  Optimization problems with uncertainty in the constraints occur in many
applications. Particularly, probability functions present a natural form to
deal with this situation. Nevertheless, in some cases, the resulting
probability functions are nonsmooth. This motivates us to propose a
regularization employing the Moreau envelope of a scalar representation of the
vector inequality. More precisely, we consider a probability function which
covers most of the general classes of probabilistic constraints:
$$\varphi(x)=\mathbb{P}(\Phi(x,\xi)\in -\mathcal{K}),$$ where $\mathcal{K}$ is
a convex cone of a Banach space. The conic inclusion $\Phi (x,\xi) \in -
\mathcal{K}$ represents an abstract system of inequalities, and $\xi$ is a
random vector. We propose a regularization by applying the Moreau envelope to
the scalarization of the function $\Phi$. In this paper, we demonstrate, under
mild assumptions, the smoothness of such a regularization and that it satisfies
a type of variational convergence to the original probability function.
Consequently, when considering an appropriately structured problem involving
probabilistic constraints, we can thus entail the convergence of the minimizers
of the regularized approximate problems to the minimizers of the original
problem. Finally, we illustrate our results with examples and applications in
the field of (nonsmooth) joint, semidefinite and probust chance constrained
optimization problems.
