Probability maximization via Minkowski functionals: convex representations and tractable resolution

In this paper, we consider the maximizing of probability $${\mathbb {P}}\left\{ \, \zeta \mid \in {\mathbf {K}}({\mathbf {x}}) \right\} $$ over a closed and convex set $${\mathcal {X}}$$ , special case chance-constrained optimization problem. Suppose $${\mathbf \triangleq \left\{ {\mathcal {K}}\, c({\mathbf {x}},\zeta ) \ge 0 $$\zeta is uniformly distributed on compact {K}}$$ $$c({\mathbf )$$ defined as either )\, 1-\left| ^T{\mathbf {x}}\right| ^m$$ where $$m\ge 0$$ (Setting A) or T{\mathbf {x}}\, - B). We show that in setting, by leveraging recent findings context non-Gaussian integrals positively homogenous functions, \,\zeta can be expressed expectation suitably continuous function $$F(\bullet ,\xi with respect to an appropriately Gaussian density (or its variant), i.e. {E}}_{{{\tilde{p}}}} \left[ F({\mathbf {x}},\xi \right] . Aided observation analysis, then develop representation original problem requiring minimization $$g\left( {\mathbb {E}}\left[ F(\bullet \right) g smooth function. Traditional stochastic approximation schemes cannot contend )\right] $$\mathcal X$$ since conditionally unbiased sampled gradients are unavailable. regularized variance-reduced (r-VRSA) scheme obviates need for such unbiasedness combining iterative regularization variance-reduction. Notably, characterized almost-sure convergence guarantees, rate {O}(1/k^{1/2-a})$$ expected sub-optimality $$a > sample complexity {O}(1/\epsilon ^{6+\delta })$$ $$\delta To best our knowledge, may first maximization problems guarantees. Preliminary numerics portfolio selection set-covering B) suggest competes well naive mini-batch SA integer programming methods.