Before revision:  We design  a goal-oriented adaptive finite element method (GOAFEM) , which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method (PCG) . We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates  with respect to the number of degrees of freedom as well as the computational cost.
After revision:  We design a goal-oriented adaptive finite element method (GOAFEM) , which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method (PCG) . We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates with respect to the number of degrees of freedom but even prove optimal complexity, i.e., optimal convergence rates with respect to the total computational cost.