  In this work we apply the S-matrix bootstrap maximization program to the 2d
bosonic O(N) integrable model which has N species of scalar particles of mass m
and no bound states. Since in previous studies theories were defined by
maximizing the coupling between particles and their bound states, the main
problem appears to be to find what other functional can be used to define this
model. Instead, we argue that the defining property of this integrable model is
that it resides at a vertex of the convex space determined by the unitarity and
crossing constraints. Thus, the integrable model can be found by maximizing any
linear functional whose gradient points in the general direction of the vertex,
namely within a cone determined by the normals to the faces intersecting at the
vertex. This is a standard problem in applied mathematics, related to
semi-definite programming and solvable by fast available numerical algorithms.
The information provided by the numerical solution is enough to reproduce the
known analytical solution without using integrability, namely the Yang-Baxter
equation. This situation seems quite generic so we expect that other theories
without continuous parameters can also be found by maximizing linear
functionals in the convex space of allowed S-matrices.
