  We analyze a goal-oriented adaptive algorithm that aims to efficiently
compute the quantity of interest $G(u^\star)$ with a linear goal functional $G$
and the solution $u^\star$ to a general second-order nonsymmetric linear
elliptic partial differential equation. The current state of the analysis of
iterative algebraic solvers for nonsymmetric systems lacks the contraction
property in the norms that are prescribed by the functional analytic setting.
This seemingly prevents their application in the optimality analysis of
goal-oriented adaptivity. As a remedy, this paper proposes a goal-oriented
adaptive iteratively symmetrized finite element method (GOAISFEM). It employs a
nested loop with a contractive symmetrization procedure, e.g., the Zarantonello
iteration, and a contractive algebraic solver, e.g., an optimal multigrid
solver. The various iterative procedures require well-designed stopping
criteria such that the adaptive algorithm can effectively steer the local mesh
refinement and the computation of the inexact discrete approximations. The main
results consist of full linear convergence of the proposed adaptive algorithm
and the proof of optimal convergence rates with respect to both degrees of
freedom and total computational cost (i.e., optimal complexity). Numerical
experiments confirm the theoretical results and investigate the selection of
the parameters.
