  The jamming transition is ubiquitous. It is present in granular matter,
colloids, glasses, and many other systems. Yet, it defines a critical point
whose properties still need to be fully understood. A major breakthrough came
about when the replica formalism was extended to build a mean-field theory that
provides an exact description of the jamming transition of spherical particles
in the infinite-dimensional limit. While such theory explains the jamming
critical behavior of both soft and hard spheres, investigating the transition
in finite-dimensional systems poses very difficult and different problems, in
particular from the numerical point of view. Soft particles are modeled by
continuous potentials; thus, their jamming point can be reached through
efficient energy minimization algorithms. In contrast, the latter methods are
inapplicable to hard-sphere (HS) systems since the interaction energy among the
particles is always zero by construction. To overcome these difficulties, here
we recast the jamming of hard spheres as a constrained optimization problem and
introduce the CALiPPSO algorithm, capable of readily producing jammed HS
packings without including any effective potential. This algorithm brings a HS
configuration of arbitrary dimensions to its jamming point by solving a chain
of linear optimization problems. We show that there is a strict correspondence
between the force balance conditions of jammed packings and the properties of
the optimal solutions of CALiPPSO, whence we prove analytically that our
packings are always isostatic and in mechanical equilibrium. Furthermore, using
extensive numerical simulations, we show that our algorithm is able to probe
the complex structure of the free-energy landscape, finding qualitative
agreement with mean-field predictions. We also characterize the algorithmic
complexity of CALiPPSO and provide an open-source implementation of it.
