Global Solutions to Non-Convex Functional Constrained Problems with Hidden Convexity

Ilyas Fatkhullin, Niao He, Guanghui Lan, Florian Wolf

Abstract: Constrained non-convex optimization is fundamentally challenging, as global solutions are generally intractable and constraint qualifications may not hold. However, in many applications, including safe policy optimization in control and reinforcement learning, such problems possess hidden convexity, meaning they can be reformulated as convex programs via a nonlinear invertible transformation. Typically such transformations are implicit or unknown, making the direct link with the convex program impossible. On the other hand, (sub)-gradients with respect to the original variables are often accessible or can be easily estimated, which motivates algorithms that operate directly in the original (non-convex) problem space using a standard (sub)-gradient oracle. In this work, we develop the first algorithms that provably solve such non-convex problems to global minima. Surprisingly, despite non-convexity, our methodology does not require constraint qualifications and achieves complexities matching those for unconstrained hidden convex optimization.