Robust approximation of chance constrained optimization with polynomial perturbation

TL;DR

This paper introduces a robust approximation approach for polynomial chance constrained optimization problems, transforming them into linear conic problems and proposing algorithms for global solutions and heuristic uncertainty set selection.

Contribution

It develops a novel robust approximation method for polynomial CCOs, including transformations to linear conic problems and semidefinite relaxation algorithms.

Findings

Efficient solution algorithms for polynomial CCOs.

Global convergence of the proposed semidefinite relaxation methods.

Heuristic uncertainty set selection improves feasibility.

Abstract

This paper proposes a robust approximation method for solving chance constrained optimization (CCO) of polynomials. Assume the CCO is defined with an individual chance constraint that is affine in the decision variables. We construct a robust approximation by replacing the chance constraint with a robust constraint over an uncertainty set. When the objective function is linear or SOS-convex, the robust approximation can be equivalently transformed into linear conic optimization. Semidefinite relaxation algorithms are proposed to solve these linear conic transformations globally and their convergent properties are studied. We also introduce a heuristic method to find efficient uncertainty sets such that optimizers of the robust approximation are feasible to the original problem. Numerical experiments are given to show the efficiency of our method.