An Inexact Primal-Dual Algorithm for Semi-Infinite Programming

TL;DR

This paper introduces an inexact primal-dual algorithm for semi-infinite programming, providing convergence guarantees, error bounds, and demonstrating its effectiveness through numerical experiments.

Contribution

It develops a new primal-dual algorithm with a novel prox function for measures, achieving convergence rates and analyzing sample complexity for SIP.

Findings

Achieves an $ ext{O}(1/ oot 2 ext{K})$ convergence rate.

Provides general error bounds for inexact primal-dual methods.

Demonstrates effectiveness through numerical experiments.

Abstract

This paper considers an inexact primal-dual algorithm for semi-infinite programming (SIP) for which it provides general error bounds. To implement the dual variable update, we create a new prox function for nonnegative measures which turns out to be a generalization of the Kullback-Leibler divergence for probability distributions. We show that under suitable conditions on the error, this algorithm achieves an $\mathcal{O}(1/\sqrt{K})$ rate of convergence in terms of the optimality gap and constraint violation. We then use our general error bounds to analyze the convergence and sample complexity of a specific primal-dual SIP algorithm based on Monte Carlo integration. Finally, we provide numerical experiments to demonstrate the performance of our algorithm.