# Corporative Stochastic Approximation with Random Constraint Sampling for Semi-Infinite Programming

## TL;DR

This paper introduces a new stochastic approximation algorithm for semi-infinite programming that handles inexact constraint solving and achieves optimal convergence rates under convexity assumptions.

## Contribution

It proposes a novel CSA algorithm with random constraint sampling schemes and provides convergence guarantees for convex and strongly convex cases.

## Key findings

- Achieves an $ ext{O}(1/\sqrt{N})$ convergence rate for convex functions.
- Improves to an $ ext{O}(1/N)$ rate for strongly convex functions.
- Provides error bounds for inexact CSA in semi-infinite programming.

## Abstract

We developed a corporative stochastic approximation (CSA) type algorithm for semi-infinite programming (SIP), where the cut generation problem is solved inexactly. First, we provide general error bounds for inexact CSA. Then, we propose two specific random constraint sampling schemes to approximately solve the cut generation problem. When the objective and constraint functions are generally convex, we show that our randomized CSA algorithms achieve an $\mathcal{O}(1/\sqrt{N})$ rate of convergence in expectation (in terms of optimality gap as well as SIP constraint violation). When the objective and constraint functions are all strongly convex, this rate can be improved to $\mathcal{O}(1/N)$.