A path-following collision-avoidance model predictive control (MPC) method is proposed which approximates obstacle shapes as convex polygons. Collision-avoidance is ensured by means of the signed distance function which is calculated efficiently as part of the MPC problem by making use of a dual formulation. The overall MPC problem can be solved by standard nonlinear programming (NLP) solvers. The dual signed distance formulation yields, besides the (dual) collision-avoidance constraints, norm, and consistency constraints. A novel approach is presented that combines the arising norm equality with the dual collision-avoidance inequality constraints to yield an alternative formulation reduced in complexity. Moving obstacles are considered using separate convex sets of linearly predicted obstacle positions in the dual problem. The theoretical findings and simplifications are compared with the often-used ellipsoidal obstacle formulation and are analyzed with regard to efficiency by the example of a simulated path-following autonomous surface vessel during a realistic maneuver and AIS obstacle data from the Kiel bay area.