We examine shape optimization problems in the context of inexact sequential quadratic
programming. Inexactness is a consequence of using adaptive finite element methods (AFEM)
to approximate the state and adjoint equations (via the dual weighted
residual method), update the boundary, and compute the geometric functional. We present a
novel algorithm that equidistributes the errors due to shape optimization and
discretization, thereby leading to coarse resolution in the early stages and fine
resolution upon convergence, and thus optimizing the computational effort. We discuss the
ability of the algorithm to detect whether or not geometric singularities such as corners
are genuine to the problem or simply due to lack of resolution – a new paradigm in
adaptivity.