Research

Research interests: mathematical optimization, data science, numerical analysis, scientific computing, machine learning, and control, with applications spanning quantum computing, biomedical science, and materials design.

Research Highlights — Explore the significance and my contributions behind six papers on derivative-free optimization.

Derivative-Free Optimization and Model-Based Trust-Region Methods

• Poisedness and local outlier factors in derivative-free trust-region optimization — preprint with Lucas Cabral, 2026.

• A Derivative-Free Method Using a New Underdetermined Quadratic Interpolation Model — SIAM Journal on Optimization, 2025.

• Derivative-Free Optimization with Transformed Objective Functions — Journal of the Operations Research Society of China, 2025.

• Privacy-Preserving Zeroth-Order Optimization (PZOO) — manuscript with Jianqing Fan, 2026; linked arXiv record is the earlier PBBO version.

• A Derivative-Free Trust-Region Method for Optimization on Ellipsoids — Journal of Physics: Conference Series, 2023.

• ReMU: Regional Minimal Updating for Model-Based Derivative-Free Optimization — Optimization Methods and Software, 2026.

• Least H^2-Norm Updating of Quadratic Interpolation Models — IMA Journal of Numerical Analysis, 2026.

• MATRO: Metric-Aware Fully Quadratic Trust Regions for Derivative-Free Optimization — arXiv preprint, 2026.

• BUP-TR: Bayesian Underdetermined Projection Trust-Region Methods — arXiv preprint, 2026.

• TOBYQA: A Trust-Region Method for Derivative-Free Optimization on Time-Varying Functions — arXiv preprint, 2026.

• Derivative-Free Trust-Region Methods in Local Coordinates on Riemannian Manifolds — preprint with Shu Zhang, 2026.

Optimization for AI

• Hybrid Neural Network Training with Two-Dimensional Model-Based Subspace Search — preprint with Churui Wang, 2026.

• Subspace Trust Region Policy Optimization with a Direct Reduced Fisher Matrix — preprint with Hongzheng Yi, 2026.

• Derivative-Free Structured Updates for Muon — arXiv preprint, 2026.

AI for Optimization

• Why and When Neural Networks Improve Local Approximation in Optimization — arXiv preprint, 2026.

• Privacy-Preserving Zeroth-Order Optimization (PZOO) — manuscript with Jianqing Fan, 2026; linked arXiv record is the earlier PBBO version.

• Objective Value Change and Shape-Based Accelerated Optimization for Neural-Network Approximation — arXiv preprint, 2025.

LLMs for Optimization

I am preparing for release a study of reusable prompts intended to improve optimization algorithms across solver families. A complementary project will benchmark LLM-designed DFO algorithms for both numerical performance and convergence guarantees under explicit assumptions. It will evaluate algorithm descriptions, proof arguments, and executable implementations, testing symbolic reasoning together with numerical judgment. Transferability and performance gains will be assessed through controlled experiments.

Approximation Theory and Neural-Network Approximation

• Local Oscillation for Convolutional Neural Networks: A Reconstructed Measure for Image Classification across Three Backbone Families — preprint with Zhixin Li and Zijian Zhou, 2026.

• Why and When Neural Networks Improve Local Approximation in Optimization — arXiv preprint, 2026.

• Objective Value Change and Shape-Based Accelerated Optimization for Neural-Network Approximation — arXiv preprint, 2025.

• A Novel Local Analysis of Objectives Approximated by Neural Networks: L-Change — conference paper/poster, 2024.

• Model-Driven Subspaces for Large-Scale Optimization with a Local Approximation Strategy — arXiv preprint, 2025.

Subspace Methods for Large-Scale Optimization

• Random-subspace Ellipsoidal Trust-region Methods for Derivative-free Optimization — arXiv preprint, 2026.

• 2D-MoSub: A Two-Dimensional Model-Based Subspace Method for Large-Scale DFO — Optimization Methods and Software, 2026.

• Model-Driven Subspaces for Large-Scale Optimization with a Local Approximation Strategy — arXiv preprint, 2025.

Parallel Computing and Optimization

• Parallel Model-Based Derivative-Free Optimization via Rank-Two KKT Updates — arXiv preprint, 2026.

• Distributed Gradient-Regularized Newton Method: Scheduled Consensus and Global Complexity — arXiv preprint, 2026.

Numerical Methods for Multiple Solutions of PDEs

• A Spectral Levenberg-Marquardt-Deflation Method for Multiple Solutions of Semilinear Elliptic Systems — Journal of Computational and Applied Mathematics, 2026.

• An Improved Adaptive Orthogonal-Basis Deflation Method for Nonlinear Elliptic Equations — Journal of Computational Mathematics, 2026.

• Second-Order H^1-Norm Error Analysis for Time-Fractional Advection-Dispersion Equations — arXiv preprint, 2026.

Trust-Region Methods for Saddle-Point Problems

With Qiang Du at Columbia University, I am developing trust-region saddle-point algorithms with global convergence as a central goal. The local analysis uses local convexity–concavity assumptions, while our current route to global convergence requires globally strong convexity–concavity. A key objective is to weaken these restrictive global curvature assumptions and address more general saddle-point problems.

Applications in Quantum Computing and Scientific Engineering

• A block reduction for stationary covariances of linear stochastic second-order systems — preprint with Lucas Cabral, 2026.

• A Two-Stage Variational Quantum Solver for Random-Feature PDE Discretizations — preprint, 2026.

• Parametric Resonant Control of Macroscopic Behaviors of Multiple Oscillators — American Control Conference, 2019.

• Simulation of Folded-Waveguide Space Traveling-Wave Tubes with Derivative-Free Mixed-Integer NEWUOA — ICCC, 2021.

• The Modeling and Optimization of a Multi-Dam System — Applied and Computational Mathematics, 2024.

AI and Mathematical Methods for Biological Modeling

I am also exploring AI and mathematical methods for biological modeling. Current projects include learning stochastic differential equations for biological networks and studying combined virotherapy–chemotherapy models. These projects emphasize model validation, uncertainty, and numerical evidence.