cuPDLPx: A GPU-Accelerated First-Order LP Solver

cuPDLPx is a GPU-accelerated linear programming solver based on a restarted Halpern PDHG method specifically tailored for GPU architectures. It incorporates a Halpern update scheme, an adaptive restart scheme, and a PID-controlled primal weight, resulting in substantial empirical improvements over its predecessor, cuPDLP, on standard LP benchmark suites.

Our work is presented in two papers:

• Computational Paper: cuPDLPx: A Further Enhanced GPU-Based First-Order Solver for Linear Programming details the practical innovations that give cuPDLPx its performance edge.

• Theoretical Paper: Restarted Halpern PDHG for Linear Programming provides the mathematical foundation for our method.

Getting started

Follow these steps to build the solver and verify its installation.

Requirements

• An NVIDIA GPU with CUDA support (≥12.4 required).
• A C toolchain (gcc) and the NVIDIA CUDA Compiler (nvcc).

Build from Source

Clone the repository and compile the project using make.

make clean
make build

This will create the solver binary at ./build/cupdlpx.

Verifying the Installation

Run a small test problem to confirm that the solver was built correctly.

# 1. Download a test instance from the MIPLIB library
wget -P test/ https://miplib.zib.de/WebData/instances/2club200v15p5scn.mps.gz

# 2. Solve the problem and write output to the current directory (.)
./build/cupdlpx test/2club200v15p5scn.mps.gz test/

If the solver runs and creates output files, your installation is successful.

Usage Guide

Command-Line Interface

The solver is invoked with the following syntax, specifying an input file and an output directory.

./build/cupdlpx [OPTIONS] <mps_file> <output_directory>

Arguments

• <mps_file>: The path to the input linear programming problem. Both plain (.mps) and gzipped (.mps.gz) files are supported.
• <output_directory>: The directory where the output files will be saved.

Solver Options

Option	Type	Description	Default
-h, --help	flag	Display the help message.	N/A
-v, --verbose	flag	Enable verbose logging.	false
--time_limit	double	Time limit in seconds.	3600.0
--iter_limit	int	Iteration limit.	2147483647
--eps_opt	double	Relative optimality tolerance.	1e-4
--eps_feas	double	Relative feasibility tolerance.	1e-4
--eps_infeas_detect	double	Infeasibility detection tolerance.	1e-10

Output Files

The solver generates three text files in the specified <output_directory>. The filenames are derived from the input file's basename. For an input INSTANCE.mps.gz, the output will be:

• INSTANCE_summary.txt: Contains solver statistics, timings, and termination information.
• INSTANCE_primal_solution.txt: The primal solution vector.
• INSTANCE_dual_solution.txt: The dual solution vector.

Python Interface

In addition to the command-line and C APIs, cuPDLPx provides a Python interface (cupdlpx)
for building and solving LPs directly with NumPy and SciPy.

• High-level, Pythonic API similar to commercial solvers.
• Supports dense and sparse matrices.
• Provides easy parameter management and solution attributes.

Install from PyPI:

pip install cupdlpx

Or build from source:

git clone https://github.com/MIT-Lu-Lab/cuPDLPx.git
cd cuPDLPx
pip install .

See the cupdlpx guide for full usage instructions, examples, and API details.

C Interface

Besides the command-line interface, cuPDLPx also provides a C API (interface.c) for directly solving LPs in memory. This is useful when integrating cuPDLPx into other C/C++ projects or when generating problems programmatically.

Functions and Parameters

The C API involves two main functions:

lp_problem_t* create_lp_problem(
    const double* c,                   // objective vector (length n)
    const matrix_desc_t* A,            // constraint matrix (m×n)
    const double* con_lb,              // constraint lower bounds (length m)
    const double* con_ub,              // constraint upper bounds (length m)
    const double* var_lb,              // variable lower bounds (length n)
    const double* var_ub,              // variable upper bounds (length n)
    const double* c0                   // scalar objective offset
);

cupdlpx_result_t* solve_lp_problem(
    const lp_problem_t* prob,
    const pdhg_parameters_t* params    // NULL → use default parameters
);

create_lp_problem parameters:

• objective_c: Objective vector. If NULL, defaults to all zeros.
• A_desc: Matrix descriptor. Supports matrix_dense, matrix_csr, matrix_csc, matrix_coo.
• con_lb: Constraint lower bounds. If NULL, defaults to all -INFINITY.
• con_ub: Constraint upper bounds. If NULL, defaults to all +INFINITY.
• var_lb: Variable lower bounds. If NULL, defaults to all 0.0.
• var_ub: Variable upper bounds. If NULL, defaults to all +INFINITY.
• objective_constant: Scalar constant term added to the objective value. If NULL, defaults to 0.0.

solve_lp_problem parameters:

• prob: An LP problem built with create_LP_problem.
• params: Solver parameters. If NULL, the solver will use default parameters.

Example: Solving a Small LP

#include "cupdlpx/interface.h"
#include <math.h>
#include <stdio.h>

int main() {
    int m = 3; // number of constraints
    int n = 2; // number of variables

    // Dense matrix A
    double A[3][2] = {
        {1.0, 2.0},
        {0.0, 1.0},
        {3.0, 2.0}
    };

    // Describe A
    matrix_desc_t A_desc;
    A_desc.m = m; A_desc.n = n;
    A_desc.fmt = matrix_dense;
    A_desc.zero_tolerance = 0.0;
    A_desc.data.dense.A = &A[0][0];

    // Objective coefficients
    double c[2] = {1.0, 1.0};

    // Constraint bounds: l <= A x <= u
    double l[3] = {5.0, -INFINITY, -INFINITY};
    double u[3] = {5.0, 2.0, 8.0};

    // Build the problem
    lp_problem_t* prob = create_lp_problem(
        &A_desc, c, NULL, NULL, NULL, l, u);

    // Solve (NULL → use default parameters)
    cupdlpx_result_t* res = solve_lp_problem(prob, NULL);

    printf("Termination reason: %d\n", res->termination_reason);
    printf("Primal objective: %.6f\n", res->primal_objective_value);
    printf("Dual objective:   %.6f\n", res->dual_objective_value);
    for (int j = 0; j < res->num_variables; ++j) {
        printf("x[%d] = %.6f\n", j, res->primal_solution[j]);
    }

    lp_problem_free(prob);
    cupdlpx_result_free(res);

    return 0;
}

Reference

If you use cuPDLPx or the ideas in your work, please cite the source below.

@article{lu2025cupdlpx,
  title={cuPDLPx: A Further Enhanced GPU-Based First-Order Solver for Linear Programming},
  author={Lu, Haihao and Peng, Zedong and Yang, Jinwen},
  journal={arXiv preprint arXiv:2507.14051},
  year={2025}
}

@article{lu2024restarted,
  title={Restarted Halpern PDHG for linear programming},
  author={Lu, Haihao and Yang, Jinwen},
  journal={arXiv preprint arXiv:2407.16144},
  year={2024}
}

License

cuPDLPx is licensed under the Apache 2.0 License. See the LICENSE file for details.