  We present a numerical linear programming bootstrap to construct dual model
scattering amplitudes. Dual models describe tree-level exchanges of higher spin
resonances in theories like string theory and large $N$ gauge theories. Despite
being very simple objects, their numerical bootstrap has proven challenging due
to slow convergence of the infinite sums over resonances. Our bootstrap
succeeds thanks to an efficient parametrization of the amplitude in terms of
Mandelstam-Regge poles and the use of combined regions that make crossing
symmetry constraining. Along the way, we discover and conjecture a property of
"super-unitarity" of the Veneziano amplitude, which we use to keep a linear
problem.
  As results, we present first the study of a class of string-like amplitudes
with linear trajectories, for which we observe that the Veneziano amplitude
lies at a preferred location, at the bottom of a pit, which minimizes crossing.
Then, we introduce a toy-model deformation to non-linear trajectories,
mimicking some features of QCD, for which our algorithm also detects a clear
pit. This gives compelling evidence that our bootstrap is able to produce
amplitudes that can exhibit non-trivial phenomenological features.
