Probability Maximization via Minkowski Functionals: Convex Representations and Tractable Resolution
Abstract
In this paper, we consider the maximization of a probability P\ ζ ζ ∈ K( x)\ over a closed and convex set X, a special case of the chance-constrained optimization problem. We define K( x) as K( x) \ ζ ∈ K c(x,ζ) ≥ 0 \ where ζ is uniformly distributed on a convex and compact set K and c(x,ζ) is defined as either c(x,ζ) 1-|ζTx|m, m≥ 0 (Setting A) or c(x,ζ) Tx -ζ (Setting B). We show that in either setting, P\ ζ ζ ∈ K(x)\ can be expressed as the expectation of a suitably defined function F(x,) with respect to an appropriately defined Gaussian density (or its variant), i.e. E p [F( x,)]. We then develop a convex representation of the original problem requiring the minimization of g(E[F(x,)]) over X where g is an appropriately defined smooth convex function. Traditional stochastic approximation schemes cannot contend with the minimization of g(E[F(·,)]) 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 both almost-sure convergence guarantees, a convergence rate of O(1/k1/2-a) in expected sub-optimality where a > 0, and a sample complexity of O(1/ε6+δ) where δ > 0.