Probability maximization via Minkowski functionals: convex representations and tractable resolution

In this paper, we consider the maximizing of the probability P ζ ∣ ζ ∈ K ( x ) over a closed and convex set X , a special case of the chance-constrained optimization problem. Suppose K ( x ) ≜ ζ ∈ K ∣ c ( x , ζ ) ≥ 0 , and ζ is uniformly distributed on a convex and compact set K and c ( x , ζ ) is defined as either c ( x , ζ ) ≜ 1 - ζ T x m where m ≥ 0 (Setting A) or c ( x , ζ ) ≜ T x - ζ (Setting B). We show that in either setting, by leveraging recent findings in the context of non-Gaussian integrals of positively homogenous functions, P ζ ∣ ζ ∈ K ( x ) can be expressed as the expectation of a suitably defined continuous function F ( ∙ , ξ ) with respect to an appropriately defined Gaussian density (or its variant), i.e. E p ~ F ( x , ξ ) . Aided by a recent observation in convex analysis, we then develop a convex representation of the original problem requiring the minimization of g E F ( ∙ , ξ ) over X , where g is an appropriately defined smooth convex function. Traditional stochastic approximation schemes cannot contend with the minimization of g E F ( ∙ , ξ ) over 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 almost-sure convergence guarantees, a convergence rate of O ( 1 / k 1 / 2 - a ) in expected sub-optimality where a > 0 , and a sample complexity of O ( 1 / ϵ 6 + δ ) where δ > 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 set-covering problem (Setting B) suggest that the scheme competes well with naive mini-batch SA schemes as well as integer programming approximation methods.