi-objective optimization problems (MOPs) is presented.First, we introduce all the necessary preliminaries, definitions, and known properties of thesedistances in order to provide a stat-of-the-art overview of their behavior from a theoretical pointof view. The presentation treats separately the definitions of the (p, q)-distances GDp,q, IGDp,q, and Δp,q for finite sets and their generalization for arbitrary measurable sets that covers as an importantexample the case of continuous sets. Among the presented results, we highlight the rigorousconsideration of metric properties of these definitions, including a proof of the triangle inequalityfor distances between disjoint subsets when p, q ≥ 1, and the study of the behavior of associatedindicators with respect to the notion of compliance to Pareto optimality. Illustration of these resultsin particular situations are also provided. Finally, we discuss a collection of examples and numericalresults obtained for the discrete and continuous incarnations of these distances that allow for anevaluation of their usefulness in concrete situations and for some interesting conclusions at the end,justifying their use and further study.

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“…Vector-valued optimisation is extremely important in applications, and there a very active current research on the topic, mostly in connection to multi-criterion optimisation and Nash equilibria, see for instance [23,24,25,26,27,28,29]. Let us also note that there exists a large number of contributions in Game Theory which are closely related to the problem considered here.…”

Section: Introductionmentioning

confidence: 99%

A review from the PDE viewpoint of Hamilton-Jacobi-Bellman equations arising in optimal control with vectorial cost

Katzourakis

Pryer

2018

J. Nonlinear Funct. Anal.

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This paper is a review of results on optimisation which are perhaps not so standard in the PDE realm. To this end, we consider the problem of deriving the PDEs associated to the optimal control of a system of either ODEs or SDEs with respect to a vector-valued cost functional. Optimisation is considered with respect to a partial ordering generated by a given cone. Since in the vector case minima may not exist, we define vectorial value functions as (Pareto) minimals of the ordering. Our main objective is the derivation of the model PDEs which turn out to be parametric families of HJB single equations instead of systems of PDEs. However, this allows the use of the theory of viscosity solutions.

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On continuation methods for non-linear bi-objective optimization: towards a certified interval-based approach

Cited by 23 publications

References 57 publications

A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems

A Set Based Newton Method for the Averaged Hausdorff Distance for Multi-Objective Reference Set Problems

The Averaged Hausdorff Distances in Multi-Objective Optimization: A Review

A review from the PDE viewpoint of Hamilton-Jacobi-Bellman equations arising in optimal control with vectorial cost

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