  Using duality in optimization theory we formulate a dual approach to the
S-matrix bootstrap that provides rigorous bounds to 2D QFT observables as a
consequence of unitarity, crossing symmetry and analyticity of the scattering
matrix. We then explain how to optimize such bounds numerically, and prove that
they provide the same bounds obtained from the usual primal formulation of the
S-matrix Bootstrap, at least once convergence is attained from both
perspectives. These techniques are then applied to the study of a gapped system
with two stable particles of different masses, which serves as a toy model for
bootstrapping popular physical systems.
