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Various first order approaches have been proposed in the literature to solve Linear Programming (LP) problems, recently leading to practically efficient solvers for large-scale LPs. From a theoretical perspective, linear convergence rates…
This paper presents a high-order deferred correction algorithm combined with penalty iteration for solving free and moving boundary problems, using a fourth-order finite difference method. Typically, when free boundary problems are solved…
We study composite optimization problems in which the smooth part of the objective function is \( p \)-times continuously differentiable, where \( p \geq 1 \) is an integer. Higher-order methods are known to be effective for solving such…
Large-scale optimization problems arising from the discretization of problems involving PDEs sometimes admit solutions that can be well approximated by low-rank matrices. In this paper, we will exploit this low-rank approximation property…
In this paper, we consider the integrating factor midpoint method for wave-type equations and derive optimal order a posteriori error estimates. We first introduce an integrating factor midpoint approximation defined by the piecewise linear…
We present a new rank-adaptive tensor method to compute the numerical solution of high-dimensional nonlinear PDEs. The method combines functional tensor train (FTT) series expansions, operator splitting time integration, and a new…
We prove that the model checking problem for the existential fragment of first-order (FO) logic on partially ordered sets is fixed-parameter tractable (FPT) with respect to the formula and the width of a poset (the maximum size of an…
Prompt tuning is a parameter-efficient tuning (PETuning) method for utilizing pre-trained models (PTMs) that simply prepends a soft prompt to the input and only optimizes the prompt to adapt PTMs to downstream tasks. Although it is…
Lyapunov functions play a fundamental role in analyzing the stability and convergence properties of optimization methods. In this paper, we propose a novel and straightforward approach for constructing Lyapunov functions for first-order…
The fundamental lemma by Jan C. Willems and co-authors enables the representation of all input-output trajectories of a linear time-invariant system by measured input-output data. This result has proven to be pivotal for data-driven…
Partially observable Markov decision processes (POMDPs) provide a modeling framework for autonomous decision making under uncertainty and imperfect sensing, e.g. robot manipulation and self-driving cars. However, optimal control of POMDPs…
We present Lift and Relax for Waveform Inversion (LRWI), an approach that mitigates the local minima issue in seismic full waveform inversion (FWI) via a combination of two convexification techniques. The first technique (Lift) extends the…
We study the performance of a Wolff-type embedding algorithm for $RP^N$ $\sigma$-models. We find that the algorithm in which we update the embedded Ising model \`a la Swendsen-Wang has critical slowing-down as $z_\chi \approx 1$. If instead…
In this work, we consider solving optimization problems with a stochastic objective and deterministic equality constraints. We propose a Trust-Region Sequential Quadratic Programming method to find both first- and second-order stationary…
Recent progress in randomized motion planners has led to the development of a new class of sampling-based algorithms that provide asymptotic optimality guarantees, notably the RRT* and the PRM* algorithms. Careful analysis reveals that the…
Parameter-efficient fine-tuning (PEFT) methods have emerged as a practical solution for adapting large foundation models to downstream tasks, reducing computational and memory costs by updating only a small subset of parameters. Among them,…
This paper introduces a novel data-driven convergence booster that not only accelerates convergence but also stabilizes solutions in cases where obtaining a steady-state solution is otherwise challenging. The method constructs a…
We consider linear two-time-scale stochastic approximation algorithms driven by martingale noise. Recent applications in machine learning motivate the need to understand finite-time error rates, but conventional stochastic approximation…
Unless special conditions apply, the attempt to solve ill-conditioned systems of linear equations with standard numerical methods leads to uncontrollably high numerical error. Often, such systems arise from the discretization of operator…
First-order methods for quadratic optimization such as OSQP are widely used for large-scale machine learning and embedded optimal control, where many related problems must be rapidly solved. These methods face two persistent challenges:…