A Proximal DC Algorithm for Sample Average Approximation of Chance Constrained Programming
Peng Wang, Rujun Jiang, Qingyuan Kong, Laura Balzano
TL;DR
This work tackles data-driven chance constrained programs by reformulating the sample average approximation via the empirical quantile $\hat{Q}_C(1-\alpha)$ into a DC-constrained DC program and solving it with a proximal DC algorithm (pDCA). The algorithm solves a sequence of convex subproblems augmented with a proximal term, with subproblems efficiently implementable as LPs/QPs; a careful construction using an auxiliary variable ensures tractable handling of the maximum constraint. The authors establish subsequential and full sequence convergence to a KKT point under a generalized MFCQ, derive an explicit iteration complexity bound by linking to a Frank–Wolfe variant, and show that the method benefits from a KL-based convergence rate depending on the exponent $\theta$. Numerical experiments on VaR-constrained portfolio optimization and probabilistic transportation problems demonstrate competitive performance against MIP, CVaR-based, and other DC/SCA methods, validating both the theory and practical value of the approach.
Abstract
Chance constrained programming (CCP) refers to a type of optimization problem with uncertain constraints that are satisfied with at least a prescribed probability level. In this work, we study the sample average approximation (SAA) of chance constraints. This is an important approach to solving CCP, especially in the data-driven setting where only a sample of multiple realizations of the random vector in the chance constraints is available. The SAA is obtained by replacing the underlying distribution with an empirical distribution over the available sample. Assuming that the functions in chance constraints are all convex, we reformulate the SAA of chance constraints into a difference-of-convex (DC) form. Moreover, considering that the objective function is a difference-of-convex function, the resulting formulation becomes a DC constrained DC program. Then, we propose a proximal DC algorithm for solving this reformulation. In particular, we show that the subproblems of the proximal DC are suitable for off-the-shelf solvers in some scenarios. Moreover, we not only prove the subsequential and sequential convergence of the proposed algorithm but also derive the iteration complexity for finding an approximate Karush-Kuhn-Tucker point. To support and complement our theoretical development, we show via numerical experiments that our proposed approach is competitive with a host of existing approaches.