Multiobjective Design Optimization Using Nash Games

This work is part of the development of a two-phase multi-objective differentiable optimization method. The first phase is classical: it corresponds to the optimization of a set of primary cost functions, subject to nonlinear equality constraints, and it yields at least one known Pareto-optimal solution xA*. This study focuses on the second phase, which is introduced to permit to reduce another set of cost functions, considered as secondary, by the determination of a continuum of Nash equilibria, {x̅ε} (ε≥ 0), in a way such that: firstly, x̅0=xA* (compatibility), and secondly, for ε sufficiently small, the Pareto-optimality condition of the primary cost functions remains O(ε2), whereas the secondary cost functions are linearly decreasing functions of ε. The theoretical results are recalled and the method is applied numerically to a Super-Sonic Business Jet (SSBJ) sizing problem to optimize the flight performance.

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“…The territory splitting is based on the orthogonal decomposition of the following n × n reduced Hessian matrix [4,5]:…”

Section: Preliminary Calculationsmentioning

confidence: 99%

Prioritized optimization by Nash games : towards an adaptive multi-objective strategy

Désidéri

Duvigneau

2021

ESAIM: ProcS

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“…Plus an arbitrary partitioning can have an unknown outcome on the solution. One option proposed in [17] is to define the partitioning based on sensitivity analysis of one main objective. Hence a perspective would be to define an optimal partitioning in the sense that it allocates variables (or a linear combination of variables) to the player it has most influence on.…”

Section: Nash Games and Equilibriamentioning

confidence: 99%

A game theoretic perspective on Bayesian multi-objective optimization

Binois,

Habbal,

Picheny

2021

Preprint

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This chapter addresses the question of how to efficiently solve many-objective optimization problems in a computationally demanding black-box simulation context. We shall motivate the question by applications in machine learning and engineering, and discuss specific harsh challenges in using classical Pareto approaches when the number of objectives is four or more. Then, we review solutions combining approaches from Bayesian optimization, e.g., with Gaussian processes, and concepts from game theory like Nash equilibria, Kalai-Smorodinsky solutions and detail extensions like Nash-Kalai-Smorodinsky solutions. We finally introduce the corresponding algorithms and provide some illustrating results.

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“…The territory splitting is based on the orthogonal decomposition of the following n × n reduced Hessian matrix [7,8]:…”

Section: Preliminary Calculationsmentioning

confidence: 99%

Adaptation by Nash Games in Gradient-Based Multi-objective/Multi-disciplinary Optimization

Désidéri

2021

Springer Proceedings in Mathematics &Amp; Statistics

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Multiobjective Design Optimization Using Nash Games

Cited by 8 publications

References 25 publications

Prioritized optimization by Nash games : towards an adaptive multi-objective strategy

Prioritized optimization by Nash games : towards an adaptive multi-objective strategy

A game theoretic perspective on Bayesian multi-objective optimization

Adaptation by Nash Games in Gradient-Based Multi-objective/Multi-disciplinary Optimization

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