Approximating Multiple Robust Optimization Solutions in One Pass via Proximal Point Methods

TL;DR

This paper introduces a proximal point method-based approach to efficiently approximate multiple robust optimization solutions simultaneously, significantly reducing computational effort and providing theoretical guarantees for linear problems.

Contribution

The paper presents a novel proximal point method that approximates many Pareto efficient robust solutions in one pass, reducing computational complexity and offering theoretical guarantees.

Findings

Reduces computation from N×T to 2×T for N solutions

Proves exact solutions for robust linear optimization

Provides high-probability approximation guarantees for general problems

Abstract

Robust optimization provides a principled and unified framework to model many problems in modern operations research and computer science applications, such as risk measures minimization and adversarially robust machine learning. To use a robust solution (e.g., to implement an investment portfolio or perform robust machine learning inference), the user has to a priori decide the trade-off between efficiency (nominal performance) and robustness (worst-case performance) of the solution by choosing the uncertainty level hyperparameters. In many applications, this amounts to solving the problem many times and comparing them, each from a different hyperparameter setting. This makes robust optimization practically cumbersome or even intractable. We present a novel procedure based on the proximal point method (PPM) to efficiently approximate many Pareto efficient robust solutions at once. This effectively reduces the total compute requirement from $N \times T$ to $2 \times T$, where $N$ is the number of robust solutions to be generated, and $T$ is the time to obtain one robust solution. We prove this procedure can produce exact Pareto efficient robust solutions for a class of robust linear optimization problems. For more general problems, we prove that with high probability, our procedure gives a good approximation of the efficiency-robustness trade-off in random robust linear optimization instances. We conduct numerical experiments to demonstrate.