Wotao Yin

Mathematical programming solvers are software tools designed to solve real-world problems using mathematical programming algorithms. This survey explores the evolution of optimization technologies, from traditional methods such as the simplex algorithm and branch-and-bound techniques to modern advancements that are facilitated by parallel computing, GPU acceleration, and AI algorithms. We also emphasize the recent emergence of mathematical programming solvers developed by research institutes and companies headquartered in China as major players, who have achieved remarkable success in benchmarks when compared to established solvers. This article provides a comprehensive overview of the theoretical foundations, historical progress, and emerging trends in mathematical programming solvers, offering valuable insights for both researchers and practitioners in the field.

Smarter Branching Speeds Up Leading Exact Vehicle Routing Solvers Researchers have introduced a learning-based branching strategy that substantially accelerates exact algorithms for vehicle routing problems, a core challenge in logistics and transportation systems. The study proposes the first learning-to-branch framework tailored for branch-price-and-cut methods, where dynamic variables and dense constraints make traditional branching decisions computationally expensive. The novel two-stage learning-based branching (2LBB) approach effectively filters promising candidates using inexpensive features and then applies selective, partial testing to reduce costly evaluations. A theoretical model further guides dynamic adjustment of branching effort, balancing decision time with solution quality. Extensive experiments show runtime reductions of 45%–50% on standard CVRP and VRPTW benchmarks and a 47% speedup over the state-of-the-art VRPSolver when integrated into the open-source RouteOpt. These results highlight the growing potential of disciplined machine learning to enhance exact optimization algorithms.

We propose ALiA, a novel adaptive variant of the alternating direction method of multipliers (ADMM). Specifically, ALiA is a variant of function-linearized proximal ADMM (FLiP ADMM), which generalizes the classical ADMM by leveraging the differentiable structure of the objective function, making it highly versatile. Notably, ALiA features an adaptive stepsize selection scheme that eliminates the need for backtracking linesearch. Motivated by recent advances in adaptive gradient and proximal methods, we establish point convergence of ALiA for convex and differentiable objectives. Furthermore, by introducing negligible computational overhead, we develop an alternative stepsize selection scheme for ALiA that improves the convergence speed both theoretically and empirically. Extensive numerical experiments on practical datasets confirm the accelerated performance of ALiA compared to standard FLiP ADMM. Additionally, we demonstrate that ALiA either outperforms or matches the practical performance of existing adaptive methods across problem classes where it is applicable.

Robust matrix completion (RMC) is a widely used machine learning tool that simultaneously tackles two critical issues in low-rank data analysis: missing data entries and extreme outliers. This paper proposes a novel scalable and learnable non-convex approach, coined Learned Robust Matrix Completion (LRMC), for large-scale RMC problems. LRMC enjoys low computational complexity with linear convergence. Motivated by the proposed theorem, the free parameters of LRMC can be effectively learned via deep unfolding to achieve optimum performance. Furthermore, this paper proposes a flexible feedforward-recurrent-mixed neural network framework that extends deep unfolding from fixed-number iterations to infinite iterations. The superior empirical performance of LRMC is verified with extensive experiments against state-of-the-art on synthetic datasets and real applications, including video background subtraction, ultrasound imaging, face modeling, and cloud removal from satellite imagery.

The Scaled Relative Graph (SRG) is a geometric tool that maps the action of a multi-valued nonlinear operator onto the 2D plane, used to analyze the convergence of a wide range of iterative methods. As the SRG includes the spectrum for linear operators, we can view the SRG as a generalization of the spectrum to multi-valued nonlinear operators. In this work, we further study the SRG of linear operators and characterize the SRG of block-diagonal and normal matrices.

Reinforcement Learning (RL)-based post-training has significantly advanced the complex reasoning capabilities of language models, fostering sophisticated self-reflection processes. However, this ``slow thinking'' paradigm presents a critical challenge to reasoning efficiency: models may expend excessive computation on simple questions and shift reasoning prematurely for complex ones. Previous mechanisms typically rely on static length budgets or predefined rules, lacking the adaptability for varying question complexities and models' evolving capabilities. To this end, we propose AdapThink, an adaptive post-training framework designed to induce more efficient thinking while maintaining the performance of reasoning language models. Specifically, AdapThink incorporates two key mechanisms: 1) A group-relative reward function that leverages model confidence and response's characteristic to dynamically adjust the preference of reflection-related transition words without resorting to a fixed length preference. 2) A diversity-aware sampling mechanism that balances the training group's solution accuracy with reasoning diversity via an entropy-guided score. Experiments on several mathematical reasoning datasets with DeepSeek-distilled models demonstrate AdapThink's advantages in enabling adaptive reasoning patterns and mitigating the inefficiencies.