Second-Order Optimality Conditions for Constrained Domain Optimization
Document Type
Article
Publication Date
9-1-2007
Identifier/URL
40993535 (Pure)
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.
Repository Citation
Miller, D. F.
(2007). Second-Order Optimality Conditions for Constrained Domain Optimization. Journal of Optimization Theory and Applications, 134 (3), 413-432.
https://corescholar.libraries.wright.edu/math/503
DOI
10.1007/s10957-007-9218-9
