  A class of convex constrained minimization problems over polyhedral cones for
geometry-dependent quadratic objective functions is considered in a functional
analysis framework. Shape differentiability of the primal minimization problem
needs a bijective property for mapping of the primal cone. This restrictive
assumption is relaxed to bijection of the dual cone within the Lagrangian
formulation as a primal-dual minimax problem. In this paper, we give results on
primal-dual shape sensitivity analysis that extends the class of
shape-differentiable problems supported by explicit formula of the shape
derivative. We apply the results to the Stokes problem under mixed
Dirichlet-Neumann boundary conditions subject to the divergence-free
constraint.
