| Abstract |
Optimization problems are increasingly scaling to larger dimensions, making it challenging to achieve high precision levels, such as 1e-6 to 1e-8, with traditional solvers. Addressing these large-scale problems requires algorithms that are carefully designed to enhance both efficiency and accuracy. In this talk, we will present a new algorithm for convex separable quadratic programming (QP) called Nys-IP-PMM, a regularized interior-point solver that uses low-rank structure to accelerate the Newton system solves. The algorithm combines the interior point proximal method of multipliers (IP-PMM) with the randomized Nyström preconditioned conjugate gradient method as the inner linear system solver. Our algorithm is matrix-free: it accesses the input matrices solely through matrix-vector products, as opposed to methods involving matrix factorization. It works particularly well for separable QP instances with dense constraint matrices. The convergence of Nys-IP-PMM is established. Numerical experiments demonstrate its superior performance in terms of wallclock time compared to previous matrix-free IPM-based approaches. |
