  In this paper, we consider the maximization of a probability $\mathbb{P}\{
\zeta \mid \zeta \in \mathbf{K}(\mathbf x)\}$ over a closed and convex set
$\mathcal X$, a special case of the chance-constrained optimization problem. We
define $\mathbf{K}(\mathbf x)$ as $\mathbf{K}(\mathbf x) \triangleq \{ \zeta
\in \mathcal{K} \mid c(\mathbf{x},\zeta) \geq 0 \}$ where $\zeta$ is uniformly
distributed on a convex and compact set $\mathcal{K}$ and $c(\mathbf{x},\zeta)$
is defined as either {$c(\mathbf{x},\zeta) \triangleq 1-|\zeta^T\mathbf{x}|^m$,
$m\geq 0$} (Setting A) or $c(\mathbf{x},\zeta) \triangleq T\mathbf{x} -\zeta$
(Setting B). We show that in either setting, $\mathbb{P}\{ \zeta \mid \zeta \in
\mathbf{K(x)}\}$ can be expressed as the expectation of a suitably defined
function $F(\mathbf{x},\xi)$ with respect to an appropriately defined Gaussian
density (or its variant), i.e. $\mathbb{E}_{\tilde p} [F(\mathbf x,\xi)]$. We
then develop a convex representation of the original problem requiring the
minimization of ${g(\mathbb{E}[F(\mathbf{x},\xi)])}$ over $\mathcal X$ where
$g$ is an appropriately defined smooth convex function. Traditional stochastic
approximation schemes cannot contend with the minimization of
${g(\mathbb{E}[F(\cdot,\xi)])}$ over $\mathcal X$, since conditionally unbiased
sampled gradients are unavailable. We then develop a regularized
variance-reduced stochastic approximation (r-VRSA) scheme that obviates the
need for such unbiasedness by combining iterative regularization with
variance-reduction. Notably, (r-VRSA) is characterized by both almost-sure
convergence guarantees, a convergence rate of $\mathcal{O}(1/k^{1/2-a})$ in
expected sub-optimality where $a > 0$, and a sample complexity of
$\mathcal{O}(1/\epsilon^{6+\delta})$ where $\delta > 0$.
