Why it matters This resolves a long-standing quasi-Newton question negatively. Classical DFP can fail even for a C² uniformly convex two-dimensional objective with global Hessian condition number at most three, while every accepted step satisfies strong Wolfe conditions.
Read first Theorem 1 and Corollary 2 in Section 2, Section 3.2 for the construction’s mechanism, and Section 6.4 for the relation to existing convergence results.
Why it matters A direct accelerated Newton scheme achieves a global O(k−3) functional-residual rate for convex functions with Lipschitz Hessian using only primal variables and one regularized linear solve per iteration. It admits Hessian-free inexact solves.
Read first Equation (3) and Table 1 for the method and comparison, then Section 4 for the accelerated-rate proof and Section 5 for composite objectives and non-Euclidean geometry.
3. A Decomposed Bilevel Search for Variable-Metric Proximal Gradient Methods
Xinpeng Li and Ya-xiang Yuan · arXiv · 26 August 2026 · Primary source ↗
Why it matters A diagonal-plus-rank-one factorization reduces a difficult scaled proximal map to a two-dimensional monotone system. The resulting oracle makes DFP/BFGS-type metrics usable with structured nonsmooth regularizers and has certified polylogarithmic accuracy dependence.
Read first Algorithm 1 and Sections 3.1–3.2 for the oracle, Sections 3.3–3.4 for its Broyden interpretation, and Sections 5.2–5.4 for SLOPE and group-lasso experiments.
4. Condition Numbers of Block Toeplitz Matrices and Stability of Space-Time IgA Approximations
Manuel Bogoya, Albrecht Böttcher, Matteo Ferrari, Sergei M. Grudsky and Stefano Serra-Capizzano · arXiv · 25 August 2026 · Primary source ↗
Why it matters This extends asymptotic conditioning theory from scalar to fixed-block Toeplitz sequences, distinguishing bounded, polynomial, and exponential growth. It links the theory directly to stability of space-time isogeometric discretizations.
Read first Section 3 for general bounds, Section 4 for banded block matrices, and Sections 5.1 and 5.3 for the wave and Schrödinger applications.
5. A Line-Search-Free Coordinate Proximal Predictor–Corrector Method for Monotone Absolute Value Equations
Haotian Wang and Yong Xia · arXiv · 25 August 2026 · Primary source ↗
Why it matters For Ax−|x|=b, CPPC replaces backtracking with an exact coordinate proximal predictor and a full-residual correction. Each iteration needs one new matrix–vector product, with whole-sequence, sublinear residual, and strong-monotonicity linear convergence guarantees.
Read first Proposition 2.1, Algorithm 1 in Section 3.1, the rate results in Sections 4.1–4.2, and Section 5’s dimension and matrix-geometry experiments.
Selection and commentary are editorial. Dates and bibliographic details link to the cited primary sources.