  The S-matrix bootstrap maps out the space of S-matrices allowed by
analyticity, crossing, unitarity, and other constraints. For the $2\rightarrow
2$ scattering matrix $S_{2\rightarrow 2}$ such space is an infinite dimensional
convex space whose boundary can be determined by maximizing linear functionals.
On the boundary interesting theories can be found, many times at vertices of
the space. Here we consider $3+1$ dimensional theories and focus on the
equivalent dual convex minimization problem that provides strict upper bounds
for the regularized primal problem and has interesting practical and physical
advantages over the primal problem. Its variables are dual partial waves
$k_\ell(s)$ that are free variables, namely they do not have to obey any
crossing, unitarity or other constraints. Nevertheless they are directly
related to the partial waves $f_\ell(s)$, for which all crossing, unitarity and
symmetry properties result from the minimization. Numerically, it requires only
a few dual partial waves, much as one wants to possibly match experimental
results. We consider the case of scalar fields which is related to pion
physics.
