Variational Bayesian Methods for Stochastically Constrained System Design Problems
This work addresses system design problems with chance constraints for engineers and researchers, offering a tractable Bayesian approach, but it is incremental as it builds on existing variational methods.
The paper tackles the challenge of solving stochastically constrained system design problems by proposing a variational Bayes-based method to approximate intractable posterior predictive integrals while preserving convexity, and demonstrates that the solution set converges to the true set as observations increase, with bounds on misclassification probability.
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.