Semidefinite Programming For Chance Constrained Optimization Over Semialgebraic Sets

TL;DR

This paper introduces a systematic approach using semidefinite programming and moment relaxations to solve chance constrained optimization problems over semialgebraic sets, with convergence guarantees and practical algorithms.

Contribution

The paper develops a hierarchy of convex relaxations based on measures and moments for chance constrained problems, providing convergence and an efficient first-order algorithm.

Findings

Convex relaxations converge to the original problem's solution.

Numerical experiments demonstrate the method's computational effectiveness.

The approach handles large-scale semidefinite relaxations efficiently.

Abstract

In this paper, "chance optimization" problems are introduced, where one aims at maximizing the probability of a set defined by polynomial inequalities. These problems are, in general, nonconvex and computationally hard. With the objective of developing systematic numerical procedures to solve such problems, a sequence of convex relaxations based on the theory of measures and moments is provided, whose sequence of optimal values is shown to converge to the optimal value of the original problem. Indeed, we provide a sequence of semidefinite programs of increasing dimension which can arbitrarily approximate the solution of the original problem. To be able to efficiently solve the resulting large-scale semidefinite relaxations, a first-order augmented Lagrangian algorithm is implemented. Numerical examples are presented to illustrate the computational performance of the proposed approach.