CAREER: Efficient computational methods for nonlinear optimization and machine learning problems with applications to power systems

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Optimization is an important tool for the design, analysis, control and operation of real-world systems, such as power systems. It also plays a central role in machine learning and artificial intelligence, particularly in deep learning, reinforcement leaning, and statistical learning. The mathematical foundation of optimization has heavily relied on the notion of convexity since convex optimization problems can be solved using fast algorithms. Nevertheless, many optimization problems in real-world applications are non-convex, and therefore it is extremely difficult to solve those problems reliably and efficiently using the existing methods. As an example, this issue is one of the main bottlenecks in the upgrade of the legacy power grids and has been incurring billions of dollars annually in the United States. This CAREER project aims to develop a set of computational tools for solving complex optimization and learning problems using efficient computational methods. This project has a significant impact on many societal problems through the development of a rich mathematical foundation for non-convex optimization, and its outcomes can be exploited in a variety of fields. The developed techniques enable solving large-scale computational problems for improving the efficiency, reliability, resiliency and sustainability of power grids, which has major societal, economical, and environmental impacts. Moreover, these tools significantly extend the application of artificial intelligence to safety-critical systems. This project has a wide range of outreach plans for K-12 and underrepresented students, and it also has several educational activities at both undergraduate and graduate levels. The state-of-the-art techniques for solving non-convex problems are based on various approximation and relaxation methods, whose practical use remains limited due to their scalability issues for real-world systems. On the other hand, the staggering advances made in artificial intelligence in the last 5 years (e.g., in deep learning) are due, in part, to handling computationally-intensive machine learning problems directly as non-convex optimization without relying on convex optimization. Motivated by the resounding success of local search methods for artificial intelligence, this CAREER project aims to design low-complexity computational methods for non-convex optimization problems. To this end, it studies the notion of spurious solutions, which are those solutions of an optimization problem that satisfy the local optimality conditions but are not globally optimal. The main property of convex optimization is the absence of spurious solutions. This project introduces the class of global functions which is far broader than the class of convex functions but benefits from the same spurious-solution-free property. Using the notions of global functions and kernel structure property, four objectives will be addressed: (i) analysis of the spurious solutions of key non-convex problems in machine learning and studying how the amount of data and the structural properties of each problem affects the inexistence of such solutions, (ii) analysis of the spurious solutions of an arbitrary polynomial optimization problem via its conversion to a machine learning problem and then discovering what structural properties guarantee the inexistence of spurious solutions, (iii) approximation of an arbitrary polynomial optimization problem having a spurious solution with a sequence of spurious-minima-free non-convex problems in a higher-dimensional space, (iv) software development and performing case studies on key problems for power systems and machine learning. This project is interdisciplinary and contributes to the areas of optimization theory, machine learning, control theory, and energy.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.