where ||·|| denotes the l2-norm and c : Pi → ℝk represents k-dimensional nonlinear constraints. In the rest of this section, we discuss adjoint sensitivity analysis of F(p), which is important for applying gradient-based optimization methods for solving (2) as well as for computing F(p) iteratively. Denote the linearization of the above map at p as F'(p). The adjoint operator F'*(p) is defined to satisfy the following: for all δp, δ ∈ ℝm,