Quaternion-based formulations for volume maximisation problems

This paper introduces a mathematical formulation for the problem of determining the optimal position for a three-dimensional item inside a convex container, where its scale can be increased the most and thus its volume maximised. Until now, no methods have been presented that guarantee optimal solutions to this volume maximisation problem while considering continuous free rotation of the item, with approaches relying on heuristics, approximations or enforcing a discrete number of rotations. We aim to find optimal solutions when considering continuous rotation, represented using quaternions. This enables modelling rotation through quadratic constraints. The resulting quadratically constrained problem can be solved to optimality by mathematical solvers. To keep the required computation time within reasonable limits, various improvements to the model such as symmetry breaking are introduced. Experiments show that the majority of our benchmark instances can be solved to optimality within minutes. The expansion to concave containers is also explored, but proves to be more challenging as the required number of quadratic constraints quickly becomes prohibitive.