The Convex Channel

An autonomous vessel must avoid other ships (collision), obey maritime rules (COLREGs), and not run aground (bathymetry). Each constraint type comes from a different domain — fluid dynamics, international law, and geology. The standard approach solves them separately. The vessel zigzags between three independent advisors.

Patil et al. solve all three simultaneously in a single convex optimization. The trick is getting everything into the same mathematical shape. Collision avoidance comes from velocity obstacles — already convex. COLREGs compliance translates to directional constraints — give way to starboard, maintain course when the stand-on vessel. These are linear and convex.

Bathymetry is the hard part. Shallow water regions are nonconvex — they have arbitrary shapes dictated by the seafloor. A vessel can’t simply stay outside a circle; it must navigate around irregular coastlines and shoals. The solution: approximate nonconvex forbidden zones with unions of convex geometries using integer linear programming. The ILP selects which convex approximation to use, and the approximation enters the main optimization cleanly.

The through-claim: the constraint conversion is the entire contribution. Once all three constraint types live in the same convex space, the optimization is straightforward. The barrier to unified planning was representational, not computational. The seafloor’s shape was the obstacle — not because it’s hard to avoid but because it’s hard to express in the same language as collision avoidance.

Three domains, one optimization. The unification cost was converting the geometry.