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On a Stochastic Differential Equation Approach for Multiobjective Optimization up to Pareto-Criticality

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Evolutionary Multi-Criterion Optimization (EMO 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6576))

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Traditional Evolutionary Multiobjective Optimization techniques, based on derivative-free dominance-based search, allowed the construction of efficient algorithms that work on rather arbitrary functions, leading to Pareto-set sample estimates obtained in a single algorithm run, covering large portions of the Pareto-set. However, these solutions hardly reach the exact Pareto-set, which means that Pareto-optimality conditions do not hold on them. Also, in problems with high-dimensional objective spaces, the dominance-based search techniques lose their efficiency, up to situations in which no useful solution is found. In this paper, it is shown that both effects have a common geometric structure. A gradient-based descent technique, which relies on the solution of a certain stochastic differential equation, is combined with a multiobjective line-search descent technique, leading to an algorithm that indicates a systematic solution for such problems. This algorithm is intended to serve as a proof of concept, allowing the comparison of the properties of the gradient-search principle with the dominance-search principle. It is shown that the gradient-based principle can be used to find solutions which are truly Pareto-critical, satisfying first-order conditions for Pareto-optimality, even for many-objective problems.

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Takahashi, R.H.C., Carrano, E.G., Wanner, E.F. (2011). On a Stochastic Differential Equation Approach for Multiobjective Optimization up to Pareto-Criticality. In: Takahashi, R.H.C., Deb, K., Wanner, E.F., Greco, S. (eds) Evolutionary Multi-Criterion Optimization. EMO 2011. Lecture Notes in Computer Science, vol 6576. Springer, Berlin, Heidelberg.

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  • Print ISBN: 978-3-642-19892-2

  • Online ISBN: 978-3-642-19893-9

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