  Strong gravitational lensing of an extended object is described by a mapping
from source to image coordinates that is nonlinear and cannot generally be
inverted analytically. Determining the structure of the source intensity
distribution also requires a description of the blurring effect due to a point
spread function. This initial study uses an iterative gravitational lens
modeling scheme based on the semilinear method to determine the linear
parameters (source intensity profile) of a strongly lensed system. Our
'matrix-free' approach avoids construction of the lens and blurring operators
while retaining the least squares formulation of the problem. The parameters of
an analytical lens model are found through nonlinear optimization by an
advanced genetic algorithm (GA) and particle swarm optimizer (PSO). These
global optimization routines are designed to explore the parameter space
thoroughly, mapping model degeneracies in detail. We develop a novel method
that determines the L-curve for each solution automatically, which represents
the trade-off between the image chi-square and regularization effects, and
allows an estimate of the optimally regularized solution for each lens
parameter set. In the final step of the optimization procedure, the lens model
with the lowest chi-square is used while the global optimizer solves for the
source intensity distribution directly. This allows us to accurately determine
the number of degrees of freedom in the problem to facilitate comparison
between lens models and enforce positivity on the source profile. In practice
we find that the GA conducts a more thorough search of the parameter space than
the PSO.
