CAREER: Foundations of semi-infinite and equilibrium constrained optimization

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Humanity is witnessing profound advances in Machine Learning and Data Science. These significant developments have been possible due to powerful computational tools and algorithms that can process large amounts of data to generate valuable insights. As the societal dependency on these systems increases in virtually all domains, the machine learning models get more complex to satisfy an increasing number of requirements. This project investigates semi-infinite and equilibrium-constrained optimization problems that directly apply to today's machine learning and engineering systems. The applications have a broad focus that includes imposing fairness requirements on machine learning models, managing large-scale inventory for better health of supply chains, and making optimal decisions when controlling chemical power plants. The research agenda naturally integrates creating educational content and mentoring undergraduate and graduate students to prepare them for the future workforce. A central component of the project is to meaningfully engage undergraduate students from under-represented and under-served backgrounds via SMU's Office of Engaged Learning.The main technical aim of the project is to develop algorithms for semi-infinite and equilibrium-constrained problems that are simple to implement, scalable for large-scale problems, and efficiently converge to the required solution. Here, the solution can be include: (1) the optimal solution when considering a convex semi-infinite constrained optimization problem or monotone equilibrium-constrained optimization problem; or (2) the first-order stationary point when considering a non-convex semi-infinite constrained optimization problem. The project focuses on proving theoretical guarantees such as bounds on the required number of iterations or sample complexities of the designed algorithms. This an important contribution since no existing methods provide such guarantees in a general setting. The approaches involve leveraging the structure of the objective, the structure of the constraint functions, and recent advances in algorithms for function-constrained optimization. The research outcomes will impact applications that include in-process fairness of machine learning models, distributionally robust constrained optimization, and controlling a chemical plant that operates with uncertainties.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.