GPU-Accelerated Conic Quadratic Programming with Local Linear Convergence under Strict Complementarity

2026-08-10

AI Summary

This work addresses the scalability and convergence challenges of large-scale conic quadratic programming (CQP) with multiple conic and affine constraints by proposing PDHCG-CQP, a GPU-accelerated first-order solver built upon a restarted averaged primal-dual hybrid gradient framework. The method integrates inexact primal updates, batched conic projections, reflected Halpern acceleration, and two-dimensional data-partitioned multi-GPU parallelism. Under strict complementarity conditions, it establishes local linear convergence guarantees. Leveraging matrix-free linear algebra and device-resident KKT residual computation, the solver achieves state-of-the-art robustness across benchmark problems—including QP, QCQP, SOCP, and Fisher market equilibria—demonstrating efficient scaling up to 8 GPUs and handling instances with up to 440 million variables.

Abstract

We present PDHCG-CQP, a GPU-accelerated first-order solver for large-scale conic convex quadratic programming. PDHCG-CQP supports affine constraints and Cartesian products of nonnegative, second-order, rotated second-order, exponential, and three-dimensional power cones. At its core is a restarted averaged primal-dual hybrid gradient (PDHG) method, whose primal update is computed inexactly by solving a conic quadratic proximal subproblem with projected gradient iterations. We establish local linear convergence of the restarted averaged scheme with both exact and inexact primal proximal evaluations under a uniform local quadratic-growth condition on the smoothed primal-dual gap. We further show that this condition holds under strict complementarity by exploiting a rotated second-order-cone lifting together with local primal and dual regularity conditions. Our C/CUDA implementation combines matrix-free linear algebra, batched cone projections, adaptive inner solves, reflected-Halpern acceleration, and fully device-resident KKT residual computations. It also supports multi-GPU execution through a two-dimensional partitioning of the problem data. Extensive experiments on standard and large-scale quadratic programming (QP), convex quadratically constrained quadratic programming (QCQP), second-order cone programming (SOCP), and quasilinear Fisher equilibrium benchmarks demonstrate that PDHCG-CQP achieves state-of-the-art robustness among first-order solvers while scaling efficiently to 8 GPUs and instances with up to $4.4\times10^8$ stored primal coordinates. PDHCG-CQP is open source and available at https://github.com/Lhongpei/PDHCG.

Problem

• conic quadratic programming
• GPU acceleration
• local linear convergence
• strict complementarity
• primal-dual methods

Innovation

• GPU acceleration
• conic quadratic programming
• local linear convergence
• strict complementarity
• primal-dual hybrid gradient