I'm looking at a paper on, and it involves an optimization problem with the following objective function, where $x_k \in R^n$, $y_j \in R^p$, $e_k \in R^p$, $T_k \in R^{n \times m}$, and $f : R^n \to R^n$, and $g : R^n\times R^m \to R^m,$
$$J = \frac{1}{2}(x_0-z)^TB^{-1}(x_0-z) + \frac{1}{2}\sum^{N-1}_{j=0}(h_j(x_j)-y_j)^TR_j^{-1}(h_j(x_j)-y_j)+\frac{1}{2}(e_0-f)^TQ^{-1}(e_0-f)$$
where
$$x_{k+1} = f_k(x_k) + T_ke_k,$$ $$e_{k+1}=g_{k}(x_k,e_k)$$
as constraints. I need to minimize $J$ with respect to $x_0$ and $e_0$, using the adjoint method, which requires you to find the gradient of $J$ with respect to both $x_0$ and $e_0$, and then to write the gradients recursively using adjoint variables. The paper says that the adjoint variables are:
$$\lambda_N = 0,\ \ \ \mu_N=0,$$ $$\lambda_k = F_k^T(x_k)\lambda_{k+1}+G_k^T(x_k,e_k)\mu_{k+1}-H_k^TR_k^{-1}(h_k(x_k)-y_k),$$ $$\mu_k = T_k^T\lambda_{k+1} + \Gamma_k^T(x_k,e_k)\mu_{k+1}$$
where $F_k$, $H_k$, and $G_k$ are the Jacobians of $f_k$, $h_k$, and $g_k$ with respect to $x_k$, and $\Gamma_k$ is the Jacobian of $g_k$ with respect to $e_k$, and the gradients are then
$$\nabla_{x_0}J = B^{-1}(x-z) - \lambda_0,$$ $$\nabla_{e_0}J = Q^{-1}(e_0-f)-\mu_0.$$
But I can't figure out how to get to this conclusion. I'm having a really hard time with coming up with this recursive formula. I've tried working both forwards and backwards from the solution, with no success. I know that to be able to do this, I should find a way to write the summation in the objective function in terms of $x_0$ and $e_0$. Knowing that, I have written:
$$h_j(x_j)-y_j = h_j\left(f_{j-1}(x_{j-1})+T_{j-1}e_{j-1}\right)-y_j$$ $$= h_j\left[f_{j-1}\left(f_{j-2}(x_2)+T_{j-2}e_{j-2}\right)+T_{j-1}g_{j-2}(x_{j-2},e_{j-2})\right]-y_j$$ $$=h_j\left\{f_{j-1}\left[f_{j-2}\left(f_{j-3}(x_{j-3})+T_{j-3}e_{k-3})\right)+T_{j-2}g_{j-3}(x_{j-3},e_{j-3})\right]+T_{j-1}g_{j-2}\left(f_{j-3}(x_{j-3}),g_{j-3}(x_{j-3},e_{j-3})\right)\right\}-y_j$$
but that's as far as I can get without getting hopelessly confused.
Can anyone walk me through the best way to go about showing this? Thanks.