Descent--Ascent Structure of Uniform Nondegeneracy

Abstract: Classical models such as smooth strongly convex minimization and strongly convex--strongly concave optimization combine two types of curvature information: curvature remains uniformly separated from zero, and its positive and negative orientations are prescribed in advance. Uniform nondegeneracy disentangles these roles, retaining the separation from zero while allowing the curvature orientation to vary across the domain. We identify a family of descent--ascent inequalities (DAI) that characterizes uniform nondegeneracy and naturally gives rise to Alternating Descent Ascent (ADA), a simple explicit method that alternates full-gradient descent and ascent. ADA converges globally and linearly to the unique stationary point from arbitrary initialization, with dimension-free first-order complexity. The framework recovers the smooth strongly convex setting as a limit and encompasses the strongly convex--strongly concave and linearly constrained settings, highlighting uniform nondegeneracy as a common first-order structure across various classical regimes.