A Projection Neural Network With Delays and Optimization Approaches for Solving Absolute Value Equations

This paper proposes a projection neural network with delays, including discrete and distributed delay, for solving absolute value equations of the form Ax− |x| =b. By reformulating the absolute value equation as an equivalent optimization problem and exploiting the associated Karush–Kuhn–Tucker (KKT) conditions, the projection neural network model is systematically constructed. Sufficient conditions for global exponential convergence to a solution of the absolute value equation are derived using Lyapunov– Krasovskii functionals, novel integral inequalities, and a linear matrix inequality (LMI) framework. Furthermore, three simulation examples are presented to demonstrate the effectiveness of the proposed approach under different conditions on ∥A−1∥, covering both ∥A−1∥ ≤ 1 and ∥A−1∥ > 1, thereby extending existing neural network-based methods that primarily focus on the case ∥A−1∥ < 1.