  We propose a sigmoidal approximation for the value-at-risk (that we call
SigVaR) and we use this approximation to tackle nonlinear programs (NLPs) with
chance constraints. We prove that the approximation is conservative and that
the level of conservatism can be made arbitrarily small for limiting parameter
values. The SigVar approximation brings scalability benefits over exact
mixed-integer reformulations because its sample average approximation can be
cast as a standard NLP. We also establish explicit connections between SigVaR
and other smooth sigmoidal approximations recently reported in the literature.
We show that a key benefit of SigVaR over such approximations is that one can
establish an explicit connection with the conditional value at risk (CVaR)
approximation and exploit this connection to obtain initial guesses for the
approximation parameters. We present small- and large-scale numerical studies
to illustrate the developments.
