Reduced Functions, Gradients and Hessians

In design optimization and parameter identi cation, the objective, or response function (s) are typically linked to the actually independent variables through equality constraints, which we will refer to as state equations. Our key assumption is that it is impossible to form and factor the corresponding constraint Jacobian, but one has instead some xed point algorithm for computing a feasible state, given any reasonable value of the independent variables. Assuming that this iteration is eventually contractive we will show how reduced gradients [Jacobians] and Hessians (in other words the total derivatives) of the response[s] with respect to the independent variables can be obtained via algorithmic, or automatic, dierentiation (AD)