Hidden critical and Morse equivalence behind duality: Theory and Applications

Abstract

The aim of this paper is to establish critical duality theory for ratios of nonnegative homogeneous convex functions (shorten for RC functions) and differences of convex functions (abbreviated as DC functions) on Banach spaces. Specifically, we establish a series of duality results on critical point theory and Morse theory for RC functions, including the homotopy type of sublevel sets, the Morse critical points and their Rothe critical groups, Lagrange critical points and their multiplicities, Lusternik-Schnirelman min-max critical values, Poincare polynomials, as well as the structure of handlebody decompositions, all of which are proved to be preserved under polarity dual. Moreover, we obtain the first critical duality theory of DC functions which does not depend on the DC decomposition. This answers a question left open from the work of Toland on DC functions and the work of Le-Pham on DC programming. We apply these results to provide a reformulation of the graph Cheeger constant using zonotopes; we introduce the contact data which serves as a geometric characterization of Lagrange criticality; and we show that the eigenproblems for 1-Laplacian and $\infty$-Laplacian on hypergraphs are equivalent to the contact problems of zonotopes, which indeed establishes a new characterization of zonotopes. We also prove a duality equivalence for certain nonlinear eigenvalue problems and bifurcation problems. Our study here reveals an intricate interaction of critical point theory with other fields such as convex analysis, combinatorial geometry, and nonlinear eigenproblems on graphs.