arXiv subjects

Yuntian Jiang

Publications and source records attributed to Yuntian Jiang.

9 recordsLinked to original sources

arXiv2026

Accelerating Trust-Region Methods: An Attempt to Balance Global and Local Efficiency

Balancing global efficiency and local convergence remains a central challenge in second-order methods for unconstrained convex optimization problems. Newton's method enjoys fast local convergence but may diverge when initialized far from the solution. In contrast, accelerated second-order methods provide global guarantees but typically suffer from slower local convergence. This raises the fundamental question of to what extent global acceleration can be achieved without sacrificing strong local convergence. In this paper, we tackle this challenge by proposing the first accelerated trust-region-type methods and leveraging their inherent primal-dual information. Our primary contribution is the Accelerated Trust-Region method with Local Detection, which utilizes the Lagrange multiplier to detect local regions and achieves a global oracle complexity of \tilde{O}(ε^{-1/3}), while maintaining quadratic local convergence. We further examine the trade-off that arises when global convergence is pushed to the limit. Specifically, we introduce the Accelerated Trust-Region Extragradient Method, which achieves a global oracle complexity of \tilde{O}(ε^{-2/7}) but no longer enjoys quadratic local convergence. This reveals a phase-transition-like phenomenon in accelerated trust-region-type methods: quadratic local convergence is preserved under moderate global acceleration, but it breaks down when pursuing extreme global efficiency. Numerical experiments are consistent with the theoretical predictions and illustrate the global-local trade-off.

math.OC↗

arXiv2026

Homogeneous second-order descent framework: a fast alternative to Newton-type methods

This paper proposes a homogeneous second-order descent framework (HSODF) for nonconvex and convex optimization based on the generalized homogeneous model (GHM). In comparison to the Newton steps, the GHM can be solved by extremal symmetric eigenvalue procedures and thus grant an advantage in ill-conditioned problems. Moreover, GHM extends the ordinary homogeneous model (OHM) (Zhang et al. 2022) to allow adaptiveness in the construction of the aggregated matrix. Consequently, HSODF is able to recover some well-known second-order methods, such as trust-region methods and gradient regularized methods, while maintaining comparable iteration complexity bounds. We also study two specific realizations of HSODF. One is adaptive HSODM, which has a parameter-free $O(ε^{-3/2})$ global complexity bound for nonconvex second-order Lipschitz continuous objective functions. The other one is homotopy HSODM, which is proven to have a global linear rate of convergence without strong convexity. The efficiency of our approach to ill-conditioned and high-dimensional problems is justified by some preliminary numerical results.

math.OC↗

arXiv2026

A homogeneous second-order descent method for nonconvex optimization

In this paper, we introduce a Homogeneous Second-Order Descent Method (HSODM) using the homogenized quadratic approximation to the original function. The merit of homogenization is that only the leftmost eigenvector of a gradient-Hessian integrated matrix is computed at each iteration. Therefore, the algorithm is a single-loop method that does not need to switch to other sophisticated algorithms and is easy to implement. We show that HSODM has a global convergence rate of $O(ε^{-3/2})$ to find an $ε$-approximate second-order stationary point, and has a local quadratic convergence rate under the standard assumptions. The numerical results demonstrate the advantage of the proposed method over other second-order methods.

math.OC↗

arXiv2026

Beyond Nonconvexity: A Universal Trust-Region Method with New Analyses

The trust-region (TR) method is renowned historically for its robustness in nonconvex problems and extraordinary numerical performance, but the study of its performance in convex optimization is somehow limited. This paper complements the existing literature by presenting a universal trust-region method that simultaneously incorporates the quadratic regularization and ball constraint. In particular, we introduce a novel descent property tailored for trust-region-type algorithms, enabling us to unify and streamline the analysis for both convex and nonconvex optimization. Our method exhibits an iteration complexity of $\tilde O(ε^{-3/2})$ to find an $ε$-approximate second-order stationary point for nonconvex optimization. Meanwhile, the analysis reveals that the universal method attains an $O(ε^{-1/2})$ complexity bound for convex optimization. Finally, we develop an adaptive universal method to address practical implementations. The numerical results show the effectiveness of our method in both nonconvex and convex problems.

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arXiv2026

Segregation-Controlled Diffusion-Induced Grain Boundary Migration in Alloy 690

Grain boundary (GB) migration accompanied by Cr depletion is widely observed in Alloy 690 and is closely linked to intergranular degradation and stress corrosion cracking. However, the fundamental driving force for GB migration and its link with Cr depletion remains unclear. In this work, hybrid molecular dynamics and semi-grand canonical Monte Carlo simulations were employed to investigate GB migration in Alloy 690 under coupled solute diffusion and segregation effects across a range of GB characters. The results show that Cr segregation at GBs, while generally considered favorable for GB stability, can facilitate diffusion-induced GB migration and Cr depletion. Cr diffusion along GBs produces localized Cr depletion zones that are energetically incompatible with positively segregating GBs, generating a chemical driving force that drives GB migration toward the Cr-rich matrix, which ultimately results in persistent GB migration accompanied by a Cr depletion. By quantifying solute-GB interaction energetics, we demonstrate that GB migration is quantitively controlled by the coupled effects of solute diffusivity and segregation strength. These mechanistic insights provide a unified framework that rationalizes experimentally observed correlations between GB character, Cr depletion, and GB migration in Cr-containing alloys.

