  We describe and analyze algorithms for shape-constrained symbolic regression,
which allows the inclusion of prior knowledge about the shape of the regression
function. This is relevant in many areas of engineering -- in particular
whenever a data-driven model obtained from measurements must have certain
properties (e.g. positivity, monotonicity or convexity/concavity). We implement
shape constraints using a soft-penalty approach which uses multi-objective
algorithms to minimize constraint violations and training error. We use the
non-dominated sorting genetic algorithm (NSGA-II) as well as the
multi-objective evolutionary algorithm based on decomposition (MOEA/D). We use
a set of models from physics textbooks to test the algorithms and compare
against earlier results with single-objective algorithms. The results show that
all algorithms are able to find models which conform to all shape constraints.
Using shape constraints helps to improve extrapolation behavior of the models.
