# Ergodic Approach to Robust Optimization and Infinite Programming Problems

## TL;DR

This paper introduces an ergodic measure-based method for solving robust optimization and infinite programming problems, demonstrating convergence of solutions and extending the scenario approach to nonconvex cases.

## Contribution

It establishes the consistency of an ergodic approach for robust optimization and infinite programming, including nonconvex problems, with convergence guarantees.

## Key findings

- Convergence of minimizers and optimal values to the original problem.
- Extension of the scenario approach to nonconvex optimization.
- Applicability to infinite programming problems.

## Abstract

In this work, we show the consistency of an approach for solving robust optimization problems using sequences of sub-problems generated by ergodic measure preserving transformations. The main result of this paper is that the minimizers and the optimal value of the sub-problems converge, in some sense, to the minimizers and the optimal value of the initial problem, respectively. Our result particularly implies the consistency of the scenario approach for nonconvex optimization problems. Finally, we show that our method can also be used to solve infinite programming problems.