Yifeng Meng | Searcharxiv

arXiv2025

Kahan's Automatic Step-Size Control for Unconstrained Optimization

The Barzilai and Borwein (BB) gradient method is one of the most widely-used line-search gradient methods. It computes the step-size for the current iterate by using the information carried in the previous iteration. Recently, William Kahan [Kahan, Automatic Step-Size Control for Minimization Iterations, Technical report, University of California, Berkeley CA, USA, 2019] proposed new Gradient Descent (KGD) step-size strategies which iterate the step-size itself by effectively utilizing the information in the previous iteration. In the quadratic model, such a new step-size is shown to be mathematically equivalent to the long BB step, but no rigorous mathematical proof of its efficiency and effectiveness for the general unconstrained minimization is available. In this paper, by this equivalence with the long BB step, we first derive a short version of KGD step-size and show that, for the strongly convex quadratic model with a Hessian matrix $H$, both the long and short KGD step-size (and hence BB step-sizes) gradient methods converge at least R-linearly with a rate $1-\frac{1}{{\rm cond}(H)}$. For the general unconstrained minimization, we further propose an adaptive framework to effectively use the KGD step-sizes; global convergence and local R-linear convergence rate are proved. Numerical experiments are conducted on the CUTEst collection as well as the practical logistic regression problems, and we compare the performance of the proposed methods with various BB step-size approaches and other recently proposed adaptive gradient methods to demonstrate the efficiency and robustness.

math.OC↗

arXiv2025

The Robin heat kernel and its expansion via Robin eigenfunctions

We prove the existence and uniqueness of the Robin heat kernel on compact Riemannian manifolds with smooth boundary for Robin parameter $\alpha\in\mathbb{R}$, expressed as a spectral expansion in terms of Robin eigenvalues and eigenfunctions. For the non-negative parameter regime ($\alpha\ge 0$), we present a direct proof based on trace Sobolev inequalities and eigenfunction estimates. The case of negative parameters ($\alpha<0$) requires novel analytical techniques to handle $L^\infty$ estimates of Robin eigenfunctions, addressing challenges not present in the non-negative case. Our result extends the the classical Dirichlet and Neumann cases to the less-studied negative parameter regime.

math.AP↗

arXiv2024

Isoperimetric inequalities for Neumann eigenvalues on bounded domains in rank-1 symmetric spaces

In this paper, we prove sharp isoperimetric inequalities for lower order eigenvalues of Neumann Laplacian on bounded domains in both compact and noncompact rank-1 symmetric spaces. Our results generalize the work of Wang and Xia for bounded domains in the hyperbolic space [13], and Szegö-Weinberger inequality in rank-1 symmetric spaces obtained by Aithal and Santhanam [1].

math.DG↗