This note proves the convergence of the Bregman Douglas-Rachford splitting method (BDRS) under certain verifiable condition, when it is treated as a matrix scaling algorithm. This is the first non-trivial convergence result of BDRS. The proof was generated by ChatGPT 5.5 and verified by the author. ArticleDownload View PDF
We propose Log-Averaged Mirror Prox (LAMP), a linear-space primal-dual method for large-scale optimal transport. LAMP implements primal mirror prox updates by tracking an averaged dual sequence, reducing storage complexity from \({O}(nm)\) to \({O}(n+m)\) while preserving dense, GPU-friendly reductions. Consequently, LAMP preserves the last-iterate \(\widetilde{{O}}( nm\varepsilon^{-1})\) arithmetic complexity of conservatively parameterized primal-dual mirror prox. We further … Read more
Can gradient descent be accelerated by just choosing better stepsizes? Surprisingly, the answer is yes. This short expository article provides an accessible introduction to this phenomenon of stepsize hedging. ArticleDownload View PDF
We study linear bilevel and pricing problems in which the upper- and lower-level constraints’ right-hand sides are perturbed. In this setting, it is an important question, also for the validity of numerical solution schemes, if the solution-set mapping of the parametric bilevel problem is calm at the zero-perturbation. We provide the complete picture both for … Read more
This work studies probability functions that appear in stochastic programming models. Although their differentiability has been widely investigated, the Lipschitz continuity of their gradients, crucial for the design and analysis of modern optimization algorithms, has received little attention. We develop a general framework that ensures differentiability and gradient Lipschitz continuity under practical conditions. Our framework … Read more
In this study, we explore how the domain of objective function values for challenging integer programs can be reduced and whether such a reduction can improve the solution process. Our work is motivated by binary search, a technique that efficiently narrows a search space by repeatedly halving it through feasibility checks. Building on this idea, … Read more
Shi et al. \cite{shi2022understanding} propose acceleration methods to solve smooth convex optimization problems. In our work, we focus on the general unconstrained composite non-smooth convex optimization problem. We provide an inertial forward-backward algorithm with subgradient correction, derived through time discretization of the ODE, as studied by Shi et al. We achieve the rate of convergence … Read more
Distributionally robust optimization (DRO) has been successful in addressing decision-making problems under uncertainty when the underlying distribution is unknown. Existing data-driven DRO frameworks, however, often impose restrictive assumptions on the data-generating process. We propose a new DRO framework based on targeted integral probability metrics. The ambiguity set is defined directly through the loss functions induced … Read more
This paper introduces Stochastic Online Scaled Gradient Methods (SOSGM), a generalization of the recently developed adaptive preconditioning framework in \cite{gao2025gradient,chu2025gradient} to stochastic optimization. Under standard assumptions, we establish convergence guarantees for SOSGM using large batchsize or variance reduction. SOSGM is compatible with popular diagonal and/or low-rank preconditioners as well as heavy-ball momentum, while maintaining memory … Read more
We design a class of additive noise mechanisms that satisfy \((\varepsilon, \delta)\)-differential privacy (DP) for scalar, real-valued query functions with known sensitivities, with a particular focus on moderate and low-privacy regimes. These mechanisms, which we call \textit{mixture mechanisms}, are constructed by mixing multiple Gaussian distributions that share the same variance but differ in their means … Read more