A First-Order Numerical Algorithm without Matrix Operations

This paper offers a matrix-free first-order numerical method to solve large-scale conic optimization problems. Solving systems of linear equations pose the most computationally challenging part in both first-order and second-order numerical algorithms. Existing direct and indirect methods are either computationally expensive or compromise on solution accuracy. Alternatively, we propose an easy-to-compute decomposition method to solve sparse linear systems that arise in conic optimization problems. Its iterations are tractable, highly parallelizable, with closed-form solutions. This algorithm can be easily implemented on distributed platforms, such as graphics processing units, with orders-of-magnitude time improvement. The performance of the proposed solver is demonstrated on large-scale conic optimization problems and is compared with the state-of-the-art first-order solvers.