PDCS: A Primal-Dual Large-Scale Conic Programming Solver with GPU Enhancements

2025-05-01

AI Summary

This work addresses scalability and computational efficiency bottlenecks of large-scale conic optimization—including linear, second-order cone, convex quadratic, and exponential cone programs—under low-accuracy requirements. We propose cuPDCS, the first GPU-accelerated primal-dual solver tailored to heterogeneous conic structures. Methodologically, it centers on sparse matrix-vector multiplication and integrates adaptive reflected Halpern restarting, dynamic step-size and weight adaptation, diagonal rescaling, and bijective conic projections. A hierarchical CUDA architecture is designed to optimize memory efficiency and parallel throughput. Experiments on Fisher market equilibrium, Lasso regression, and multi-period portfolio optimization demonstrate superior performance over leading commercial and first-order solvers. Furthermore, benchmarking on CBLIB confirms substantial improvements in solving speed, scalability, and robustness for large-scale, low-accuracy conic problems.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: The sharing economy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

In this paper, we introduce the"Primal-Dual Conic Programming Solver"(PDCS), a large-scale conic programming solver with GPU enhancements. Problems that PDCS currently supports include linear programs, second-order cone programs, convex quadratic programs, and exponential cone programs. PDCS achieves scalability to large-scale problems by leveraging sparse matrix-vector multiplication as its core computational operation, which is both memory-efficient and well-suited for GPU acceleration. The solver is based on the restarted primal-dual hybrid gradient method but further incorporates several enhancements, including adaptive reflected Halpern restarts, adaptive step-size selection, adaptive weight adjustment, and diagonal rescaling. Additionally, PDCS employs a bijection-based method to compute projections onto rescaled cones. Furthermore, cuPDCS is a GPU implementation of PDCS and it implements customized computational schemes that utilize different levels of GPU architecture to handle cones of different types and sizes. Numerical experiments demonstrate that cuPDCS is generally more efficient than state-of-the-art commercial solvers and other first-order methods on large-scale conic program applications, including Fisher market equilibrium problems, Lasso regression, and multi-period portfolio optimization. Furthermore, cuPDCS also exhibits better scalability, efficiency, and robustness compared to other first-order methods on the conic program benchmark dataset CBLIB. These advantages are more pronounced in large-scale, lower-accuracy settings.

Problem

• Solves large-scale conic programs efficiently using GPU acceleration
• Supports various cone types including linear and second-order cones
• Enhances primal-dual methods with adaptive techniques for scalability

Innovation

• GPU-enhanced primal-dual conic programming solver
• Sparse matrix-vector multiplication for scalability
• Adaptive restarts and step-size selection