Second-Order Optimality Conditions for Constrained Domain Optimization

D. F. Miller ()
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D. F. Miller: Wright State University

Journal of Optimization Theory and Applications, 2007, vol. 134, issue 3, No 4, 413-432

Abstract: 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.

Keywords: Domain optimization; Second variations; Domain constraints; Optimality conditions (search for similar items in EconPapers)
Date: 2007
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DOI: 10.1007/s10957-007-9218-9

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