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Second-Order Optimality Conditions for Constrained Domain Optimization

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  • D. F. Miller

    (Wright State University)

Abstract

This paper develops boundary integral representation formulas for the second variations of cost functionals for elliptic domain optimization problems. From the collection of all Lipschitz domains Ω which satisfy a constraint ∫ Ω g(x) dx=1, a domain is sought which maximizes either $\mathcal{F}_{x_{0}}(\Omega )=F(x_{0},u(x_{0}))$ , fixed x 0∈Ω, or ℱ(Ω)=∫ Ω F(x,u(x)) dx, where u solves the Dirichlet problem Δu(x)=−f(x), x∈Ω, u(x)=0, x∈∂Ω. Necessary and sufficient conditions for local optimality are presented in terms of the first and second variations of the cost functionals $\mathcal{F}_{x_{0}}$ and ℱ. The second variations are computed with respect to domain variations which preserve the constraint. After first summarizing known facts about the first variations of u and the cost functionals, a series of formulas relating various second variations of these quantities are derived. Calculating the second variations depends on finding first variations of solutions u when the data f are permitted to depend on the domain Ω.

Suggested Citation

  • D. F. Miller, 2007. "Second-Order Optimality Conditions for Constrained Domain Optimization," Journal of Optimization Theory and Applications, Springer, vol. 134(3), pages 413-432, September.
    • Handle: RePEc:spr:joptap:v:134:y:2007:i:3:d:10.1007_s10957-007-9218-9
    DOI: 10.1007/s10957-007-9218-9
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    References listed on IDEAS

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    1. C. Huang & D. Miller, 2001. "Variations of Constrained Domain Functionals Associated with Boundary-Value Problems," Journal of Optimization Theory and Applications, Springer, vol. 108(3), pages 587-615, March.
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