CAREER: Foundations of semi-infinite and equilibrium constrained optimization

人类正在见证机器学习和数据科学的深刻进步。这些重大的发展之所以成为可能，是因为强大的计算工具和算法可以处理大量数据，产生有价值的见解。随着社会对这些系统的依赖在几乎所有领域都在增加，机器学习模型变得更加复杂，以满足越来越多的需求。该项目研究直接应用于当今机器学习和工程系统的半无限和平衡约束优化问题。这些应用程序的关注点很广泛，包括对机器学习模型施加公平性要求、管理大规模库存以改善供应链的健康状况，以及在控制化学发电厂时做出最佳决策。研究议程自然整合了创建教育内容和指导本科生和研究生，为他们未来的劳动力做好准备。该项目的一个核心组成部分是有意义地吸引来自代表性不足和服务不足背景的本科生通过SMU的办公室学习。该项目的主要技术目标是开发半无限和平衡约束问题的算法，这些算法易于实现，可扩展为大规模问题，并有效地收敛到所需的解决方案。这里，解可以包括：（1）当考虑凸半无限约束优化问题或单调平衡约束优化问题时的最优解;或者（2）当考虑非凸半无限约束优化问题时的一阶不动点。该项目的重点是证明理论保证，如所需的迭代次数或设计算法的样本复杂性的界限。这是一个重要的贡献，因为没有现有的方法在一般情况下提供这样的保证。这些方法涉及利用目标的结构、约束函数的结构以及函数约束优化算法的最新进展。该研究成果将影响包括机器学习模型的过程中公平性、分布鲁棒约束优化以及控制不确定性运行的化工厂等应用。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.