# A stochastic approximation method for approximating the efficient frontier of chance-constrained nonlinear programs

## TL;DR

This paper introduces a stochastic approximation method that efficiently approximates the efficient frontier of chance-constrained nonlinear programs by smoothing and stochastic subgradient techniques, avoiding poor local solutions.

## Contribution

It develops a novel smoothing-based stochastic approximation approach for chance-constrained problems, providing convergence guarantees and improved solution quality over existing sampling methods.

## Key findings

- Efficiently approximates the efficient frontier in test problems.
- Converges to stationary solutions of smooth approximations.
- Outperforms traditional sample average approximation methods.

## Abstract

We propose a stochastic approximation method for approximating the efficient frontier of chance-constrained nonlinear programs. Our approach is based on a bi-objective viewpoint of chance-constrained programs that seeks solutions on the efficient frontier of optimal objective value versus risk of constraint violation. To this end, we construct a reformulated problem whose objective is to minimize the probability of constraints violation subject to deterministic convex constraints (which includes a bound on the objective function value). We adapt existing smoothing-based approaches for chance-constrained problems to derive a convergent sequence of smooth approximations of our reformulated problem, and apply a projected stochastic subgradient algorithm to solve it. In contrast with exterior sampling-based approaches (such as sample average approximation) that approximate the original chance-constrained program with one having finite support, our proposal converges to stationary solutions of a smooth approximation of the original problem, thereby avoiding poor local solutions that may be an artefact of a fixed sample. Our proposal also includes a tailored implementation of the smoothing-based approach that chooses key algorithmic parameters based on problem data. Computational results on four test problems from the literature indicate that our proposed approach can efficiently determine good approximations of the efficient frontier.