We establish sharp asymptotic rates for the two Barzilai--Borwein (BB) rules on uniformly positive quadratics and local nonlinear problems. In finite dimensions, for either fixed rule and an arbitrary positive first step, the gradient root factor is bounded by $(b_0-a_0)/(b_0+a_0)$, where $[a_0,b_0]$ is the initially active spectral interval. Hence the worst trajectory factor is $c_H=(\kappa(H)-1)/(\kappa(H)+1)$. When $H$ has at least two distinct eigenvalues, matched initialization and a balanced endpoint trajectory attain this value. Under matched initialization, the same constant is the optimal uniform-envelope threshold. For bounded, self-adjoint, uniformly positive operators on Hilbert space, scalar spectral measures yield the corresponding active-support bound and optimal matched uniform-envelope threshold, including continuous endpoint spectrum. Finally, if the gradient is strictly Fr\'echet differentiable at a stationary point and its derivative is self-adjoint and uniformly positive, every $\gamma\in(c_*,1)$, where $c_*=(\kappa(A_*)-1)/(\kappa(A_*)+1)$, is a uniform local envelope rate for either pure BB rule. Every well-defined trajectory converging to the stationary point has error and gradient root factors at most $c_*$ and objective-gap root factor at most $c_*^2$. Over the class of objectives with prescribed distinct derivative endpoints $m_*
Barzilai--Borwein (BB) method has shown strong practical performance in continuous optimization, yet its convergence dynamics remains poorly understood. In particular, a central unresolved question is whether BB converges superlinearly for almost every strictly convex quadratic problem and initialization. We provide a negative answer to this question. Specifically, for every finite dimension $n\geq4$, we construct a nonempty open, hence positive-Lebesgue-measure, family of strictly convex quadratic problems and initial points for which the long Barzilai--Borwein method (BB1) converges but cannot converge root-superlinearly. More precisely, with the explicit constants $\rho_{\min}=10^{-6},\rho_{\max}=0.61$, every spectral component of the gradient is bounded above and below by the corresponding geometric sequence. Consequently, the gradient norm and the energy norm of the error satisfy two-sided geometric estimates with the same rates, while the objective gap satisfies the corresponding estimates with squared rates. In particular, all three quantities are bounded below by geometric sequences, ruling out superlinear convergence. The construction is highly nontrivial, based on a computer-assisted proof of a nonresonant, attracting seven-cycle of the projectivized BB dynamics in dimension four.
We study one-dimensional Jacobi operators of divergence-gradient type on $\ell^2(\mathbb{Z}_{\geq 0})$, with coefficients generated by the doubling map, $a_n(x)=a(2^n x \mathrm{mod} 1)$. For continuous positive sampling functions, we show that the almost-sure essential spectrum is an interval containing the bottom of the spectrum. Under the assumption $a\in C^1(\mathbb{T})$, we prove square-root asymptotics for the integrated density of states and obtain a small-energy expansion for the Lyapunov exponent as $E\to0^+$. The leading term of the Lyapunov exponent is linear and positive precisely under a nondegeneracy condition on the sampling function. In this case, we prove a large-deviation estimate for the transfer matrices with an explicit rate depending on $E$. As consequences, we obtain local H\"older continuity of the Lyapunov exponent and the integrated density of states near the bottom of the spectrum, as well as the Anderson localization. Some of these results extend the small-coupling results of Chulaevsky--Spencer and Bourgain--Schlag for Schr\"odinger operators generated by the doubling map to divergence-gradient type operators. The arguments adapt methods from those works, but require uniform control in the small-energy parameter and refined correlation estimates reflecting the divergence-gradient structure.
