# Solving Chance-Constrained Problems via a Smooth Sample-Based Nonlinear Approximation

## TL;DR

This paper presents a novel approach for solving chance-constrained optimization problems by reformulating probabilistic constraints as quantile functions and approximating them with a differentiable sample average, enabling the use of standard solvers.

## Contribution

The authors introduce a differentiable sample-based approximation of the quantile function for chance constraints, with theoretical guarantees and a new trust-region method for joint constraints.

## Key findings

- Method scales well with sample size.
- Smoothing reduces bias in chance constraint approximation.
- Empirical results demonstrate effectiveness on various problems.

## Abstract

We introduce a new method for solving nonlinear continuous optimization problems with chance constraints. Our method is based on a reformulation of the probabilistic constraint as a quantile function. The quantile function is approximated via a differentiable sample average approximation. We provide theoretical statistical guarantees of the approximation, and illustrate empirically that the reformulation can be directly used by standard nonlinear optimization solvers in the case of single chance constraints. Furthermore, we propose an S$\ell_1$QP-type trust-region method to solve instances with joint chance constraints. We demonstrate the performance of the method on several problems, and show that it scales well with the sample size and that the smoothing can be used to counteract the bias in the chance constraint approximation induced by the sample approximation.