arXiv subjects

Edward Rothberg

Publications and source records attributed to Edward Rothberg.

5 recordsLinked to original sources

arXiv2026

Using a MIP Solver as a PDHG-Based MIP Heuristic

The PDHG algorithm provides a new capability for solving difficult linear programming (LP) problems: the ability to find low-accuracy solutions quickly. While such solutions may not be applicable in all applications of LP, one possible use is to accelerate heuristics for finding feasible solutions to Mixed-Integer Programming (MIP) problems, where these approximate LP solutions can hopefully guide a heuristic towards accurate and high-quality MIP solutions. While a monolithic heuristic that exploits PDHG solutions would be useful, we pose a broader question here: could we replace the default LP solver in a modern MIP solver with PDHG to accelerate most (or all) of its existing heuristics? We find that doing so does have some obvious drawbacks, preventing us from using several powerful MIP techniques, but it leaves most others unaffected, ultimately resulting in a heuristic that often finds high-quality, high-accuracy solutions faster than current state-of-the-art strategies.

math.OC↗

arXiv2026

Empirical Asymptotic Runtime Analysis of Linear Programming Algorithms

This paper takes an empirical look at asymptotic runtime growth rates for the most widely used algorithms for solving linear programming (LP) problems across a set of six optimization application areas that are known to produce large and difficult LP models. On the algorithm side, we consider the simplex method, interior-point methods, and PDHG. On the model side, we use a large language model (LLM) to create families of instances in different application areas, allowing us to study model types and sizes that are simultaneously synthetic and realistic. The results indicate that simple regression models typically predict observed runtimes quite well within a model class, and that asymptotic behavior can vary significantly between the different algorithms. This may have a significant impact on which algorithms will be most effective for solving large LP models in the future.

math.OC↗

arXiv2026

Hybridizing PDHG and Interior-Point Methods

The Primal-Dual Hybrid Gradient (PDHG) algorithm is a first-order method that can exploit GPUs to solve large-scale linear programming problems. The approach can often be faster than the alternatives, simplex and interior-point methods, typically at the cost of much lower accuracy. This paper looks at whether PDHG can be hybridized with an interior-point method to retain some of the speed advantages of the former while capturing the accuracy advantages of the latter.

math.OC↗

arXiv2025

Backing PDHG into a Corner

Recent enhancements to the Primal-Dual Hybrid Gradient (PDHG) algorithm have enabled GPUs to efficiently solve large linear programming problems, often faster than the long-dominant simplex and interior-point methods. The solutions found by PDHG are typically of much lower quality than those found by the alternatives, which can be remedied by following the PDHG iterations with a crossover step to obtain an accurate optimal basic solution. However, the cost of this highly sequential crossover step can be quite significant. This paper examines whether PDHG iterations can be enhanced to push the solution into a corner of the optimal LP face, thereby providing crossover a better starting point and hopefully reducing its runtime.

math.OC↗

arXiv2025

Concurrent Crossover for PDHG

First-order methods based on the PDHG algorithm have recently emerged as a viable option for efficiently solving large-scale linear programming problems. One highly desirable property of these methods is that they can make effective use of GPUs. One undesirable property is that, as first-order methods, their convergence can be extremely slow. This property forces one to decide how much accuracy is truly necessary when solving an LP problem. This paper looks at whether a parallel, concurrent crossover scheme can help to obtain highly accurate solutions without sacrificing the benefits of these new approaches.

math.OC↗