Computational Mathematics Research Digest · Issue 4
Weekly computational mathematics digest
Published · Curated by Hassan Mohammad
Coverage: substantive postings from 28 August–3 September 2026. Four papers met the quality threshold.
1. DOFFO_TR: A Decentralized Objective Function-Free Optimization Method with Trust Region
Stefania Bellavia, Greta Malaspina and Benedetta Morini · arXiv · 1 September 2026 · Primary source ↗
Why it matters This brings adaptive trust-region globalization to decentralized, potentially nonconvex optimization without requiring shared objective values or gradient vectors. Agents communicate mainly scalar gradient norms, reducing privacy exposure and communication cost. Certain variants match the iteration-complexity order of centralized trust-region methods.
Read first Section 3, especially Algorithms 3.1 and 3.2, for selective flooding and the DOFFO_TR iteration. Then examine Section 4’s complexity results and Section 5.3’s comparison with the Gradient Tracking Algorithm.
2. Optimal Gradient-Norm Minimization in Non-Euclidean Hölder-Smooth Convex Optimization
Nico Pelleriti, Maryam Shiran, David Martínez-Rubio, Max Zimmer and Sebastian Pokutta · arXiv · 1 September 2026 · Primary source ↗
Why it matters Gradient norm is a computable stationarity certificate and can be more informative than objective residual. The paper develops near-optimal oracle-complexity methods for Hölder-smooth convex objectives across the full range of ℓp geometries, closing previously open complexity gaps.
Read first The main complexity table and theorem statements in the introduction, followed by the first algorithm family based on mirror duality and controlled inexactness. Then inspect the matching lower bounds.
3. Further Analysis and Extension of the Higher-Order Newton Method of Ahmadi, Chaudhry, and Zhang
Lucas ter Voert and Etienne de Klerk · arXiv · 1 September 2026 · Primary source ↗
Why it matters The authors extend a d-th-order Newton method to problems with SOS-convex polynomial constraints. Each regularized Taylor subproblem can be represented by a semidefinite program. Under strong convexity and Lipschitz continuity of the d-th derivative tensor, the method has local convergence order d and identifies the optimal active set locally in one iteration.
Read first The constrained SDP formulation, the theorem proving order-d local convergence, and the active-set identification theorem. Then examine the performance-estimation analysis of the unconstrained third-order method.
4. Two Adjoint Perspectives on Fokker–Planck Optimization: A Microscopic–Macroscopic Correspondence
Kathrin Hellmuth, Qin Li and Yunan Yang · arXiv · 2 September 2026 · Primary source ↗
Why it matters The paper proves that macroscopic density and microscopic stochastic-trajectory adjoints agree in the continuum. Although their gradients differ after discretization, both consistently approximate the same continuum gradient with explicit rates.
Read first The continuum adjoint-correspondence theorem, followed by the two discrete-gradient convergence theorems and the numerical convergence plots comparing Eulerian and particle formulations.
Selection and commentary are editorial. Dates and bibliographic details link to the cited primary sources.