in this talk, we consider the maximization of a probability $\mathbb{p}\{ \zeta \mid \zeta \in k(x)\}$ over a closed and convex set $\mathcal x$, a special case of the chance-constrained optimization problem. we define $k(x)$ as $k(x) \triangleq \{ {\zeta} \in к \mid c(x,\zeta) \geq 0 \}$ where $\zeta$ is uniformly distributed on a convex and compact set $к$ and $c(x,\zeta)$ is defined as either {$c(x,\zeta) \triangleq  1-|\zeta^tx|m$, $m\geq 0$} (setting a) or $c(x,\zeta) \triangleq tx - \zeta$ (setting b). we show that in either setting, by leveraging recent findings in the context of non-gaussian integrals of positively homogenous functions,  $\mathbb{p}\{ \zeta \mid \zeta \in k(x)\}$ can be expressed as the expectation of a suitably defined ${continuous}$ function $f({\bullet},\xi)$ with respect to an appropriately defined gaussian density (or its variant), i.e. $\mathbb{e}_{\tilde p} [f(x,\xi)]$. aided by a recent observation in convex analysis, we then develop a convex representation of the original problem requiring the minimization of ${g(\mathbb{e} [f(x,\xi)])}$ over $\mathcal x$  where ${g}$ is an appropriately defined smooth convex function. traditional stochastic approximation schemes cannot contend with the minimization of ${g(\mathbb{e} [f(\bullet,\xi)])}$ over $\mathcal x$, since conditionally unbiased sampled gradients are unavailable. we then develop a regularized variance-reduced stochastic approximation (${\bf r-vrsa}$) scheme that obviates the need for such unbiasedness by combining iterative ${regularization}$ with variance-reduction. notably,  (${\bf r-vrsa}$) 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 $\delta > 0$.

to the best of our knowledge, this may be the first such scheme for probability maximization problems with convergence and rate guarantees. preliminary numerics on a portfolio selection problem (setting a) and a vehicle routing problem (setting b) suggest that the scheme competes well with naivemini-batch sa schemes as well as integer programming approximation methods. this is joint work with ibrahim bardakci, afrooz jalilzadeh, and constantino lagoa. time permitting, a brief summary of ongoing work will be provided on ongoing research in hierarchical optimization and games under uncertainty.