Bayesian Joint Chance Constrained Optimization: Approximations and Statistical Consistency

TL;DR

This paper investigates Bayesian methods for chance-constrained stochastic optimization, proving statistical consistency and convergence of approximate solutions, and demonstrating practical utility in queueing system staffing.

Contribution

It establishes the statistical consistency and convergence rates of Bayesian approximations in chance-constrained optimization, with theoretical proofs and practical validation.

Findings

Proves convergence of approximate Bayesian optimal values to true values.

Establishes probabilistic rates of convergence.

Demonstrates convex feasibility of the approximate Bayesian problem.

Abstract

This paper considers data-driven chance-constrained stochastic optimization problems in a Bayesian framework. Bayesian posteriors afford a principled mechanism to incorporate data and prior knowledge into stochastic optimization problems. However, the computation of Bayesian posteriors is typically an intractable problem, and has spawned a large literature on approximate Bayesian computation. Here, in the context of chance-constrained optimization, we focus on the question of statistical consistency (in an appropriate sense) of the optimal value, computed using an approximate posterior distribution. To this end, we rigorously prove a frequentist consistency result demonstrating the convergence of the optimal value to the optimal value of a fixed, parameterized constrained optimization problem. We augment this by also establishing a probabilistic rate of convergence of the optimal value. We also prove the convex feasibility of the approximate Bayesian stochastic optimization problem. Finally, we demonstrate the utility of our approach on an optimal staffing problem for an M/M/c queueing model.