Good and Bad Optimization Models: Insights from Rockafellians

TL;DR

This paper explores the importance of model stability in optimization, analyzing how perturbations affect solutions, and introduces a framework for improving models through sensitivity analysis and dual problem tuning.

Contribution

It introduces a comprehensive perspective on model stability, sensitivity analysis, and the role of dual problems in tuning optimization models for better reliability.

Findings

Sensitivity analysis reveals unstable solutions in 'bad' models.

Embedding problems in families aids in deriving optimality conditions.

Tuning models via dual problems enhances stability and solution quality.

Abstract

A basic requirement for a mathematical model is often that its solution (output) shouldn't change much if the model's parameters (input) are perturbed. This is important because the exact values of parameters may not be known and one would like to avoid being mislead by an output obtained using incorrect values. Thus, it's rarely enough to address an application by formulating a model, solving the resulting optimization problem and presenting the solution as the answer. One would need to confirm that the model is suitable, i.e., "good," and this can, at least in part, be achieved by considering a family of optimization problems constructed by perturbing parameters of concern. The resulting sensitivity analysis uncovers troubling situations with unstable solutions, which we referred to as "bad" models, and indicates better model formulations. Embedding an actual problem of interest within a family of problems is also a primary path to optimality conditions as well as computationally attractive, alternative problems, which under ideal circumstances, and when properly tuned, may even furnish the minimum value of the actual problem. The tuning of these alternative problems turns out to be intimately tied to finding multipliers in optimality conditions and thus emerges as a main component of several optimization algorithms. In fact, the tuning amounts to solving certain dual optimization problems. In this tutorial, we'll discuss the opportunities and insights afforded by this broad perspective.