  The rapid growth of deep learning applications in real life is accompanied by
severe safety concerns. To mitigate this uneasy phenomenon, much research has
been done providing reliable evaluations of the fragility level in different
deep neural networks. Apart from devising adversarial attacks, quantifiers that
certify safeguarded regions have also been designed in the past five years. The
summarizing work of Salman et al. unifies a family of existing verifiers under
a convex relaxation framework. We draw inspiration from such work and further
demonstrate the optimality of deterministic CROWN (Zhang et al. 2018) solutions
in a given linear programming problem under mild constraints. Given this
theoretical result, the computationally expensive linear programming based
method is shown to be unnecessary. We then propose an optimization-based
approach \textit{FROWN} (\textbf{F}astened C\textbf{ROWN}): a general algorithm
to tighten robustness certificates for neural networks. Extensive experiments
on various networks trained individually verify the effectiveness of FROWN in
safeguarding larger robust regions.
