  In this article we develop a convergence theory for goal-oriented adaptive
finite element algorithms designed for a class of second-order semilinear
elliptic equations. We briefly discuss the target problem class, and introduce
several related approximate dual problems that are crucial to both the analysis
as well as to the development of a practical numerical method. We then review
some standard facts concerning conforming finite element discretization and
error-estimate-driven adaptive finite element methods (AFEM). We include a
brief summary of a priori estimates for this class of semilinear problems, and
then describe some goal-oriented variations of the standard approach to AFEM
(GOAFEM). Following the recent approach of Mommer-Stevenson and Holst-Pollock
for increasingly general linear problems, we first establish a quasi-error
contraction result for the primal problem. We then develop some additional
estimates that make it possible to establish contraction of the combined
primal-dual quasi-error, and subsequently show convergence with respect to the
quantity of interest. Finally, a sequence of numerical experiments are then
carefully examined. It is observed that the behavior of the implementation
follows the predictions of the theory.
