ON RICCATI EQUATIONS IN NONCONVEX OPTIMIZATION

Convex optimization serves as a foundational pillar for modern engineering and data science, providing highly efficient algorithms that reliably converge to a global optimal solution across diverse modeling applications. However many problems in engineering and data-science fall outside of the convex paradigm. This leads to severe difficulties in guaranteeing optimality and in ensuring constraint satisfaction, which can cause serious challenges when they serve as the foundation for safety mechanisms or for ensuring dependable operation. Moreover, without convexity, even basic questions of feasibility and tractability can become unclear. Historically, convexity has delineated the frontier between problems we can provably solve quickly and reliably and those where we must compromise accept heuristics, approximations, or methods lacking worst-case guarantees. Nonconvexity does not always invalidate the intuition and methods developed for convex problems, especially when especially when some of the structures leading to convexity remain intact. It is often possible to preserve feasibility through reformulations that build the constraints into the problem parameterization, even when optimality can no longer be guaranteed. These cases include problems such as low-rank matrix recovery, dictionary learning, and certain formulations of optimal control, which have optimization landscapes that are well-behaved in the sense where every local minimum is global and critical points are connected through predictable symmetries. These instances suggest that some forms of nonconvexity are more forgiving than others, revealing a richer spectrum between easy and hard problems than the classical perspective of convex/nonconvex would suggest. Understanding when such well-structured nonconvexity arises helps bridge the gap between classical convex theory and the empirical success of modern nonconvex methods. In this thesis, we study situations where nonconvexity arises due to the presence of a nonlinear Riccati equation, in the objective or constraints. We find that these Riccati equations play a large role in guaranteeing global optimality, and it certifying constraints. This suggests that when faced with nonconvexity due to the presence of Riccati equations, it is reasonable to have optimism that the nonconvexity remains structured enough to admit efficient solution methods with meaningful guarantees.

Author dc:creator

• Nguyen-Le, Alexian

Advisor dc:contributor.advisor

• Matni, Nikolai

• Electrical Engineering
• Mathematics

Language dc:language.iso

en

Repository record dc:identifier.uri

https://repository.upenn.edu/handle/20.500.14332/62368

OAI identifier oai:identifier

oai:repository.upenn.edu:20.500.14332/62368