  We present an algorithm for learning parametric constraints from
locally-optimal demonstrations, where the cost function being optimized is
uncertain to the learner. Our method uses the Karush-Kuhn-Tucker (KKT)
optimality conditions of the demonstrations within a mixed integer linear
program (MILP) to learn constraints which are consistent with the local
optimality of the demonstrations, by either using a known constraint
parameterization or by incrementally growing a parameterization that is
consistent with the demonstrations. We provide theoretical guarantees on the
conservativeness of the recovered safe/unsafe sets and analyze the limits of
constraint learnability when using locally-optimal demonstrations. We evaluate
our method on high-dimensional constraints and systems by learning constraints
for 7-DOF arm and quadrotor examples, show that it outperforms competing
constraint-learning approaches, and can be effectively used to plan new
constraint-satisfying trajectories in the environment.
