  Given its wide spectrum of applications, the classical problem of
all-terminal network reliability evaluation remains a highly relevant problem
in network design.
  The associated optimization problem -- to find a network with the best
possible reliability under multiple constraints -- presents an even more
complex challenge, which has been addressed in the scientific literature but
usually under strong assumptions over failures probabilities and/or the network
topology.
  In this work, we propose a novel reliability optimization framework for
network design with failures probabilities that are independent but not
necessarily identical. We leverage the linear-time evaluation procedure for
network reliability in the series-parallel graphs of Satyanarayana and
Wood(1985) to formulate the reliability optimization problem as a mixed-integer
nonlinear optimization problem. To solve this nonconvex problem, we use
classical convex envelopes of bilinear functions, introduce custom cutting
planes, and propose a new family of convex envelopes for expressions that
appear in the evaluation of network reliability. Furthermore, we exploit the
refinements produced by spatial branch-and-bound to locally strengthen our
convex relaxations. Our experiments show that, using our framework, one can
efficiently obtain optimal solutions in challenging instances of this problem.
