Optimization problems with constraints in machine learning

• Grant number: 25/27566-1
• Support Opportunities: Scholarships in Brazil - Master
• Start date: March 01, 2026
• End date: February 29, 2028
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Ernesto Julián Goldberg Birgin
• Grantee: Emerson Wilfredy Torres Benitez
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 23/08706-1 - Numerical optimization, AP.TEM

Abstract

Machine learning is a subfield of artificial intelligence that focuses on developing algorithms and models that enable systems to learn from data. Instead of being explicitly programmed to perform tasks, systems are trained to recognize patterns, make predictions, and make decisions based on examples of data. Machine learning relies on large volumes of data to train models. This data can be structured (such as tables) or unstructured (such as text and images). A machine learning model is a mathematical representation that captures patterns in the data. Common models include decision trees, neural networks, support vector machines, and regression. The training process involves feeding data to the model, allowing it to adjust its parameters to minimize errors in predictions. This is done using optimization algorithms. After training, the model is evaluated using a separate dataset (validation and testing) to measure its accuracy and ability to generalize to new data.The above description suggests that optimization plays a key role in adjusting model parameters in machine learning. The optimization problems that must be solved are generally problems whose objective function depends on a huge amount of data. This type of objective function is costly to evaluate. A practical way to deal with them is to use low-cost approximations that can be considered inexact evaluations of the original objective function. This approach has driven the development of specific optimization methods in recent decades to deal with inaccuracy in the evaluation of functions that represent an optimization problem.As artificial intelligence systems are increasingly deployed in safety-critical domains, including credit scoring, medical diagnosis, and autonomous systems, there is a growing demand to ensure their fairness, safety, robustness, and interpretability. These demands can be translated into constraints that model parameters must satisfy, which leads us to consider optimization problems with constraints. This master's project aims to identify concrete examples of optimization problems with constraints that are used to illustrate the scientific literature on machine learning. The goal is to identify the problems, access the data associated with these problems, and test the application of established optimization methods with constraints, such as coordinate block search methods, augmented Lagrangian methods, and/or inexact restoration. (AU)