Variational Bayesian Methods for Stochastically Constrained System Design Problems

TL;DR

This paper introduces a variational Bayesian approach for solving stochastically constrained system design problems, ensuring tractability, convexity, and convergence to true solutions as data increases.

Contribution

It develops a variational Bayes method to approximate posterior integrals in stochastic programs, maintaining convexity and providing convergence guarantees.

Findings

Proposes a variational Bayes method for tractable approximation.

Ensures convexity of the feasible set in the approximate solution.

Provides convergence and probability bounds for infeasible points.

Abstract

We study system design problems stated as parameterized stochastic programs with a chance-constraint set. We adopt a Bayesian approach that requires the computation of a posterior predictive integral which is usually intractable. In addition, for the problem to be a well-defined convex program, we must retain the convexity of the feasible set. Consequently, we propose a variational Bayes-based method to approximately compute the posterior predictive integral that ensures tractability and retains the convexity of the feasible set. Under certain regularity conditions, we also show that the solution set obtained using variational Bayes converges to the true solution set as the number of observations tends to infinity. We also provide bounds on the probability of qualifying a true infeasible point (with respect to the true constraints) as feasible under the VB approximation for a given number of samples.