  For the minimization of a nonlinear cost functional $j$ under convex
constraints the relaxed projected gradient process $\varphi_{k+1} = \varphi_{k}
+ \alpha_k(P_H(\varphi_{k}-\lambda_k \nabla_H j(\varphi_{k}))-\varphi_{k})$ is
a well known method. The analysis is classically performed in a Hilbert space
$H$. We generalize this method to functionals $j$ which are differentiable in a
Banach space. Thus it is possible to perform e.g. an $L^2$ gradient method if
$j$ is only differentiable in $L^\infty$. We show global convergence using
Armijo backtracking in $\alpha_k$ and allow the inner product and the scaling
$\lambda_k$ to change in every iteration. As application we present a
structural topology optimization problem based on a phase field model, where
the reduced cost functional $j$ is differentiable in $H^1\cap L^\infty$. The
presented numerical results using the $H^1$ inner product and a pointwise
chosen metric including second order information show the expected mesh
independency in the iteration numbers. The latter yields an additional, drastic
decrease in iteration numbers as well as in computation time. Moreover we
present numerical results using a BFGS update of the $H^1$ inner product for
further optimization problems based on phase field models.
