Editing Shape optimization (section)

==Definition==
[[Mathematics|Mathematically]], shape optimization can be posed as the problem of finding a [[bounded set]] <math>\Omega</math>, [[maxima and minima|minimizing]] a [[functional (mathematics)|functional]]
:<math>\mathcal{F}(\Omega)</math>, 
possibly subject to a [[constraint (mathematics)|constraint]] of the form
:<math>\mathcal{G}(\Omega)=0.</math> 
Usually we are interested in sets <math>\Omega</math> which are [[Lipschitz continuity|Lipschitz]] or C<sup>1</sup> [[Boundary (topology)|boundary]] and consist of finitely many [[connected component (analysis)|components]], which is a way of saying that we would like to find a rather pleasing shape as a solution, not some jumble of rough bits and pieces. Sometimes additional constraints need to be imposed to that end to ensure well-posedness of the problem and uniqueness of the solution.

Shape optimization is an [[infinite-dimensional optimization]] problem. Furthermore, the space of allowable shapes over which the optimization is performed does not admit a [[vector space]] structure, making application of traditional optimization methods more difficult.