arXiv subjects

Shutai Yang

Publications and source records attributed to Shutai Yang.

4 recordsLinked to original sources

arXiv2026

Lyapunov Functions and $R$-Linear Convergence for Quadratic Barzilai-Borwein Dynamics

The Barzilai--Borwein (BB) method is $R$-linearly convergent on strongly convex quadratics, although neither the objective value nor the gradient norm is generally monotone. We construct an explicit current-point Lyapunov function for the quadratic dynamics after the warm-up step, assuming that both endpoints of the initial active spectrum survive this step. If $a<b$ are these endpoints and $P_a,P_b$ are the corresponding spectral projectors, then \[ \overline{f}(x) :=\lVert P_a\nabla f(x)\rVert^{\frac{2b}{a+b}} \lVert P_b\nabla f(x)\rVert^{\frac{2a}{a+b}} \] satisfies, for BB1, BB2, and every fixed positive weighted delayed Rayleigh rule, \[ \overline{f}(x_{k+1}) =\left(\frac{b-a}{b+a}\right)^2 e^{-2D_{a,b}(τ_k)}\overline{f}(x_k), \qquad D_{a,b}(τ_k)\ge0, \] where $τ_k=1/α_k$ is the reciprocal step. The function is obtained by denormalizing the endpoint coboundary certificate in the sharp-rate analysis of Yang and Yuan. We prove that the endpoint exponents are the unique minimax choice among endpoint monomials and illustrate the Lyapunov law on a nonmonotone BB1 trajectory. Using this law, we also give a proof of $R$-linear convergence of the method in finite dimensions.

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arXiv2026

$R$-Linear Convergence of Barzilai-Borwein Methods on Strictly Convex Quadratics via Closed-Cone Homogeneous Dynamics

We give an $R$-linear convergence proof for BB1, BB2, and fixed positive spectral-weight variants on finite-dimensional strictly convex quadratics. The delayed recurrence is written as a first-order system on the closed cone of compatible consecutive-gradient pairs. Its transition is continuous and positively homogeneous of degree one, including at finite termination. A compactness theorem shows that pointwise convergence of every cone orbit implies a uniform finite-step contraction and hence a global $R$-linear estimate. By a realization lemma, every compatible state initiates a BB1 trajectory, so Raydan's theorem yields pointwise stability on the entire cone. Finally, the transformation $g\mapsto W^{1/2}g$ conjugates every fixed positive spectral-weight rule to BB1. Consequently, BB1, BB2, and all such weighted rules have the same homogeneous growth radius.

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arXiv2026

Global and Unconditional $R$-Linear Convergence of Rank-Compressed Weighted LMSD Sweeps for Strictly Convex Quadratics

We study limited memory steepest descent (LMSD) with exact algebraic rank compression for strictly convex quadratic optimization. At each cycle, the method restricts the gradient history to its column space and applies the reciprocals of all Ritz values of the compressed projection in the next sweep. If the block-start gradient has at most $p$ active distinct eigenvalues, the delayed sweep terminates finitely. For nonterminating trajectories, a determinant and Cauchy--Binet formula for the complete-sweep polynomial yields global convergence without a full-rank history or a run-wise normalized-conditioning bound. Consecutive sweep endpoints define a continuous positively homogeneous map on a closed cone of compatible states. Compactness then gives an $R$-linear endpoint estimate, uniform over compatible initial states, with constants depending only on $H$ and $p$; the estimate extends to inner gradients, iterate errors, and objective gaps. Every fixed positive spectral weight $W=ω(H)$ is linearly conjugate to the standard method. The weighted methods therefore share its decay factor, while the Euclidean-norm prefactor is increased by at most $\sqrt{κ_2(W)}$. This includes harmonic-Ritz LMSD, fixed power weights, and the delayed BB1 and BB2 recurrences.

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arXiv2026

The Sharp Worst-Case Asymptotic Rate of the Barzilai--Borwein Method in $\mathbb R^d$ and Hilbert Spaces

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 $c_H=(κ(H)-1)/(κ(H)+1)$. 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 Fréchet differentiable at a stationary point and its derivative is self-adjoint and uniformly positive, every $γ\in(c_*,1)$, where $c_*=(κ(A_*)-1)/(κ(A_*)+1)$, 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 $m_*<M_*$, matched endpoint trajectories for quadratic and $C^\infty$ genuinely nonquadratic examples in $\mathbb R^2$ attain these factors.

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