  We present a novel algorithm that allows us to gain detailed insight into the
effects of sparsity in linear and nonlinear optimization, which is of great
importance in many scientific areas such as image and signal processing,
medical imaging, compressed sensing, and machine learning (e.g., for the
training of neural networks). Sparsity is an important feature to ensure
robustness against noisy data, but also to find models that are interpretable
and easy to analyze due to the small number of relevant terms. It is common
practice to enforce sparsity by adding the $\ell_1$-norm as a weighted penalty
term. In order to gain a better understanding and to allow for an informed
model selection, we directly solve the corresponding multiobjective
optimization problem (MOP) that arises when we minimize the main objective and
the $\ell_1$-norm simultaneously. As this MOP is in general non-convex for
nonlinear objectives, the weighting method will fail to provide all optimal
compromises. To avoid this issue, we present a continuation method which is
specifically tailored to MOPs with two objective functions one of which is the
$\ell_1$-norm. Our method can be seen as a generalization of well-known
homotopy methods for linear regression problems to the nonlinear case. Several
numerical examples - including neural network training - demonstrate our
theoretical findings and the additional insight that can be gained by this
multiobjective approach.
