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

ICLR 2025 Conference Submission13062 Authors

Keywords: Robust optimization, Robustness-accuracy tradeoff

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 approximate many Pareto-efficient robust solutions using the PPM trajectory. Compared with the existing method with computation cost $N\times T_{\mathrm{RC}}$, the cost of our method is $T_{\mathrm{RC}} + (N-1)\times T_{\mathrm{\widetilde{PPM}}}$, where $N$ is the number of robust solutions to be generated, $T_{\mathrm{RC}}$ is the cost of solving a single robust optimization problem, and $T_{\mathrm{\widetilde{PPM}}}$ is cost of a single step of an approximate PPM. We prove exact PPM can produce exact Pareto efficient robust solutions for a class of robust linear optimization problems. For robust optimization problems with nonlinear and differentiable objective functions, compared with the existing method, our method equipped with first-order approximate PPMs is computationally cheaper and generates robust solutions with comparable performance.

Primary Area: optimization

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Submission Number: 13062