Transit network design — choosing bus routes, frequencies, and capacities — is a nonconvex optimization problem. Continuous approximation models make it tractable by replacing discrete stops and routes with smooth density functions, but the resulting optimization still has multiple local minima. Coordinate descent and general nonlinear solvers find solutions, but without guarantees that they're globally optimal.
Geometric programming solves this. By reformulating the transit cost function as a posynomial — a sum of products of variable powers — the problem becomes convex after a logarithmic change of variables. The global optimum falls out of the convex program, no iterative guessing required.
The improvement is modest in magnitude — 1-4% cost reduction over coordinate descent — but universal in direction. Across six demand distributions, four demand levels, and three value-of-time parameters, geometric programming never does worse. The gap widens under the hardest conditions: high demand and highly heterogeneous networks, exactly where local solvers most often find suboptimal basins.
General nonlinear solvers perform comparably in easy cases (low demand, homogeneous networks) but degrade as complexity increases. The geometric programming approach is indifferent to complexity because the convex reformulation eliminates the landscape features that cause degradation. The nonconvexity wasn't in the physics of transit — it was in the mathematical formulation. Changing the representation to one where the structure is convex doesn't change what the optimal network looks like. It changes whether you can find it.