Projection Neural Dynamics for Inverse Variational Inequality Problems: Stability Analysis and Applications to Sparse Signal Recovery

In this work, we develop a projection neural network based on a second-order dynamical model (SO-PDM) for solving inverse variational inequality problems (IVIPs) in Hilbert spaces. The proposed framework incorporates inertial and damping components, resulting in improved convergence behavior while ensuring feasibility through a projection operator. Under the Lipschitz continuity assumption on the operator, the proposed SO-PDM admits a unique global trajectory. Under the additional strong monotonicity assumption and suitable parameter conditions, convergence to the unique solution of the IVIP is established. A discrete-time formulation is derived via a finite-difference scheme, leading to a projection-based inertial algorithm with relaxation. Under suitable parameter conditions, the algorithm is shown to converge linearly to the unique solution of the IVIP, and under an additional parameter condition, the global asymptotic stability of the continuous-time SO-PDM is established via Lyapunov analysis. Furthermore, a numerical comparison in a higher-dimensional setting shows that the proposed algorithm converges faster and attains higher accuracy than the existing first-order projection method. Numerical experiments further confirm the effectiveness and stability of the proposed SO-PDM, including its application to sparse signal recovery in compressed sensing.