We develop a fixed-reference weighted $L^2$ theory for terminal-value fully nonlinear parabolic equations. A prescribed uniformly elliptic diffusion starting from a point determines both the linear reference operator and a space--time occupation measure; time weights of order $\alpha$ accommodate the point-start singularity. For the associated linear backward equation, we identify the optimal source-to-intrinsic-Hessian norm $C_w(X)$ and derive bounds on it from a moving-measure Bochner identity. We compute the sharp Brownian benchmark $C_\alpha^{\mathrm{Br}}=\sup_{n\in\mathbb{N}, n\geq 2}2\sqrt{n(n-1)}/(n-1+\alpha)$, prove that it is a universal tangent lower bound, and show that it is attained by regular weights for time-inhomogeneous affine Gaussian diffusions and is the infimum over such weights for normalized state-dependent covariances. Under the standing data and lower-order assumptions, the Hessian estimate yields our main nonlinear result: if the driver's Hessian Lipschitz constant satisfies $C_w(X)L_H<1$, then the equation has a unique weighted Sobolev solution and the Picard iteration on the source converges geometrically, with no smallness condition on the value or gradient channels. For the Brownian model at order $1/2$, counterexamples in dimensions $d\geq7$ show that this strict threshold cannot be uniformly improved within the occupation-Sobolev class. For weights of order below one, additional spatial regularity and unweighted integrability of the data yield an unweighted spatial jet and a unique solution in the regular fixed-reference Markovian class for second-order backward stochastic differential equations (2BSDEs) defined here.
We establish the bound for the second-order Hankel determinant $H_{2,2}(F) = A_2 A_4 - A_3^2$ associated with the class $\mathcal{C}_{B}^{\beta}(\mathbb{B})$ of normalized $\beta$-spirallike quasi-convex mappings of type $B$ on the open unit ball $\mathbb{B}$ of a complex Banach space. By utilizing a generalized framework based on a directional slice homogeneous polynomial expansion, we eliminate the standard, restrictive assumption that the mapping is of the form $F(x) = g(x)x$. Under these weaker operational conditions, we parameterize the targeted scalar invariants $A_n$ via the classical Carath\'{e}odory functional parameters. A rigorous optimization analysis proves that the established upper bound is strictly sharp for the classical non-spirallike case $\beta = 0$, yielding a maximal value of $1/8$. This sharp bound is verified by constructing explicit multi-dimensional extremal mappings that lift the corresponding single-variable convex profile. Finally, an unresolved open question regarding the exact variational behavior for $\beta \neq 0$ is formulated.
M. B. Ahamed, Nabadwip Sarkar, Pradip Das· 0 citations
We explore fast convergence of the Riemannian Frank--Wolfe method for smooth geodesically convex optimization over compact feasible sets. Hadamard manifolds are the main setting. On general complete manifolds, the analysis accounts for all feasible minimizing geodesics. We consider the open loop step-size $\eta_k=a/(k+a)$, which only uses the iteration index. Under a local H\"olderian error bound and local length-normalized directional scaling, every $a>2$ gives the eventual rate $O(k^{-1/(1-\theta)})$ for $\theta\in(0,1/2]$. An interior-ball condition yields $O(k^{-2})$ for strongly geodesically convex objectives. Under an exact Riemannian scaling inequality and a uniform positive lower bound on the gradient norm, every $a\geq2$ gives $O(k^{-a})$ after an explicit threshold index. The same rate holds for the smallest Frank--Wolfe gap over the most recent half of the iterates. For geodesic balls of radius \(R<\pi/2\) in the unit sphere, we establish the scaling inequality with $\alpha_R=\tfrac12\cot R$, yielding $O(k^{-a})$ primal error and recent-window gap rates. We also analyze the standard gap-feedback short step under the local error-bound conditions, obtaining $O(k^{-1/(1-2\theta)})$ for $\theta<1/2$ and a linear rate for $\theta=1/2$. Numerical experiments illustrate the predicted rates and compare iteration-only and feedback-based step selection.
Let $A$ be Hermitian positive definite and let $f_m$ denote the Lanczos approximation to $f(A)b$. We prove that if $f(z)$ or $f(z) / z$ is Stieltjes, then the $A^\alpha$-norm error of the Lanczos approximation is within a factor $\tfrac{1}{2}(\kappa(A)^{E/2} + \kappa(A)^{-E/2})$ of the the best possible Krylov Subspace Method, where $\kappa(A)$ is the condition number of $A$ and $E = \max\{\alpha,1-\alpha\}$. Our result strengthens and generalizes the upper bound of [Schweitzer; SIMAX, 46.3 (2025)]. Moreover, we prove that the constant $\tfrac{1}{2}(\kappa(A)^{E/2} + \kappa(A)^{-E/2})$ is optimal.
