MFPI-Net: Learning to Solve Absolute Value Equations via MFPI Unrolling

Absolute value equations (AVEs) offer a unified characterization for optimality conditions across a wide range of problems, including linear and convex quadratic programming, bimatrix games, among others. Due to the non-smooth, non-convex structure and NP-hard nature induced by the absolute-value operator, solving AVEs poses significant challenges, for which many classical numerical algrithms have been proposed, and the modified fixed-point iteration (MFPI) ranks among the most effective. By combining a learnable smooth approximation of the absolute-value operator with paired neural modules regularized to preserve mutual-inverse consistency, we propose MFPI-Net, a neural solver that unrolls MFPI into learnable residual updates. Building on MFPI’s convergence theory, we derive approximation error bounds for the unrolled network and establish layerwise stability against input perturbations. To understand how network depth affects the trade-off between computational cost and approximation accuracy, we analyze MFPI-Net from a continuous-time dynamical systems perspective. This motivates a two-stage framework: MFPI-Net is first leveraged to perform sign estimation to eliminate the absolute-value operator, and conventional solvers are then deployed to obtain high-accuracy solutions. Experiments demonstrate approximately 3$\times$ faster high-accuracy AVE solving than MFPI, with successful solution recovery on tested instances outside the standard convergence regime. Beyond AVEs, MFPI-Net provides warm starts for various linear programming solvers through AVE reformulations.