# Curiosities and counterexamples in smooth convex optimization

arXiv: https://arxiv.org/abs/2001.07999 · HTML: https://mlsanity.com/paper/2001.07999/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jan 2020.

## Built on by (most cited first, 7 of 7)

- [Regularized Newton Method with Global O(1/k2) Convergence](https://mlsanity.com/paper/2112.02089/) (2021)
- [Implicit regularization in AI meets generalized hardness of approximation in optimization -- Sharp results for diagonal linear networks](https://mlsanity.com/paper/2307.07410/) (2023)
- [Sketch-and-Project Meets Newton Method: Global mathcal O(k-2) Convergence with Low-Rank Updates](https://mlsanity.com/paper/2305.13082/) (2023)
- [Deep Learning as the Disciplined Construction of Tame Objects](https://mlsanity.com/paper/2509.18025/) (2025)
- [Glocal Smoothness: Line search and adaptive step sizes can help in theory too!](https://mlsanity.com/paper/2506.12648/) (2025)
- [Non-KKT Accumulation in Entropic Mirror Descent](https://mlsanity.com/paper/2608.01658/) (2026)
- [Establishing Boundary KKT Convergence of Mirror Descent through Reparameterization](https://mlsanity.com/paper/2608.07248/) (2026)

## Abstract (arXiv)

Counterexamples to some old-standing optimization problems in the smooth convex coercive setting are provided. We show that block-coordinate, steepest descent with exact search or Bregman descent methods do not generally converge. Other failures of various desirable features are established: directional convergence of Cauchy's gradient curves, convergence of Newton's flow, finite length of Tikhonov path, convergence of central paths, or smooth Kurdyka-Lojasiewicz inequality. All examples are planar. These examples are based on general smooth convex interpolation results. Given a decreasing sequence of positively curved C k convex compact sets in the plane, we provide a level set interpolation of a C k smooth convex function where k ≥ 2 is arbitrary. If the intersection is reduced to one point our interpolant has positive definite Hessian, otherwise it is positive definite out of the solution set. Furthermore , given a sequence of decreasing polygons we provide an interpolant agreeing with the vertices and whose gradients coincide with prescribed normals.