Yau's Affine Normal Descent: Algorithmic Framework and Convergence Analysis

• URL: http://arxiv.org/abs/2603.28448v2
• Date: Fri, 03 Apr 2026 01:05:10 GMT
• Title: Yau's Affine Normal Descent: Algorithmic Framework and Convergence Analysis
• Abstract summary: Yau's Affine Normal Descent (YAND) is a geometric framework for smooth unconstrained optimization.<n>We show that affinenormal directions are robust under affine scalings, remaining insensitive to arbitrarily ill-conditioned transformations.
• Score: 0.30222726571099656
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We propose Yau's Affine Normal Descent (YAND), a geometric framework for smooth unconstrained optimization in which search directions are defined by the equi-affine normal of level-set hypersurfaces. The resulting directions are invariant under volume-preserving affine transformations and intrinsically adapt to anisotropic curvature. Using the analytic representation of the affine normal from affine differential geometry, we establish its equivalence with the classical slice-centroid construction under convexity. For strictly convex quadratic objectives, affine-normal directions are collinear with Newton directions, implying one-step convergence under exact line search. For general smooth (possibly nonconvex) objectives, we characterize precisely when affine-normal directions yield strict descent and develop a line-search-based YAND. We establish global convergence under standard smoothness assumptions, linear convergence under strong convexity and Polyak-Lojasiewicz conditions, and quadratic local convergence near nondegenerate minimizers. We further show that affine-normal directions are robust under affine scalings, remaining insensitive to arbitrarily ill-conditioned transformations. Numerical experiments illustrate the geometric behavior of the method and its robustness under strong anisotropic scaling.

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