Lagrangian Dual Sections: A Topological Perspective on Hidden Convexity
Venkat Chandrasekaran, Timothy Duff, Jose Israel Rodriguez, Kevin Shu
TL;DR
The paper develops a topological framework for hidden convexity using Lagrangian dual sections, showing that when a Lagrangian dual section exists on a compact domain, the constrained optimum $V_f$ is concave and admits a natural convex reformulation via the convex hull of the image ${\mathrm{conv}}(f({\mathcal{M}}))$. It specializes this framework to linear inverse spectral problems, establishing criteria based on (singular-)noncrossing subspaces that guarantee the tightness of natural SDP relaxations, and extends these ideas through Kostant convexity to a Lie-theoretic generalization. The Unbalanced Procrustes Problem is analyzed under this lens, with results for the $(3,2)$ case and related Grassmannian reformulations, complemented by algorithmic developments on Riemannian manifolds. Finally, the work introduces CHORD, a path-tracking method that leverages the Lagrangian dual section to obtain global optima, and discusses ellipsoid-based strategies to solve constrained problems, alongside future directions toward approximate convexity and hierarchical relaxations.
Abstract
Hidden convexity is a powerful idea in optimization: under the right transformations, nonconvex problems that are seemingly intractable can be solved efficiently using convex optimization. We introduce the notion of a Lagrangian dual section of a nonlinear program defined over a topological space, and we use it to give a sufficient condition for a nonconvex optimization problem to have a natural convex reformulation. We emphasize the topological nature of our framework, using only continuity and connectedness properties of a certain Lagrangian formulation of the problem to prove our results. We demonstrate the practical consequences of our framework in a range of applications and by developing new algorithmic methodology. First, we present families of nonconvex problem instances that can be transformed to convex programs in the context of spectral inverse problems -- which include quadratically constrained quadratic optimization and Stiefel manifold optimization as special cases -- as well as unbalanced Procrustes problems. In each of these applications, we both generalize prior results on hidden convexity and provide unifying proofs. For the case of the spectral inverse problems, we also present a Lie-theoretic approach that illustrates connections with the Kostant convexity theorem. Second, we introduce new algorithmic ideas that can be used to find globally optimal solutions to both Lagrangian forms of an optimization problem as well as constrained optimization problems when the underlying topological space is a Riemannian manifold.