cond-mat.mtrl-sci↗

arXiv2025

History-Aware Adaptive High-Order Tensor Regularization

In this paper, we develop a new adaptive regularization method for minimizing a composite function, which is the sum of a $p$th-order ($p \ge 1$) Lipschitz continuous function and a simple, convex, and possibly nonsmooth function. We use a history of local Lipschitz estimates to adaptively select the current regularization parameter, an approach we shall term the {\it history-aware adaptive regularization method}. We explore how the selection of an appropriate volume of historical information affects both the theoretical and practical performance. By using all the historical information, our method matches the complexity guarantees of the standard $p$th-order tensor methods that require a known Lipschitz constant, for both convex and nonconvex objectives. In the nonconvex case, the number of iterations required to find an $(ε_g,ε_H)$-approximate second-order stationary point is bounded by $\mathcal{O}(\max\{ε_g^{-(p+1)/p}, ε_H^{-(p+1)/(p-1)}\})$. For convex functions, we establish an $\mathcal{O}(ε^{-1/p})$ iteration complexity for finding an $ε$-approximate optimal point and further propose an accelerated variant attaining an iteration complexity of $\mathcal{O}(ε^{-1/(p+1)})$. For practical consideration, we propose several variants of this method with only part of historical information. We introduce cyclic and sliding-window strategies for choosing historical Lipschitz estimates, which mitigate the limitation of overly conservative updates. As long as a rough upper bound of the Lipschitz constant is known, these two variants achieve the same iteration complexity guarantees in terms of the input accuracy as the method using full historical information. Finally, extensive numerical experiments are conducted to demonstrate the effectiveness of our adaptive approach.

math.OC↗

arXiv2024

An Enhanced ADMM-based Interior Point Method for Linear and Conic Optimization

The ADMM-based interior point (ABIP, Lin et al. 2021) method is a hybrid algorithm that effectively combines interior point method (IPM) and first-order methods to achieve a performance boost in large-scale linear optimization. Different from traditional IPM that relies on computationally intensive Newton steps, the ABIP method applies the alternating direction method of multipliers (ADMM) to approximately solve the barrier penalized problem. However, similar to other first-order methods, this technique remains sensitive to condition number and inverse precision. In this paper, we provide an enhanced ABIP method with multiple improvements. Firstly, we develop an ABIP method to solve the general linear conic optimization and establish the associated iteration complexity. Secondly, inspired by some existing methods, we develop different implementation strategies for ABIP method, which substantially improve its performance in linear optimization. Finally, we conduct extensive numerical experiments in both synthetic and real-world datasets to demonstrate the empirical advantage of our developments. In particular, the enhanced ABIP method achieves a 5.8x reduction in the geometric mean of run time on $105$ selected LP instances from Netlib, and it exhibits advantages in certain structured problems such as SVM and PageRank. However, the enhanced ABIP method still falls behind commercial solvers in many benchmarks, especially when high accuracy is desired. We posit that it can serve as a complementary tool alongside well-established solvers.

math.OC↗

arXiv2024

Inexact and Implementable Accelerated Newton Proximal Extragradient Method for Convex Optimization

In this paper, we investigate the convergence behavior of the Accelerated Newton Proximal Extragradient (A-NPE) method when employing inexact Hessian information. The exact A-NPE method was the pioneer near-optimal second-order approach, exhibiting an oracle complexity of $\Tilde{O}(ε^{-2/7})$ for convex optimization. Despite its theoretical optimality, there has been insufficient attention given to the study of its inexact version and efficient implementation. We introduce the inexact A-NPE method (IA-NPE), which is shown to maintain the near-optimal oracle complexity. In particular, we design a dynamic approach to balance the computational cost of constructing the Hessian matrix and the progress of the convergence. Moreover, we show the robustness of the line-search procedure, which is a subroutine in IA-NPE, in the face of the inexactness of the Hessian. These nice properties enable the implementation of highly effective machine learning techniques like sub-sampling and various heuristics in the method. Extensive numerical results illustrate that IA-NPE compares favorably with state-of-the-art second-order methods, including Newton's method with cubic regularization and Trust-Region methods.

math.OC↗

arXiv2023

DRSOM: A Dimension Reduced Second-Order Method

In this paper, we propose a Dimension-Reduced Second-Order Method (DRSOM) for convex and nonconvex (unconstrained) optimization. Under a trust-region-like framework, our method preserves the convergence of the second-order method while using only curvature information in a few directions. Consequently, the computational overhead of our method remains comparable to the first-order such as the gradient descent method. Theoretically, we show that the method has a local quadratic convergence and a global convergence rate of $O(ε^{-3/2})$ to satisfy the first-order and second-order conditions if the subspace satisfies a commonly adopted approximated Hessian assumption. We further show that this assumption can be removed if we perform a corrector step using a Krylov-like method periodically at the end stage of the algorithm. The applicability and performance of DRSOM are exhibited by various computational experiments, including $L_2 - L_p$ minimization, CUTEst problems, and sensor network localization.

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