  Robotic planning in real-world scenarios typically requires joint
optimization of logic and continuous variables. A core challenge to combine the
strengths of logic planners and continuous solvers is the design of an
efficient interface that informs the logical search about continuous
infeasibilities. In this paper we present a novel iterative algorithm that
connects logic planning with nonlinear optimization through a bidirectional
interface, achieved by the detection of minimal subsets of nonlinear
constraints that are infeasible. The algorithm continuously builds a database
of graphs that represent (in)feasible subsets of continuous variables and
constraints, and encodes this knowledge in the logical description. As a
foundation for this algorithm, we introduce Planning with Nonlinear Transition
Constraints (PNTC), a novel planning formulation that clarifies the exact
assumptions our algorithm requires and can be applied to model Task and Motion
Planning (TAMP) efficiently. Our experimental results show that our framework
significantly outperforms alternative optimization-based approaches for TAMP.
